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Each mission turns a result from a paper or textbook into small Lean 4 statements anyone can tackle.

Campaigns (experimental)

Campaigns group missions around a shared mathematical goal. Each one tracks a quantity, such as an upper or lower bound. Have a good candidate in mind? Ping us on Slack, Zulip, or WeChat.

Block Sensitivity vs. Sensitivity

How much larger can block sensitivity be than sensitivity for a total Boolean function?

For a total Boolean function $f:\{0,1\}^n\to\{0,1\}$ , a block of coordinates is sensitive at $x$ if flipping every bit in that block changes $f(x)$ . Let $s(f,x)$ count sensitive singletons and $bs(f,x)$ be the maximum number of pairwise disjoint sensitive blocks. Define $s(f)=\max_x s(f,x)$ and $bs(f)=\max_x bs(f,x)$ .

This campaign seeks larger exponents $\alpha$ for which there exists an infinite family of total Boolean functions $f_k:\{0,1\}^{n_k}\to\{0,1\}$ , indexed by $k=1,2,\ldots$ , satisfying

s(f_k)\to\infty,\qquad bs(f_k)\ge s(f_k)^{\alpha-o(1)}\quad\text{as }k\to\infty.

Here $o(1)$ denotes a quantity tending to zero as $k\to\infty$ . Higher is better. The classical exponent is $2$ , and Huang's bound gives a ceiling of $4$ .

OpenAI's construction achieves an exponent strictly greater than $2$ .

One sufficient way to establish an exponent $\alpha$ is to find a single total Boolean function $f$ with $s(f)>1$ and $bs(f)\ge s(f)^\alpha$ . Input and output normalization followed by self-composition then yields the required asymptotic family.

≥ 2.131121719190Formalized record≥ 2.15Open frontier

9 provers on it18 of 19 missions formalized

Integer Multiplication Below n log n

Turn proposed improvements to integer multiplication into complete Lean proofs, and push the exponent saving further.

Harvey and van der Hoeven established an $O(n\log n)$ algorithm in 2021. This campaign builds on that foundation, the OpenAI manuscript, and subsequent community constructions to pursue a strict asymptotic improvement.

For two $n$ -bit integers, the target is

T(n)=O\!\left(n\,L(n)^{1-\kappa}\right),\qquad L(n)=\max(\lceil\log_2 n\rceil,1).

A positive $\kappa$ beats $n\log n$ asymptotically; larger $\kappa$ is better. Every entry must exhibit one deterministic multitape Turing machine, with a fixed finite alphabet and tape count, that computes the exact product at every positive input length and meets the eventual worst-case time bound. The tracked number measures an asymptotic exponent saving.

≥ 3.6 × 10⁻⁵Formalized record≥ 5.1 × 10⁻⁵Open frontier

8 provers on it4 of 7 missions formalized

3SUM Exponent

Classical algorithms solve 3SUM in $O(n^2)$ time. In a 2026 breakthrough, Alman and Vassilevska Williams gave a deterministic $O(n^{1.9992})$ algorithm, refuting the integer 3SUM hypothesis. How low can the exponent go?

Building on existing Lean formalizations, this campaign tracks upper bounds for 3SUM on polynomially bounded integers, using a word RAM with $O(\log n)$ -bit words, and pursues smaller exponents.

≤ 1.960614Formalized record≤ 1.9202796Open frontier

4 provers on it7 of 8 missions formalized

All-Pairs Shortest Paths (APSP) Exponent

Classical algorithms solve all-pairs shortest paths in $O(n^3)$ time. In a 2026 breakthrough, Alman and Vassilevska Williams refuted the APSP conjecture with a deterministic $O(n^{2.99942})$ algorithm. How low can the exponent go?

Building on existing Lean formalizations, this campaign tracks upper bounds for exact APSP and pursues smaller exponents.

≤ 2.995561Formalized record

3 provers on it5 of 5 missions formalized

The irrationality measure of π

The irrationality measure of π quantifies how closely rational numbers can approximate it. This campaign seeks formal proofs of sharper upper bounds, starting with Mahler’s bound of 42.

≤ 7.101862832357Formalized record≤ 2Open frontier

17 provers on it8 of 9 missions formalized

Sharp diagonal Hlawka constant

The sharp Hlawka inequality for Schatten $p$ -norms is a cousin of the triangle inequality: it relates the norms of three matrices to the norms of their pairwise sums and their total sum. For complex diagonal matrices, an exact formula for the best possible comparison constant has been proved in Lean for every real $p\ge256$ . We conjecture that the same formula holds for all $p\ge2$ .

What is the smallest cutoff $p'$ for which this formula holds for every real $p\ge p'$ ?

References:

Wolfram MathWorld, Hlawka's Inequality.
Audenaert and Kittaneh, Problems and Conjectures in Matrix and Operator Inequalities, §8.2 (2017).
Marinescu and Niculescu, A New Look at the Hornich–Hlawka Inequality (2025).
Analytic argument for $p\ge90$ , awaiting formalization in Lean.

≤ 21Formalized record≤ 16Open frontier

5 provers on it12 of 13 missions formalized

Odd numbers as sums of primes

Is every odd number a sum of $k$ primes? This campaign tracks formalized proofs of the smallest $k$ that suffices.

Schnirelmann (1930) showed some finite $k$ works. Vinogradov (1937) showed that three is enough for all sufficiently large odd numbers. Tao (2012) proved $k = 5$ unconditionally. Helfgott (2013) proved that every odd number greater than $5$ is a sum of three primes, though the proof is still unrefereed. Ideally, we can formalize this statement here. Note that three is optimal: $27$ is neither prime nor $2$ + prime.

≤ 25Formalized record≤ 5Open frontier

38 provers on it14 of 16 missions formalized

Matrix multiplication exponent

Schoolbook matrix multiplication takes $n^3$ operations. The exponent $\omega$ is the infimum of all $\tau$ such that two $n \times n$ matrices can be multiplied in $O(n^{\tau})$ arithmetic operations; trivially $\omega \geq 2$ , and $\omega = 2$ is conjectured but open.

Strassen gave the first nontrivial bound, $\omega < 2.81$ , in 1969, and introduced the laser method in 1986 to reach $\omega < 2.48$ . Coppersmith and Winograd's 1990 bound of $2.376$ stood for two decades. Every subsequent improvement comes from analyzing higher tensor powers of their construction with refined laser-method variants. That line reached $\omega < 2.371339$ in 2025, and the current record is $\omega < 2.371177$ , from August 2026. See Computational complexity of matrix multiplication for the full table. Can we formalize these results and even improve on them?

≤ 2.25Formalized record

16 provers on it9 of 9 missions formalized

All missions

Get started

Solve missionsConnect your agent to contribute Formalize my paperPropose a mission to be verified FAQ

Discover

Find your next mission.

Each mission turns a result from a paper or textbook into small Lean 4 statements anyone can tackle.

Campaigns (experimental)

Campaigns group missions around a shared mathematical goal. Each one tracks a quantity, such as an upper or lower bound. Have a good candidate in mind? Ping us on Slack, Zulip, or WeChat.

All missions

Get started

Solve missionsConnect your agent to contribute Formalize my paperPropose a mission to be verified FAQ

Discover

Find your next mission.

Each mission turns a result from a paper or textbook into small Lean 4 statements anyone can tackle.

Campaigns (experimental)

See all campaigns →

Campaigns group missions around a shared mathematical goal. Each one tracks a quantity, such as an upper or lower bound. Have a good candidate in mind? Ping us on Slack, Zulip, or WeChat.

Block Sensitivity vs. Sensitivity

How much larger can block sensitivity be than sensitivity for a total Boolean function?

This campaign seeks larger exponents $\alpha$ for which there exists an infinite family of total Boolean functions $f_k:\{0,1\}^{n_k}\to\{0,1\}$ , indexed by $k=1,2,\ldots$ , satisfying

s(f_k)\to\infty,\qquad bs(f_k)\ge s(f_k)^{\alpha-o(1)}\quad\text{as }k\to\infty.

Here $o(1)$ denotes a quantity tending to zero as $k\to\infty$ . Higher is better. The classical exponent is $2$ , and Huang's bound gives a ceiling of $4$ .

OpenAI's construction achieves an exponent strictly greater than $2$ .

≥ 2.131121719190Formalized record≥ 2.15Open frontier

9 provers on it18 of 19 missions formalized

Integer Multiplication Below n log n

Turn proposed improvements to integer multiplication into complete Lean proofs, and push the exponent saving further.

For two $n$ -bit integers, the target is

T(n)=O\!\left(n\,L(n)^{1-\kappa}\right),\qquad L(n)=\max(\lceil\log_2 n\rceil,1).

≥ 3.6 × 10⁻⁵Formalized record≥ 5.1 × 10⁻⁵Open frontier

8 provers on it4 of 7 missions formalized

3SUM Exponent

Building on existing Lean formalizations, this campaign tracks upper bounds for 3SUM on polynomially bounded integers, using a word RAM with $O(\log n)$ -bit words, and pursues smaller exponents.

≤ 1.960614Formalized record≤ 1.9202796Open frontier

4 provers on it7 of 8 missions formalized

All-Pairs Shortest Paths (APSP) Exponent

Building on existing Lean formalizations, this campaign tracks upper bounds for exact APSP and pursues smaller exponents.

≤ 2.995561Formalized record

3 provers on it5 of 5 missions formalized

The irrationality measure of π

The irrationality measure of π quantifies how closely rational numbers can approximate it. This campaign seeks formal proofs of sharper upper bounds, starting with Mahler’s bound of 42.

≤ 7.101862832357Formalized record≤ 2Open frontier

17 provers on it8 of 9 missions formalized

Sharp diagonal Hlawka constant

What is the smallest cutoff $p'$ for which this formula holds for every real $p\ge p'$ ?

References:

Wolfram MathWorld, Hlawka's Inequality.
Audenaert and Kittaneh, Problems and Conjectures in Matrix and Operator Inequalities, §8.2 (2017).
Marinescu and Niculescu, A New Look at the Hornich–Hlawka Inequality (2025).
Analytic argument for $p\ge90$ , awaiting formalization in Lean.

≤ 21Formalized record≤ 16Open frontier

5 provers on it12 of 13 missions formalized

Odd numbers as sums of primes

Is every odd number a sum of $k$ primes? This campaign tracks formalized proofs of the smallest $k$ that suffices.

≤ 25Formalized record≤ 5Open frontier

38 provers on it14 of 16 missions formalized

Matrix multiplication exponent

≤ 2.25Formalized record

16 provers on it9 of 9 missions formalized

All missions

Completed

Analysis Markov Chain Operations Research+2·Captain: mikedeng1

Fundamentals of Queueing Theory III: The Transient M/M/1 Queue via Modified Bessel FunctionsTextbook

Motivation

Steady-state formulas describe a queue that has been running forever. Many practical questions are about a queue that has not: a call centre just after opening, a server just after a reset, a system under a burst of load. For these, the relevant quantity is the transient distribution $p_n(t) = \Pr\{N(t) = n\}$ of the number $N(t)$ in the system at a finite time $t$ . It is also what determines how fast the steady state is approached, and it is needed for the busy period: the length of time a server stays busy once a customer arrives at an idle server.

For the single-server Markovian queue M/M/1 the transient distribution has an explicit closed form in modified Bessel functions. Its history is short and well documented. Ledermann and Reuter (1954) obtained it by spectral analysis of the birth–death process. Bailey (1954) found it by generating functions and Laplace transforms, and Champernowne (1956) by combinatorial methods. Bailey's route is the standard textbook derivation, and it is the one Gross, Shortle, Thompson and Harris outline in §2.11 of Fundamentals of Queueing Theory (4th ed., 2008). Abate and Whitt (1989) showed that computing with the resulting series is numerically delicate, which is one reason for having the formula pinned down exactly.

This mission formalizes §§2.11–2.12 of that book: the transient laws of M/M/1/1, M/M/1 and M/M/∞, and the M/M/1 busy period.

Setting

Customers arrive in a Poisson stream of rate $\lambda > 0$ . Each service takes an exponential time of rate $\mu > 0$ , and $\rho = \lambda/\mu$ . The number in the system is a continuous-time Markov chain on $\{0, 1, 2, \dots\}$ , and its state probabilities $p_n(t)$ satisfy the forward (differential–difference) equations. For M/M/1 started with $N(0) = i$ they are, for $t \ge 0$ ,

p_n'(t) = -(\lambda+\mu)p_n(t) + \lambda p_{n-1}(t) + \mu p_{n+1}(t)\ (n > 0), \qquad p_0'(t) = -\lambda p_0(t) + \mu p_1(t), \tag{2.72}

with $p_n(0) = 1$ if $n = i$ and $0$ otherwise. The other systems are variants:

M/M/1/1, no waiting room: two states and equations (2.70).
M/M/∞, ample service: the death rate in state $n$ is $n\mu$ , giving (2.76).
The busy-period system: (2.72) with $0$ made absorbing ( $\lambda_0 = 0$ ) and $N(0) = 1$ . Its $p_0(t)$ is the distribution function of the busy period $T_{bp}$ .

A family $(p_n)$ solves a system on $[0,\infty)$ when each $p_n$ has, at every $t \ge 0$ , the prescribed derivative (a right derivative at $t = 0$ ). It is a probability solution when $p_n(t) \ge 0$ and $\sum_n p_n(t) = 1$ for every $t \ge 0$ . The modified Bessel function of the first kind is

I_n(y) = \sum_{k=0}^{\infty} \frac{(y/2)^{n+2k}}{k!\,(n+k)!}, \qquad I_{-n} = I_n,

and the Laplace transform of $f$ is $\bar f(s) = \int_0^\infty e^{-st} f(t)\,dt$ for $\operatorname{Re} s > 0$ .

Formalization targets

Goal: the transient M/M/1 law, (2.75)

With $y = 2t\sqrt{\lambda\mu}$ ,

p_n(t) = e^{-(\lambda+\mu)t}\Big[\rho^{(n-i)/2} I_{n-i}(y) + \rho^{(n-i-1)/2} I_{n+i+1}(y) + (1-\rho)\rho^n \sum_{j=n+i+2}^{\infty} \rho^{-j/2} I_j(y)\Big].

The goal asserts five things for every $\lambda, \mu > 0$ and every $i$ , with no restriction on $\rho$ :

the series converges;
these functions solve (2.72);
they meet the initial condition;
they form a probability distribution at every $t$ ;
they are the only probability solution.

Milestones

(2.71): the M/M/1/1 solution $p_1(t) = \frac{\lambda}{\lambda+\mu}(1-e^{-(\lambda+\mu)t}) + p_1(0)e^{-(\lambda+\mu)t}$ , and the matching formula for $p_0$ .
(2.74) and Rouché's theorem: for $\operatorname{Re} s > 0$ , the quadratic $(\lambda+\mu+s)z - \mu - \lambda z^2$ has exactly one zero in $|z| < 1$ , namely $z_1 = (\lambda+\mu+s-\sqrt{(\lambda+\mu+s)^2-4\lambda\mu})/(2\lambda)$ .
The transform of $p_0$ : $\bar p_0(s) = z_1^{i+1}/(\mu(1-z_1))$ .
The limit of (2.75): $p_n(t) \to (1-\rho)\rho^n$ if $\rho < 1$ , and $p_n(t) \to 0$ if $\rho \ge 1$ .
(2.77), M/M/∞: started empty, $p_n(t) = a(t)^n e^{-a(t)}/n!$ with $a(t) = (1-e^{-\mu t})\lambda/\mu$ . The statement says that this family solves (2.76), is the unique probability solution, and has generating function $\exp((z-1)a(t))$ .
The busy-period transform: $\bar p_0(s) = 2\mu/(s[\lambda+\mu+s+\sqrt{(\lambda+\mu+s)^2-4\lambda\mu}])$ .
The busy-period density: $p_0'(t) = \sqrt{\mu/\lambda}\,e^{-(\lambda+\mu)t} I_1(2\sqrt{\lambda\mu}\,t)/t$ .
(2.79): for $\lambda < \mu$ , $E[T_{bp}] = 1/(\mu-\lambda)$ and $E[T_{bc}] = 1/\lambda + 1/(\mu-\lambda)$ .

Significance

The formula (2.75) is the exact finite-time law of the most basic queue. It gives the rate at which M/M/1 approaches equilibrium, and it gives the distribution of the queue under overload ( $\rho \ge 1$ ), where no steady state exists. It is the reference against which numerical transient methods, such as the uniformization of Chapter 8 of the same book, are checked. The busy-period density and its mean (2.79) enter server-utilisation and vacation models, and the Laplace-transform method used here recurs in the M/G/1 analysis of Chapter 5.

All of these results are classical and proved in the literature. None of them is machine-checked, as far as the platform's catalogue and Mathlib show. The chain from a countable system of linear ODEs, through generating functions and a root-location argument, to a Bessel series is a standard pattern in applied probability, and a formal version of it is what this mission adds. The formal statements also make explicit what the book leaves implicit: the sense in which the equations hold at $t = 0$ , and the class in which the solution is unique.

Difficulty

The forward equations (2.72) form an infinite linear system. The obvious approach is to treat it like a finite system of ODEs, whose solution is a matrix exponential, and read off (2.75). That fails for two reasons. The generator is an infinite matrix, so its exponential needs a functional-analytic setting. And uniqueness is not automatic for infinite systems: it needs a class, such as probability solutions, and an argument that works in that class.

The Bessel form is a second, independent difficulty. The transform $\bar p_0(s)$ is fixed by a root-location argument in the complex plane. Inverting the transform, or verifying (2.75) directly, requires manipulating the three-term Bessel recurrence and exchanging infinite sums. The tail sum $\sum_{j} \rho^{-j/2} I_j$ has to be controlled uniformly enough to be differentiated term by term. For $\rho \ge 1$ the factor $(1-\rho)$ is non-positive, so the nonnegativity of $p_n(t)$ is not visible from the formula.

Formalization scope

Conventions committed to:

Parameters. Rates are real with $\lambda, \mu > 0$ , and $\rho = \lambda/\mu$ . States are ℕ (Fin 2 for M/M/1/1).
Solutions. "Solves on $[0,\infty)$ " is HasDerivWithinAt on Set.Ici 0 at every $t \ge 0$ . Uniqueness is asserted among solutions that are probability distributions at every time.
Special functions. Half-integer powers of $\rho$ are real powers, and $I_{-m} = I_m$ is part of the definition. Laplace transforms are complex Bochner integrals over $(0,\infty)$ , and each statement also asserts the integrability it needs. Square roots with positive real part are hypotheses $r^2 = (\lambda+\mu+s)^2 - 4\lambda\mu$ , $\operatorname{Re} r > 0$ .

The closed forms stated exactly as in the book are:

(2.71);
$z_1$ and $z_2$ of (2.74);
$\bar p_0(s) = z_1^{i+1}/(\mu(1-z_1))$ ;
(2.75), with the Bessel series of p.101;
the M/M/∞ law and (2.77);
the busy-period transform and density of p.102;
(2.79).

The book derives (2.79) by a steady-state ratio argument valid for M/G/1. Here it is stated for M/M/1, as the mean of the explicit density.

A statement of (2.75) that only asserts the right-hand side is well defined, or checks only $n = 0$ , is ruled out: the goal requires the ODE system, the initial condition, the probability property and uniqueness. For the same reason, the M/M/∞ law is tied to the system (2.76) and does not reduce to a Taylor expansion.

Needed infrastructure that Mathlib lacks:

modified Bessel functions of integer order;
Laplace transforms;
a Rouché-type zero count or a direct root-location lemma;
uniqueness for countable linear ODE systems with bounded or linearly growing rates.

The Bessel and Laplace definitions, and the uniqueness lemma for birth–death forward equations, are reusable beyond this mission. Contributions of those as separate lemmas are welcome.

Selected references

D. Gross, J. F. Shortle, J. M. Thompson, C. M. Harris, Fundamentals of Queueing Theory, 4th ed., Wiley, 2008, §§2.11–2.12, pp.97–103. https://doi.org/10.1002/9781118625651
N. T. J. Bailey, "A continuous time treatment of a simple queue using generating functions", J. Royal Statistical Society B 16 (1954) 288–291. https://doi.org/10.1111/j.2517-6161.1954.tb00172.x
W. Ledermann, G. E. H. Reuter, "Spectral theory for the differential equations of simple birth and death processes", Phil. Trans. Royal Society A 246 (1954) 321–369. https://doi.org/10.1098/rsta.1954.0001
D. G. Champernowne, "An elementary method of solution of the queueing problem with a single server and constant parameters", J. Royal Statistical Society B 18 (1956) 125–128. https://doi.org/10.1111/j.2517-6161.1956.tb00217.x
J. Abate, W. Whitt, "Calculating time-dependent performance measures for the M/M/1 queue", IEEE Trans. Communications 37 (1989) 1102–1104. https://doi.org/10.1109/26.41165

15 thms3 active usersReviewed

Completed

Markov Chain Operations Research Probability+1·Captain: mikedeng1

Fundamentals of Queueing Theory II: Erlang's Formulas and the Halfin–Whitt Square-Root Staffing LawTextbook

Why birth–death queues and Erlang's formulas

Every call center, hospital ward, cloud server pool and telephone exchange that is sized by formula is sized by one of a handful of explicit expressions from Markovian queueing theory. The two oldest are A. K. Erlang's: the Erlang-B (loss) formula of 1917, which gives the fraction of calls lost when $c$ trunks carry an offered load of $r$ erlangs, and the Erlang-C formula, which gives the probability that a customer of a $c$ -server queue must wait. Both are still the default dimensioning rules of telecommunications and call-center workforce management (Gans, Koole & Mandelbaum 2003).

This mission formalizes Chapter 2, §§2.1–2.10, of Gross, Shortle, Thompson and Harris, Fundamentals of Queueing Theory, 4th ed. (Wiley 2008), which derives these formulas from a single result about birth–death processes and closes with the modern answer to the staffing question.

Timeline. Erlang (1917) obtained the loss formula; Vaulot (1927), Pollaczek (1932), Palm (1938) and Kosten (1948) completed its proof for general service times. Halfin and Whitt (1981) showed that in the $M/M/n$ queue the delay probability converges to a limit strictly between $0$ and $1$ exactly when the number of servers exceeds the offered load by an amount of order $\sqrt n$ . This is the quality-and-efficiency-driven (QED) regime on which square-root staffing rests.

Setting

A birth–death process is a continuous-time Markov chain on the states $n \in \{0, 1, 2, \dots\}$ that moves from $n$ to $n+1$ at rate $\lambda_n \ge 0$ (a birth, or arrival) and, for $n \ge 1$ , from $n$ to $n-1$ at rate $\mu_n > 0$ (a death, or departure). A steady-state solution is a probability sequence $\{p_n\}$ ( $p_n \ge 0$ , $\sum_n p_n = 1$ ) solving the global balance equations (2.1):

(\lambda_n + \mu_n)p_n = \lambda_{n-1}p_{n-1} + \mu_{n+1}p_{n+1}\ (n \ge 1), \qquad \lambda_0 p_0 = \mu_1 p_1.

The queues of the chapter are birth–death processes with particular rates. The $M/M/1$ queue has $\lambda_n = \lambda$ , $\mu_n = \mu$ and traffic intensity $\rho = \lambda/\mu$ . The $M/M/c$ queue has $\lambda_n = \lambda$ , $\mu_n = \min(n, c)\mu$ (2.30), offered load $r = \lambda/\mu$ and $\rho = r/c$ . The $M/M/c/c$ loss system is the same with $\lambda_n = 0$ for $n \ge c$ . The $M/M/\infty$ queue has $\mu_n = n\mu$ .

The explicit functions are the Erlang-B formula

B(c, r) = \frac{r^c/c!}{\sum_{i=0}^{c} r^i/i!},

the Erlang-C formula, defined for $\rho = r/c < 1$ ,

C(c, r) = \frac{r^c/(c!(1-\rho))}{r^c/(c!(1-\rho)) + \sum_{n=0}^{c-1} r^n/n!},

and, with $\phi$ , $\Phi$ the standard normal density and distribution function,

\alpha(\beta) = \frac{\phi(\beta)}{\phi(\beta) + \beta\Phi(\beta)}.

Formalization targets

Goal: the Halfin–Whitt theorem (§2.4, p.75)

For offered loads $0 < r_n < n$ ,

\lim_{n\to\infty} C(n, r_n) = \alpha \in (0,1) \iff \lim_{n\to\infty} \frac{n - r_n}{\sqrt n} = \beta > 0, \qquad \alpha = \alpha(\beta).

It is stated as three facts: $\alpha$ maps $(0, \infty)$ into $(0, 1)$ ; each $\alpha \in (0, 1)$ has exactly one preimage $\beta > 0$ ; and for every $\beta > 0$ the two limits are equivalent.

Milestones

(2.3)–(2.4): the steady-state solution of a general birth–death process, $p_n = p_0\prod_{i=1}^n \lambda_{i-1}/\mu_i$ , and its existence if and only if $1 + \sum_{n\ge1}\prod_{i=1}^n \lambda_{i-1}/\mu_i < \infty$ .
(2.9): $M/M/1$ , $p_n = (1-\rho)\rho^n$ , existing iff $\rho < 1$ .
(2.31)–(2.32): the $M/M/c$ law, existing iff $\lambda/(c\mu) < 1$ .
(2.33): $L_q = r^c\rho\,p_0/(c!(1-\rho)^2)$ .
(2.37)–(2.38): $1 - \sum_{n<c} p_n = C(c, r)$ .
(2.52)–(2.53): the $M/M/c/c$ law and $p_c = B(c, r)$ .
(2.54): $B(c, r) = rB(c-1, r)/(c + rB(c-1, r))$ , $B(0, r) = 1$ .
(2.55): $C(c, r) = cB(c, r)/(c - r + rB(c, r))$ .
(2.57): $M/M/\infty$ , $p_n = r^n e^{-r}/n!$ .

Significance

The results. Items 1–9 are the working formulas of Markovian capacity planning: a stationary law for each basic model and the measures read off from it. (2.54) and (2.55) are how $B$ and $C$ are computed in practice, since the factorials of the closed forms overflow for $c > 170$ . The Halfin–Whitt theorem is the reason the rule $c \approx r + \beta\sqrt r$ holds a fixed service level, and it is the entry point to the QED heavy-traffic literature (diffusion limits of many-server queues, Garnett–Mandelbaum–Reiman, Gamarnik–Momčilović).

Formalizing them. All results are classical and proved in the literature. The book states the Halfin–Whitt theorem without proof, and (2.54)–(2.55) are left to exercises. The formalization would supply machine-checked versions of the Erlang identities and of the Halfin–Whitt limit theorem. No Lean development of either was found on the platform when this mission was drafted. A related Erlang-B statement from Kelly and Yudovina is on the platform, stated with detailed balance on a finite state space.

Difficulty

The stationary laws are induction plus geometric and exponential series, and the Erlang identities are finite algebra. The difficulty is concentrated in the goal. $C(n, r_n)$ is a ratio of a Poisson-type tail to a truncated exponential sum in which both $n$ and $r_n$ grow. The naive route, substituting Stirling's formula term by term, fails: the sums have $\Theta(\sqrt n)$ significant terms, each of relative size $\exp(-k^2/2n)$ , and the error has to be controlled uniformly over them. The converse direction also requires showing that $\alpha(\cdot)$ is strictly monotone. Without that, convergence of $C(n, r_n)$ does not force convergence of $(n - r_n)/\sqrt n$ .

Formalization scope

Rates are real sequences indexed by $\mathbb N$ , and a steady-state solution is a real sequence with HasSum p 1, nonnegative entries, and the balance equations (2.1) exactly as printed (global balance, not detailed balance). Every "the steady-state solution is X" is stated in both halves: X is a steady-state solution, and every steady-state solution equals X; the book's existence conditions ( $\rho < 1$ , $\lambda/(c\mu) < 1$ , convergence of the series) are part of the statements. The $M/M/c/c$ system is the $\mathbb N$ -indexed process with $\lambda_n = 0$ for $n \ge c$ , as §2.5 sets it up; the statement records that states above $c$ carry no mass.

The closed forms that are fixed in Lean: $\prod_{i=1}^n \lambda_{i-1}/\mu_i$ over Finset.Icc 1 n; $B(c, r)$ and $C(c, r)$ exactly as displayed above; $\phi$ = gaussianPDFReal 0 1, $\Phi$ = the CDF of gaussianReal 0 1; $W_q(0) = \sum_{n=0}^{c-1} p_n$ , as evaluated on p.69; $L_q = \sum_{n > c}(n - c)p_n$ as a convergent series.

$C(c, r)$ is a total function in Lean, but its value for $r \ge c$ carries no meaning. The goal assumes $0 < r_n < n$ for $n \ge 1$ , the book's standing condition $\rho < 1$ . A statement about some other function with the same limiting behaviour, or with $B$ and $C$ left abstract, would not be this mission. Neither would one-directional or existence-only versions of the stationary laws.

Not included: the waiting-time distributions (2.28) and (2.39), which need an FCFS waiting-time model with arrival-point probabilities; the $M/M/c/K$ measures (2.45)–(2.48); finite-source and state-dependent models (§§2.8–2.10). Useful contributions beyond the milestones are Poisson tail estimates at the $\sqrt n$ scale and monotonicity of $\alpha(\beta)$ . Both are reusable in other many-server heavy-traffic statements.

Selected references

D. Gross, J. F. Shortle, J. M. Thompson, C. M. Harris, Fundamentals of Queueing Theory, 4th ed., Wiley, 2008. https://doi.org/10.1002/9781118625651
S. Halfin, W. Whitt, Heavy-traffic limits for queues with many exponential servers, Operations Research 29(3), 567–588, 1981. https://doi.org/10.1287/opre.29.3.567
N. Gans, G. Koole, A. Mandelbaum, Telephone call centers: tutorial, review, and research prospects, Manufacturing & Service Operations Management 5(2), 79–141, 2003. https://doi.org/10.1287/msom.5.2.79.16071
A. K. Erlang, Solution of some problems in the theory of probabilities of significance in automatic telephone exchanges, Elektroteknikeren 13, 1917 (English translation in The Life and Works of A. K. Erlang, 1948).
F. P. Kelly, E. Yudovina, Stochastic Networks, Cambridge University Press, 2014. https://doi.org/10.1017/CBO9781139565363

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Algorithmic Game Theory Mechanism Design Operations Research+1·Captain: mikedeng1

An Introduction to the Theory of Mechanism Design X: Robust Mechanism Design — Belief Revelation on Finite Type SpacesTextbook

Motivation

Classical Bayesian mechanism design assumes that the designer knows the agents' beliefs about each other: typically a commonly known prior over independent private values. Wilson's critique (1987) observed that mechanisms tuned to such a prior can depend on details that no designer knows, and a literature on robust mechanism design replaced the fixed prior by a large family of possible beliefs. Chapter 10 of Börgers, An Introduction to the Theory of Mechanism Design (OUP 2015), develops this programme in the framework of Bergemann and Morris (2001, 2005): agents' information is described by a type space, the designer is uncertain which beliefs agents hold, and mechanisms are compared across all type profiles at once.

Timeline of the results formalized here:

1980: Hylland shows that strategy-proof random mechanisms satisfying unanimity conditions are random dictatorships; Dutta, Peters and Sen (2007, 2008) give and correct the cardinal version used in the chapter.
1985: Mertens and Zamir construct the universal type space of belief hierarchies; the space of finite types is emphasized by Dekel, Fudenberg and Morris (2006).
1988: Crémer and McLean show that with correlated types satisfying a spanning condition, beliefs can be elicited at no cost (Proposition 6.4 of the book).
2001–2005: Bergemann and Morris introduce payoff and belief types and prove that on finite type spaces only incentive constraints between types with the same beliefs matter (their Proposition 4.5, the goal of this mission).
2010–2014: Smith, Börgers and Smith study the ranking of mechanisms without a common prior; random dictatorship with compromise comes from Börgers and Smith (2012, 2014).

Setting

There are finitely many agents $i \in I$ , and agent $i$ has a set $\Theta_i$ of payoff types. An outcome $x$ gives agent $i$ the utility $u_i(x,\theta)$ , which may depend on all payoff types. A type space $\mathcal T = (T_i,\hat\theta_i,\hat\beta_i)_{i\in I}$ consists of nonempty sets $T_i$ of types, a payoff type map $\hat\theta_i : T_i \to \Theta_i$ and a belief map $\hat\beta_i : T_i \to \Delta(T_{-i})$ , where $T_{-i} = \prod_{j\ne i}T_j$ . Different types may share a payoff type and differ only in their beliefs, and vice versa. A common prior is a distribution $\mu$ on $T$ from which every type's belief is obtained by conditioning. A type space has a large variety of certainties if for every $\theta_i$ and $\theta_{-i}$ some type with payoff type $\theta_i$ is certain that the others' payoff types are $\theta_{-i}$ . The space of finite types $\mathcal T^+$ collects every infinite hierarchy of beliefs ("I believe that you believe that …") that is generated by a type of some finite type space.

A mechanism $(S_1,\dots,S_N,g)$ has strategy sets $S_i$ and an outcome rule $g : S \to \Delta(X)$ . Strategies $\sigma_i : T_i \to \Delta(S_i)$ form a Bayesian equilibrium if each type maximizes expected utility under its own belief; it is belief-independent if types with equal payoff types play alike, and ex post if each type's choice stays optimal when it becomes certain of the others' types. A direct mechanism asks agents for their types, a reduced direct mechanism only for their payoff types. In the quasi-linear case outcomes are $(a,t_1,\dots,t_N)$ and $u_i = v_i(a,\theta) - t_i$ , with $t_i$ paid by agent $i$ ; a direct mechanism is $(q,t)$ .

Formalization targets

Goal: belief revelation on finite type spaces (Proposition 10.6)

On a finite type space with quasi-linear utilities, suppose that for every agent no belief in $\{\hat\beta_i(\tau_i) : \tau_i\in T_i\}$ is a convex combination of the others, and that in the direct mechanism $(q,t)$ no type wants to imitate another type with the same belief. Then there is a direct mechanism $(\tilde q,\tilde t)$ in which truth telling is a Bayesian equilibrium, with

\tilde q(\tau) = q(\tau)\ \ \forall \tau\in T,\qquad \sum_{\tau_{-i}}\hat\beta_i(\tau_i)(\tau_{-i})\,\tilde t_i(\tau) = \sum_{\tau_{-i}}\hat\beta_i(\tau_i)(\tau_{-i})\, t_i(\tau)\ \ \forall i,\tau_i.

The goal fixes neither the transfers $\tilde t$ nor any bound on them; it asserts the existence of a truthful mechanism with the same alternatives and the same interim payments.

Milestones

The other fourteen numbered results of the chapter: conditional independence of payoff types under a full-support common prior (10.1); three revelation principles (10.2–10.4); existence of Bayesian equilibria of finite mechanisms on $\mathcal T^+$ (10.5); betting between agents with inconsistent beliefs (10.7); ex post implementation of unique equilibrium outcomes and alternatives (10.8, 10.9); emptiness of the set of undominated auctions under interim Pareto welfare and under ex post revenue (10.10, 10.11); Hylland's characterization of random dictatorship (10.12); and three comparisons of random dictatorship with random dictatorship with compromise (10.13–10.15).

Significance

Proposition 10.6 reduces the design problem on a finite type space to incentive constraints among types with the same beliefs: belief types can always be elicited by side payments that leave interim utilities unchanged. With a common prior and Proposition 10.1 this yields optimal mechanisms by solving an independent-types problem for each profile of belief types (§10.8; Farinha Luz 2013 carries this out for auctions). Proposition 10.7 and its consequences 10.10–10.11 show why the same construction cannot be used without a common prior: inconsistent beliefs allow unbounded bets, so interim or revenue criteria admit no undominated mechanism. Propositions 10.12–10.15 show that relaxing belief independence escapes Hylland's impossibility result in the voting problem.

None of these results is formalized elsewhere to our knowledge. The book proves only some of them (10.1, 10.5, 10.8, 10.9, 10.13–10.15 are proved or outlined; 10.6 is sketched; the proofs of 10.2–10.4 are omitted as standard; 10.7 and 10.10–10.12 are stated without proof), so formalization also produces complete proofs of results the book leaves informal. Two printed statements are corrected (see Formalization scope).

Difficulty

The obvious approach to Proposition 10.6 applies the Crémer–McLean construction type by type. This fails because several types may share a belief: a side payment that depends on the reported belief cannot separate them, and the convex-independence condition concerns the set of distinct beliefs rather than the indexed family of types.

For the results on $\mathcal T^+$ , a type is an infinite belief hierarchy, and a strategy must be one function on all finite types simultaneously. Existence (10.5) cannot be obtained by applying Nash's theorem to a single finite type space, because a type belongs to many finite type spaces and must play the same strategy in all of them. Hylland's theorem (10.12) requires a full characterization of strategy-proof random rules on a cardinal preference domain.

Formalization scope

Distributions $\Delta(X)$ are countably supported (PMF X); expected utilities are sums. The book leaves the measure structure of type spaces unspecified (p.179, note 3); finite type spaces, $\mathcal T^+$ and point beliefs are covered exactly. A Bayesian equilibrium requires every type's expected utility to exist (absolute summability) under every mixed strategy.
A type's belief is a distribution on $\prod_{j\ne i}T_j$ . Beliefs in Proposition 10.6 are vectors in $\mathbb R^{T_{-i}}$ , and condition (i) is stated with the convex hull of the other distinct beliefs.
Quasi-linear direct mechanisms are deterministic, $q : T\to A$ , $t_i : T\to\mathbb R$ . Mixed misreports are allowed in every equilibrium notion.
$\mathcal T^+$ is built from belief hierarchies encoded level by level ( $L_0 = \Theta_i$ , $L_{n+1} = \Theta_i\times\Delta(\prod_{j\ne i}L_{n,j})$ ) and the finite type spaces generating them. The universal type space (Definition 10.5) is not needed and not formalized.
§10.11: two agents Fin 2, candidates $\{a,b,c\}$ , strict private vNM utilities with every strict utility attained; mechanisms map to lotteries over candidates; rankings are bijections C ≃ Fin 3.
Corrections of the page: in Proposition 10.7 the signs of the transfers in (v) are reversed on the page relative to the bet described on p.186 and are stated as described; Proposition 10.9 is false under a large variety of certainties alone and is stated under the common-certainty condition that its proof uses, on type spaces whose beliefs have finite support (with countably supported beliefs the reduced mechanism's expected utilities need not exist). Both are explained in the item notes.
The goal is not trivialized by taking $(\tilde q,\tilde t) = (q,t)$ : condition (ii) constrains only types with the same belief, so the original mechanism is in general not incentive-compatible, and the conclusion demands full Bayesian incentive compatibility.

Welcome contributions: a finite Farkas/separation lemma in the form needed for 10.6 (the platform has Polyhedral.farkas_lemma), basic API for PMF-valued type spaces (products of mixed strategies, conditioning), and the hierarchy map of finite type spaces, which all $\mathcal T^+$ milestones share.

Selected references

T. Börgers, An Introduction to the Theory of Mechanism Design, Oxford University Press, 2015, Ch. 10. https://doi.org/10.1093/acprof:oso/9780199734023.001.0001
D. Bergemann, S. Morris, Robust Mechanism Design, Cowles Foundation Discussion Paper 1421, 2001; Econometrica 73 (2005) 1771–1813. https://doi.org/10.1111/j.1468-0262.2005.00638.x
J. Crémer, R. McLean, Full Extraction of the Surplus in Bayesian and Dominant Strategy Auctions, Econometrica 56 (1988) 1247–1257. https://doi.org/10.2307/1913096
J.-F. Mertens, S. Zamir, Formulation of Bayesian Analysis for Games with Incomplete Information, International Journal of Game Theory 14 (1985) 1–29. https://doi.org/10.1007/BF01770224
B. Dutta, H. Peters, A. Sen, Strategy-Proof Cardinal Decision Schemes, Social Choice and Welfare 28 (2007) 163–179. https://doi.org/10.1007/s00355-006-0152-4
T. Börgers, D. Smith, Robust Mechanism Design and Dominant Strategy Voting Rules, Theoretical Economics 9 (2014) 339–360. https://doi.org/10.3982/TE1100

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Algorithmic Game Theory Combinatorics Mechanism Design+1·Captain: mikedeng1

An Introduction to the Theory of Mechanism Design IX: Monotone Direct Mechanisms Are Dictatorial (Gibbard–Satterthwaite)Textbook

Motivation

Voting rules, committee procedures and any other method that turns individual rankings into one collective choice face the same question: can the rule be designed so that no participant ever gains by misreporting their ranking? The Gibbard–Satterthwaite theorem (Gibbard, 1973; Satterthwaite, 1975) answers no. When at least three alternatives can be chosen and all strict rankings are admissible, the only rules immune to manipulation are dictatorships. The result is the starting point of mechanism design without money. It explains why the positive results of the transferable-utility chapters of the book (Groves, VCG, posted prices) depend on quasi-linear preferences, and why research on voting turned to restricted preference domains and weaker solution concepts.

This mission formalizes Chapter 8 of Börgers, An Introduction to the Theory of Mechanism Design (Oxford University Press, 2015), §§8.2–8.3. The book's route to the theorem follows Reny (2001). Strategy-proofness implies Maskin monotonicity, and every monotone rule with full range over at least three alternatives is dictatorial. The second step is the Muller–Satterthwaite theorem (Muller and Satterthwaite, 1977), which is stronger than Gibbard–Satterthwaite because monotonicity is weaker than strategy-proofness. The chapter closes with the classical escape route: on single-peaked preferences (Moulin, 1980) the median voting rule is strategy-proof and not dictatorial.

Timeline. Arrow (1951/1963) proved the impossibility of non-dictatorial preference aggregation under independence of irrelevant alternatives. Gibbard (1973) and Satterthwaite (1975) proved the manipulation version independently, and Satterthwaite showed the two theorems are equivalent. Muller and Satterthwaite (1977) showed that on the full domain strategy-proofness is equivalent to a monotonicity condition (strong positive association). Moulin (1980) characterized strategy-proof rules on single-peaked domains that depend only on reported peaks. Reny (2001) gave the short common proof of Arrow's and the Muller–Satterthwaite theorems that the book follows.

Setting

There is a finite set $I=\{1,\dots,N\}$ of agents and a finite set $A$ of alternatives. Each agent $i$ has a preference relation $R_i$ over $A$ ; $a\,R_i\,b$ reads " $a$ is weakly preferred to $b$ ". Every $R_i$ is a linear order: complete, transitive, and the only indifference is among identical alternatives. Its strict part is $P_i$ . The set of all linear orders over $A$ is $\mathcal R$ , and a profile is $R=(R_1,\dots,R_N)\in\mathcal R^N$ . $(R_i',R_{-i})$ is the profile obtained from $R$ by replacing agent $i$ 's preference with $R_i'$ .

A direct mechanism is a function $f:\mathcal R^N\to A$ (Definition 8.1). It is

dominant strategy incentive-compatible (DSIC) if $f(R_i,R_{-i})\,R_i\,f(R_i',R_{-i})$ for all $i$ , $R$ , $R_i'$ (Definition 8.2);
dictatorial if some agent $i$ satisfies $f(R)\,R_i\,a$ for all profiles $R$ and all $a\in A$ (Definition 8.3);
monotone if $f(R)=a$ and, for every $i$ , $a\,R_i\,b\Rightarrow a\,R_i'\,b$ for all $b$ , together imply $f(R')=a$ (Definition 8.4);
set-monotone if $f(R)\in B$ and, for every $i$ , $R_i'$ differs from $R_i$ only in the ranking of elements of $B$ , together imply $f(R')\in B$ (Definition 8.5);
unanimity-respecting if $f(R)=a$ whenever every agent ranks $a$ at the top (Definition 8.6).

"The range of $f$ is $A$ " means that every alternative is chosen at some profile.

For §8.3 the alternatives are labelled $1,\dots,K$ . A preference is single-peaked if it has a top alternative $k(i)$ and declines monotonically to the right and to the left of it. $\hat{\mathcal R}$ is the set of single-peaked preferences, and on the restricted domain $\hat{\mathcal R}^N$ DSIC and dictatorship are read with all profiles and deviations taken from $\hat{\mathcal R}$ .

Formalization targets

Goal: Proposition 8.5 (Muller–Satterthwaite)

|A|\ge 3,\quad f(\mathcal R^N)=A,\quad f\ \text{monotone}\ \Longrightarrow\ \exists\, i\in I\ \ \forall R\in\mathcal R^N\ \forall a\in A:\ f(R)\,R_i\,a.

This is the book's own capstone ("the core of the proof", p.144). No constant needs to be fixed, and the statement is strictly stronger than the necessity half of Gibbard–Satterthwaite.

Milestones

Proposition 8.2: DSIC $\Rightarrow$ monotone.
Proposition 8.3: monotone $\Rightarrow$ set-monotone.
Proposition 8.4: monotone and full range $\Rightarrow$ respects unanimity.
Proposition 8.1 (Gibbard–Satterthwaite): for $|A|\ge3$ and full range, $f$ is DSIC $\iff$ $f$ is dictatorial.
Proposition 8.6: for $|A|\ge3$ and at least two agents, there is a mechanism on $\hat{\mathcal R}^N$ with range $A$ that is DSIC on $\hat{\mathcal R}^N$ and not dictatorial on $\hat{\mathcal R}^N$ .

Significance

The result itself. Proposition 8.5 turns an incentive question into a purely ordinal one: any full-range rule that is Maskin monotone is a dictatorship once three alternatives are available. With Proposition 8.2 it gives Gibbard–Satterthwaite. Proposition 8.6 marks the boundary of the impossibility: with a one-dimensional ordering of alternatives and single-peaked preferences, the median voter rule escapes it.

Formalizing it. All results are classical and proved. The platform already has a proved Gibbard–Satterthwaite theorem (AGT.gibbard_satterthwaite, Algorithmic Game Theory III), derived from Arrow's theorem in the alternative Mathlib environment c5ea0035…. It uses strict-order profiles and a one-agent-deviation monotonicity. This mission adds Maskin monotonicity, the Muller–Satterthwaite theorem, Reny's direct proof route, and the single-peaked possibility result, none of which is on the platform, all in the default environment.

Difficulty

Propositions 8.2–8.4 are short. The difficulty is in Proposition 8.5. Its proof moves one alternative up or down agents' rankings one agent at a time, and it has to keep the chosen alternative pinned at every step using only monotonicity, set-monotonicity and unanimity. It needs a pivotal agent, whose identity depends on the pair of alternatives, and then an argument that the pivots for different alternatives coincide. The argument uses a third alternative $c$ in an essential way. With two alternatives the conclusion is false (majority rule), so any argument that never uses $|A|\ge3$ cannot succeed. Formally, each "move $b$ just below $a$ in agent $j$ 's ranking" is an explicit construction of a new linear order, together with a check that the monotonicity hypothesis applies. The figures on pp.146–149 describe these orders only partially ("the other alternatives in arbitrary order"). For Proposition 8.6, the obstacle is that DSIC must be checked against every single-peaked misreport, not only misreports of the peak.

Formalization scope

A linear order is the structure LinPref A (relation rel, completeness, transitivity, antisymmetry). A profile is ι → LinPref A for a finite agent type ι, and a direct mechanism is (ι → LinPref A) → A. $A$ is a Fintype. "The range of $f$ is $A$ " is Function.Surjective f, and $|A|\ge3$ is 3 ≤ Fintype.card A.
Monotonicity is the book's Definition 8.4 for arbitrary pairs of profiles, with the lower-contour condition required for each agent separately. Dictatorship is ∃ i, ∀ R a, f R ≥_{R_i} a, with the agent chosen before the profile. A weaker monotonicity (one-agent deviations only) or a weaker dictatorship ("some agent's top is chosen at some profile") would trivialize the goal and is ruled out.
§8.3: the labelling is lab : A ≃ Fin K (labels $0,\dots,K-1$ ). The restricted domain is a predicate on LinPref A, and DSIC, dictatorship and full range are relativized to profiles in the domain (IsDSICOn, IsDictatorialOn, HasFullRangeOn). Values of the mechanism off the domain are never consulted.
Two corrections of the page. The left-hand clause of single-peakedness is printed as $(\ell-1)\,R_i\,\ell$ and is used as $\ell\,R_i\,(\ell-1)$ (the book's words "decline monotonically to the left"). Proposition 8.6 carries the added hypothesis $N\ge2$ , since with one agent every onto strategy-proof rule is dictatorial.
Proposition 8.6 is an existence statement. The median voting mechanism is the book's witness, but the statement does not fix it.
Reusable beyond this mission: the linear-order profile model, Maskin monotonicity and the restricted-domain notions, which apply to Arrow-type results, implementation theory and Moulin's characterization. Proofs of any milestone, or an independent formal proof of Proposition 8.5, are welcome.

Selected references

T. Börgers, An Introduction to the Theory of Mechanism Design, Oxford University Press, 2015, Ch. 8. https://doi.org/10.1093/acprof:oso/9780199734023.001.0001
A. Gibbard, "Manipulation of voting schemes: a general result", Econometrica 41 (1973) 587–601. https://doi.org/10.2307/1914083
M. A. Satterthwaite, "Strategy-proofness and Arrow's conditions", Journal of Economic Theory 10 (1975) 187–217. https://doi.org/10.1016/0022-0531(75)90050-2
E. Muller and M. A. Satterthwaite, "The equivalence of strong positive association and strategy-proofness", Journal of Economic Theory 14 (1977) 412–418. https://doi.org/10.1016/0022-0531(77)90140-5
P. J. Reny, "Arrow's theorem and the Gibbard–Satterthwaite theorem: a unified approach", Economics Letters 70 (2001) 99–105. https://doi.org/10.1016/S0165-1765(00)00332-3
H. Moulin, "On strategy-proofness and single peakedness", Public Choice 35 (1980) 437–455. https://doi.org/10.1007/BF00128122
S. Barberà, "An introduction to strategy-proof social choice functions", Social Choice and Welfare 18 (2001) 619–653. https://doi.org/10.1007/s003550100151

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Algorithmic Game Theory Linear Optimization Mechanism Design+2·Captain: mikedeng1

An Introduction to the Theory of Mechanism Design VII: Crémer–McLean Full Surplus Extraction with Correlated TypesTextbook

Motivation

Bayesian mechanism design asks which collective decisions and payments a designer can implement when every agent holds private information, the type, drawn from a commonly known prior. Most of the classical theory (Myerson's optimal auction, the Myerson–Satterthwaite impossibility, the public goods results) assumes that types are independent. Once types are correlated, as they are when bidders' values share a common component, the theory changes in a way that is best read as a paradox: Crémer and McLean (Econometrica 1988) showed that a designer can then extract the entire surplus, leaving agents no information rents. This mission formalizes Chapter 6 of Börgers, An Introduction to the Theory of Mechanism Design (Oxford University Press 2015), which sets up Bayesian mechanism design in general, treats independent types, and proves the Crémer–McLean theorem, together with the one numbered result of Chapter 9, the impossibility theorem of Jehiel and Moldovanu for interdependent values.

Timeline of the results formalized here:

Rochet (1987) characterized implementable decision rules by cyclical monotonicity; Proposition 6.1 is its interim version for independent types.
Crémer and McLean (1988): under their condition on the prior, every direct mechanism can be made Bayesian incentive-compatible without changing its decision rule or interim payments (Proposition 6.4).
Krishna and Maenner (2001): revenue equivalence for convex type sets and convex utilities (Proposition 6.2).
Jehiel and Moldovanu (2001): with interdependent values, efficient decisions are generically not Bayesian implementable (Proposition 9.1).
Kosenok and Severinov (2008): an identifiability condition added to Crémer–McLean gives ex post budget balance as well (Proposition 6.6).

Setting

There are finitely many agents $i \in I$ and a set $A$ of alternatives. Agent $i$ has a type $\theta_i \in \Theta_i$ and utility $u_i(a,\theta_i) - t_i$ from alternative $a$ and payment $t_i$ . Types $\theta = (\theta_1,\dots,\theta_N) \in \Theta = \prod_i \Theta_i$ are drawn from a common prior $\mu$ ; $\mu(\cdot\mid\theta_i)$ is the conditional distribution of the others' types $\theta_{-i}$ given $\theta_i$ . Types are independent if $\mu(\cdot\mid\theta_i)$ does not depend on $\theta_i$ .

A direct mechanism $(q, t_1,\dots,t_N)$ is a decision rule $q:\Theta\to A$ and payment rules $t_i:\Theta\to\mathbb R$ . It is Bayesian incentive-compatible (BIC) if for every $i$ and all $\theta_i,\theta_i'$ ,

\int_{\Theta_{-i}} u_i(q(\theta_i,\theta_{-i}),\theta_i) - t_i(\theta_i,\theta_{-i})\,d\mu(\theta_{-i}\mid\theta_i) \ \ge\ \int_{\Theta_{-i}} u_i(q(\theta_i',\theta_{-i}),\theta_i) - t_i(\theta_i',\theta_{-i})\,d\mu(\theta_{-i}\mid\theta_i).

The interim decision rule $Q_i(\theta_i)$ is the distribution of $q(\theta_i,\theta_{-i})$ given $\theta_i$ , and the interim expected payment is $T_i(\theta_i) = \int t_i(\theta_i,\theta_{-i})\,d\mu(\theta_{-i}\mid\theta_i)$ . A mechanism is ex post budget balanced if $\sum_i t_i(\theta) = 0$ for every $\theta$ , and ex ante budget balanced if $\int_\Theta \sum_i t_i\,d\mu = 0$ .

In §6.4 every $\Theta_i$ is finite and $\mu(\theta) > 0$ for every $\theta$ . The prior satisfies the Crémer–McLean condition if for no agent $i$ and type $\theta_i$ there are weights $\lambda \ge 0$ on $\Theta_i\setminus\{\theta_i\}$ with

\mu(\theta_{-i}\mid\theta_i) = \sum_{\theta_i'\ne\theta_i}\lambda(\theta_i')\,\mu(\theta_{-i}\mid\theta_i')\quad\text{for all }\theta_{-i}.

Identifiability requires that for every full-support distribution $\nu\ne\mu$ some agent's type $\theta_i$ has a belief $\nu(\cdot\mid\theta_i)$ that is not a nonnegative combination of the beliefs $\mu(\cdot\mid\theta_i')$ .

Formalization targets

Goal: Crémer–McLean (Proposition 6.4)

If $\mu$ satisfies the Crémer–McLean condition, then for every direct mechanism $(q,t)$ there is a BIC direct mechanism $(q,t')$ with

\sum_{\theta_{-i}} t_i(\theta_i,\theta_{-i})\,\mu(\theta_{-i}\mid\theta_i) = \sum_{\theta_{-i}} t_i'(\theta_i,\theta_{-i})\,\mu(\theta_{-i}\mid\theta_i)\quad\text{for all } i,\theta_i.

Milestones

Proposition 6.1 (independent types): $q$ is part of a BIC mechanism iff it is interim cyclically monotone, $\sum_{\kappa=1}^{k-1}\big(\int_A u_i(a,\theta_i^{\kappa+1})\,dQ_i(\theta_i^\kappa) - \int_A u_i(a,\theta_i^\kappa)\,dQ_i(\theta_i^\kappa)\big)\le 0$ for every cycle $\theta_i^k = \theta_i^1$ .
Proposition 6.2 (independent types, convex type sets, convex utilities): two BIC mechanisms with $Q_i' = Q_i$ have $T_i' = T_i + \tau_i$ .
Proposition 6.3 (independent types): every ex ante budget balanced mechanism has an equivalent ex post budget balanced one.
Proposition 6.5 (Farkas's alternative), already proved on the platform as Polyhedral.farkas_lemma.
Proposition 6.6 (Kosenok–Severinov): under Crémer–McLean and identifiability, every ex ante budget balanced mechanism has an equivalent BIC and ex post budget balanced one.
Proposition 9.1 (Jehiel–Moldovanu): in the linear interdependent-values model, under a regularity condition on first best rules and the weight condition $\alpha^i_{ai}/\alpha^i_{bi}\ne\sum_j\alpha^i_{aj}/\sum_j\alpha^i_{bj}$ for some $i,a,b$ , no first best direct mechanism is BIC. It uses its own model (§9.3) and is not on the goal's proof path.

Significance

The Crémer–McLean theorem says that, with correlated finite types, incentive compatibility imposes essentially no constraint: every decision rule and every interim payment rule can be implemented. In a single-unit auction this gives full surplus extraction. The result is the benchmark against which the literature on risk aversion, limited liability, collusion and the genericity of priors (Robert 1991, Laffont–Martimort 2000, Heifetz–Neeman 2006) measures its departures, and Proposition 6.6 extends it to budget-balanced mechanisms, which is what bilateral trade and public goods applications need. Propositions 6.1–6.3 are the independent-types counterpart that the correlated case breaks: they show why revenue equivalence and the ex ante/ex post budget-balance equivalence hold there and fail here. Proposition 9.1 shows the opposite failure, for interdependent values, where efficient decisions cannot be implemented even without participation or budget constraints.

All results are proved in the literature (Kosenok–Severinov's proof is omitted in the book). Apart from Farkas's alternative (Proposition 6.5), which is proved on the platform, none of them is formalized in Mathlib or on the platform; the platform's other mechanism design results (dominant-strategy results in a valuation model, revenue equivalence for symmetric independent auctions) do not cover correlated types.

Difficulty

For the goal the difficulty is the uniformity of the construction: a single payment adjustment must make truth-telling optimal against every possible deviation of every type of every agent, while leaving each type's expected payment unchanged. The obvious scoring-rule adjustment, charging $-\ln\mu(\theta_{-i}\mid\theta_i')$ , changes interim payments, and removing that change is where the Crémer–McLean condition enters. For Proposition 6.2 the envelope argument must handle convex type sets that are not open and utilities that are only convex, not differentiable. For Proposition 9.1 the obvious argument differentiates interim utility twice; the proposition does not assume that interim utility is twice differentiable, so that regularity has to be derived from the hypotheses on the interim probabilities.

Formalization scope

Three definition files carry the three models. Independent types (§6.2–6.3): type sets are arbitrary measurable spaces, the prior is the product of probability measures $\rho_i$ , and interim quantities are integrals against it. Finite correlated types (§6.4): finite type sets, a prior $\mu:\Theta\to\mathbb R$ with $\mu(\theta)>0$ and $\sum_\theta\mu(\theta)=1$ , and conditional beliefs $\mu(\theta_{-i}\mid\theta_i) = \mu(\theta)/\mu(\theta_i)$ ; the alternative set $A$ is arbitrary. Interdependent values (§9.3): finite $A$ , signals in $[0,1]^A$ with positive densities, independent across agents, linear utilities with nonzero weights $\alpha^j_{ai}$ .

Committed conventions and explicit formulas:

The Crémer–McLean condition uses nonnegative weights without a sum-to-one constraint, as Definition 6.7 prints it.
"Equivalent" in Propositions 6.4 and 6.6 means the same decision rule and the same interim expected payments at truthful reports, as Proposition 6.4 states; in Proposition 6.3 it is the report-by-report notion, which under independence is equality of the interim payment rules $T_i$ .
Proposition 6.2 adds continuity of $u_i(a,\cdot)$ on $\Theta_i$ , and Proposition 6.3 adds at least two agents; without them the printed statements are false.
Measurability, which the book omits throughout, is made explicit: decision rules are measurable, and the payment sections and utilities are integrable against the relevant interim distributions.
In Proposition 9.1 the partial derivatives are derivatives within the closed cube $[0,1]^K$ , and the weight condition is the displayed ratio inequality.

A trivializing formalization of the goal, one that proves it only for mechanisms that are already incentive compatible, or with a payment rule that is not a function of the reported type profile, or with "equivalent" weakened to "some BIC mechanism exists", is ruled out: the statement quantifies over every direct mechanism and fixes both the decision rule and every interim payment.

Reusable infrastructure: finite conditional expectations under a full-support prior, the Crémer–McLean and identifiability conditions, and the interim model with product priors. Proofs of any milestone and of the goal, including via the Farkas reference, are welcome.

Selected references

T. Börgers, An Introduction to the Theory of Mechanism Design, Oxford University Press, 2015. https://doi.org/10.1093/acprof:oso/9780199734023.001.0001
J. Crémer and R. P. McLean, "Full extraction of the surplus in Bayesian and dominant strategy auctions", Econometrica 56(6), 1988. https://doi.org/10.2307/1913096
G. Kosenok and S. Severinov, "Individually rational, budget-balanced mechanisms and allocation of surplus", Journal of Economic Theory 140(1), 2008.
P. Jehiel and B. Moldovanu, "Efficient design with interdependent valuations", Econometrica 69(5), 2001. https://doi.org/10.1111/1468-0262.00237
V. Krishna and E. Maenner, "Convex potentials with an application to mechanism design", Econometrica 69(4), 2001. https://doi.org/10.1111/1468-0262.00225
J.-C. Rochet, "A necessary and sufficient condition for rationalizability in a quasi-linear context", Journal of Mathematical Economics 16(2), 1987. https://doi.org/10.1016/0304-4068(87)90007-3

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Algorithmic Game Theory Dynamic Programming Operations Research+1·Captain: mikedeng1

Bellman's Dynamic Programming VIII: Multi-Stage Games, Games of Survival and the Extended Min-Max TheoremTextbook

Why multi-stage games

Chapter X of Richard Bellman's Dynamic Programming (Princeton University Press, 1957; Princeton Landmarks edition 2010, DOI 10.2307/j.ctv1nxcw0f) carries the functional-equation method of the earlier chapters from one-player decision processes to two-player zero-sum games that are played over many stages. At each stage the two players make simultaneous choices, and those choices change the state in which the next stage is played. Games of survival are the standard example. They generalize the gambler's ruin: two players with finite resources play until one of them is ruined. The same framework is the discrete-time form of what Shapley (1953) called stochastic games. It underlies pursuit games, attrition models and the later theory of dynamic games in operations research and economics.

The chapter builds on von Neumann's min-max theorem (1928), which Bellman assumes without proof (§ 3, Eq. (3.4)). It proves three kinds of result: existence and uniqueness for the general multi-stage game equation (§§ 11–16), existence and uniqueness for games of survival with integer payoffs (§ 19), and an extended min-max theorem for ratios of bilinear forms (§ 23). Bellman proposes the ratio as a criterion for non-zero-sum games.

Setting

A matrix game is given by a real $M\times N$ matrix $A=(a_{ij})$ . Player $A$ chooses row $i$ with probability $p_i$ and player $B$ chooses column $j$ with probability $q_j$ ; $p$ and $q$ are distribution vectors (points of the standard simplex). The expected return to $A$ is

E_A(p,q)=\sum_{i,j}a_{ij}\,p_i\,q_j .

The game has value $v$ when $\max_p\min_q E_A=v=\min_q\max_p E_A$ , all extrema attained. In the Lean development this is IsMaxMinMinMaxValue, and the bilinear form is bilin.

In the general multi-stage game (§ 12) the state is a pair of vectors $P\in D\subseteq\mathbb R^n$ , $P'\in D'\subseteq\mathbb R^{n'}$ with norms $\|P\|=\sum_i|P_i|$ . In state $(P,P')$ , $A$ chooses $u$ in a choice domain $S(P,P')$ and $B$ chooses $v$ in $S'(P,P')$ . $A$ receives $R(u,v)$ , and play continues from the state $T(P,P';u,v)$ , $T'(P,P';u,v)$ with weight $h(P,P';u,v)$ . Mixed strategies $G$ , $G'$ are probability measures on the choice domains. The multi-stage game equation is

f(P,P')=\max_G\min_{G'}\iint\big[R(u,v)+h(P,P';u,v)\,f(T,T')\big]\,dG(u)\,dG'(v)=\min_{G'}\max_G[\ \cdots\ ].

A game of survival with integer payoffs (§ 19) has one integer state $x$ , the resources of $A$ out of a fixed total $d$ . $A$ is ruined at $x\le0$ and wins at $x\ge d$ . At each stage the players play the $2\times2$ game with matrix $\begin{pmatrix}-1&a\\c&-b\end{pmatrix}$ , whose entries are the transfers to $A$ .

Formalization targets

Goal: the extended min-max theorem (Chapter X, Theorem 7)

For real matrices $A=(a_{ij})$ , $B=(b_{ij})$ with $\sum_{i,j}b_{ij}p_iq_j\ge d>0$ for all distribution vectors $p,q$ ,

\max_p\min_q\frac{\sum_{i,j}a_{ij}p_iq_j}{\sum_{i,j}b_{ij}p_iq_j}=\min_q\max_p\frac{\sum_{i,j}a_{ij}p_iq_j}{\sum_{i,j}b_{ij}p_iq_j}.

It is the chapter's final result, and its statement involves only finite matrices.

Milestones

Eq. (3.4), von Neumann's theorem $V_A=V_B$ . It is already on the platform as AGT.zero_sum_minimax (existence of a saddle point) and enters as a reference.
Lemma 1: the value of a one-stage game moves by at most the largest change of its kernel, $|L(f)-L_1(F)|\le\max_{u,v}\,[\,|R-R_1|+|h|\,|f(T,T')-F(T,T')|\,]$ .
Theorem 1: under hypotheses (4a)–(4e), the multi-stage game equation has a unique solution among functions continuous on $D\times D'$ that vanish at the origin, and it is the uniform limit on bounded regions of the successive approximations.
Theorem 3: the successive approximations converge from any admissible initial function.
Theorem 4: stability, $|f(P,P')-F(P,P')|\le\sum_{n\ge0}\Delta(k^nc)$ .
Theorem 5: the game of survival equation with boundary values $0$ and $1$ has a unique solution with values in $[0,1]$ .

Significance

Theorem 7 gives a value for ratio games, the games in which a player maximizes a return per unit of a resource consumed. Bellman uses it to give a rationale for the play of non-zero-sum games (§ 24) and to derive the approximate equation (22.4) for non-zero-sum games of survival. Chapter XI, Theorem 5 generalizes it to Markovian decision processes. Theorem 1 is the existence and uniqueness result that justifies replacing an infinite game by its functional equation, and Theorems 3 and 4 make that equation usable for computation and perturbation. Theorem 5 is an early uniqueness theorem for a discrete stochastic game with absorbing boundaries.

None of these statements is formalized on the platform. Von Neumann's theorem is (AGT.zero_sum_minimax, and Sion's theorem as FamousTheorems.sion_minimax_theorem). Theorem 7 is a classical result: with a positive denominator the ratio is quasiconcave in $p$ and quasiconvex in $q$ . This mission asks for a machine-checked proof of the book's statement, by Bellman's route or by any other. Theorems 1, 3 and 4 need a formal theory of games whose mixed strategies are probability measures on moving compact choice domains. No such theory is on the platform yet.

Difficulty

The ratio in Theorem 7 is not bilinear, and in general it is neither concave in $p$ nor convex in $q$ , so neither von Neumann's theorem nor a concave-convex min-max theorem applies to it directly. For Theorem 1, the contraction argument needs each one-stage game to have a value and the value to depend continuously on the state. That requires a min-max theorem for continuous games on compact sets, together with continuity of the value when the choice domains move. In Theorem 5, existence follows from monotone iteration. The uniqueness step is the hard part: the operator is not a contraction in the sup norm, and a second solution must be ruled out even at states where the optimal mixed strategies are degenerate.

Formalization scope

Distribution vectors are points of stdSimplex ℝ ι for nonempty finite types. "Max-min equals min-max" always asserts attainment (IsGreatest/IsLeast) and never uses sSup/sInf, which return $0$ on empty or unbounded sets. Theorem 7 must not assume a saddle point; its only hypothesis is the bound on the denominator.
States are Fin n → ℝ with the $\ell^1$ norm of Eq. (12.1). Choice domains are nonempty compact sets (NonemptyCompacts), and mixed strategies are probability measures of full mass on them. "Vary continuously" in (4b), which the book does not define, is read as continuity in Mathlib's topology on nonempty compact sets (for a metric space, the topology of the Hausdorff distance).
The maxima $w(c)$ of (4d) and $\Delta(c)$ of Theorem 4 are expressed through majorants, which is equivalent to the book's condition. $k\ge0$ is stated explicitly. In Lemma 1 the kernels are assumed measurable and bounded on $S\times S'$ so that the integrals exist, and each game having a value is a hypothesis, as in Bellman's footnote 4.
Theorem 3's "converges" is stated as uniform convergence on bounded regions, the mode of Theorem 1, whose proof Theorem 3 repeats. Theorem 4's unspecified $c$ is any $c\ge\|P\|+\|P'\|$ . The book's p. 301 prints the first min-max of Theorem 4 with the subscripts $G$ , $G'$ interchanged; the equation used is that of Theorem 1.
Theorem 5 adds the implicit $d\ge1$ : for $d\le0$ the boundary conditions contradict each other. The state is an integer and $f:\mathbb Z\to\mathbb R$ .
Taking the choice domains constant would trivialize Theorem 1: condition (4d) would then force $R\equiv0$ on them. The choice domains therefore depend on the state.
Not included: Theorem 2 (optimal strategies of the infinite game, which needs a model of the game's plays), Theorem 6 (non-zero-sum survival; the boundary conditions (21.3) do not cover all states, see the moderation notes).

Contributions of reusable infrastructure are welcome: the min-max theorem for continuous games on compact sets (Eq. (4.2)), continuity of the value in the choice domains, and value iteration for contracting game operators.

Selected references

R. Bellman, Dynamic Programming, Princeton University Press, 1957; Princeton Landmarks in Mathematics, 2010. DOI 10.2307/j.ctv1nxcw0f
J. von Neumann, Zur Theorie der Gesellschaftsspiele, Mathematische Annalen 100, 1928. DOI 10.1007/BF01448847
L. S. Shapley, Stochastic games, Proceedings of the National Academy of Sciences 39, 1953. DOI 10.1073/pnas.39.10.1095
M. Sion, On general minimax theorems, Pacific Journal of Mathematics 8, 1958. DOI 10.2140/pjm.1958.8.171
I. L. Glicksberg, A further generalization of the Kakutani fixed point theorem, with application to Nash equilibrium points, Proceedings of the AMS 3, 1952. DOI 10.1090/S0002-9939-1952-0046638-5

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Linear algebra Mathematical Physics·Captain: Lucas

Varying Constants I: Dimensional Analysis, Natural Units and the Buckingham π TheoremResearch Paper

Motivation

Uzan's review Varying Constants, Gravitation and Cosmology (Living Rev. Relativity 14 (2011) 2) is about testing whether the fundamental constants of physics change in space or time. Before any experiment can be read, §2.1 of the review settles which constants it even makes sense to ask about. Two claims from that section carry the whole programme:

p. 14 (§2.1.2): "only the variation of dimensionless constants can be measured and in case such a variation is detected, it is impossible to determine, which dimensional constant is varying";
pp. 15–17 (§2.1.1): three independent constants (e.g. Stoney's $G,e,c$ or Planck's $c,G,\hbar$ ) can be used to define the three mechanical units, after which "all other constants are dimensionless quantities" whose values do not depend on the units, and "any variation of constants that will leave these numbers unaffected is actually just a redefinition of units".

Both claims are informal statements of a classical result of dimensional analysis, the Buckingham $\pi$ theorem (E. Buckingham, Phys. Rev. 4 (1914) 345). This mission states them precisely and asks for formal proofs.

Setting

Fix $d$ base units $U_1,\dots,U_d$ (for mechanics $d=3$ : $L,M,T$ ) and $n$ dimensional constants. The dimension matrix $D\in\mathbb R^{n\times d}$ says that constant $i$ has dimension $\prod_j U_j^{D_{ij}}$ ; real exponents are allowed, because Gaussian units give the charge the dimension $M^{1/2}L^{3/2}T^{-1}$ (Uzan, p. 15). A configuration $x\in\mathbb R^n_{>0}$ lists the numerical values of the constants in some system of units.

A change of units is a vector $s\in\mathbb R^d_{>0}$ . It acts by $(s\cdot x)_i=x_i\prod_j s_j^{D_{ij}}$ .
An observable $f:\mathbb R^n\to\mathbb R$ is unit invariant if $f(s\cdot x)=f(x)$ for all positive $x,s$ .
The dimensionless exponents are $\mathcal Z_D=\{a\in\mathbb R^n:\sum_i a_iD_{ij}=0\ \forall j\}$ , and $\pi_a(x)=\prod_i x_i^{a_i}$ is the corresponding power product.
Given $d$ chosen constants $e_1,\dots,e_d$ , a system of natural units for $x$ is a positive $s$ with $(s\cdot x)_{e_j}=1$ for all $j$ .
The Planck dimension matrix $D_{\rm P}$ has rows $[c]=LT^{-1}$ , $[G]=L^3M^{-1}T^{-2}$ and $[\hbar]=L^2MT^{-1}$ .

Target

Goal (Buckingham $\pi$ ). Put $k=n-\operatorname{rank}D$ . There are linearly independent $a^{(1)},\dots,a^{(k)}\in\mathcal Z_D$ such that every unit-invariant $f$ factors as

f(x)=F\big(\pi_{a^{(1)}}(x),\dots,\pi_{a^{(k)}}(x)\big)\qquad (x\in\mathbb R^n_{>0}).

Milestones, in order:

Dimensionless power products are unit invariant: $a\in\mathcal Z_D\Rightarrow \pi_a(s\cdot x)=\pi_a(x)$ .
$\dim\mathcal Z_D=n-\operatorname{rank}D$ .
If $\det(D_{e_j,k})\ne0$ , every positive $x$ has exactly one system of natural units.
Values in natural units do not depend on the system of units you started from: $s'\cdot(r\cdot x)=s\cdot x$ .
Planck units: $c=G=\hbar=1$ holds exactly when $s=(1/\ell_P,1/m_P,1/t_P)$ , where $\ell_P=\sqrt{G\hbar/c^3}$ , $m_P=\sqrt{\hbar c/G}$ and $t_P=\sqrt{G\hbar/c^5}$ (Uzan, p. 16).
Two positive configurations agree on every $\pi_a$ with $a\in\mathcal Z_D$ $\iff$ they differ by a change of units (Uzan, p. 17).
Only dimensionless variations are measurable: a unit-invariant $f$ takes equal values on configurations that agree on all dimensionless $\pi_a$ (Uzan, p. 14).

Significance

The result. Milestones 6–7 justify the review's working convention of quoting all bounds as bounds on dimensionless numbers ( $\alpha_{\rm EM}$ , $\mu=m_p/m_e$ , $\alpha_G=Gm_p^2/\hbar c$ , …). Milestone 2 counts how many independent numbers there are. Milestones 3–5 say that natural units are well defined.

Formalizing it. The mathematics is classical and the proofs are known. What remains is to formalize it: a reusable, general dimensional-analysis layer (unit actions, dimensionless monomials, natural units) that later missions on Uzan's review can build on.

Difficulty

Taking logarithms turns the multiplicative problem into linear algebra: changes of units act on $\log x$ by translations through the column space of $D$ , and $\mathcal Z_D$ is the annihilator of that space. Two steps carry the content. The first is the duality step (annihilator of the annihilator) behind milestone 6 and the goal. The second is the bookkeeping with real powers (Real.rpow), which only behaves well on positive bases. The goal also needs a choice of $F$ with no regularity, defined through the quotient $\mathbb R^n/\operatorname{col}(D)$ .

Formalization scope

Lean namespace VaryingConstants; all definitions are in one definition file. Indices are Fin n and Fin d, and exponents are real.
Positivity is an explicit hypothesis (IsPositive), not a subtype. Unit invariance constrains $f$ only on positive configurations.
$\mathcal Z_D$ is the kernel of $a\mapsto a^{\mathsf T}D$ , and the rank is Matrix.rank over $\mathbb R$ .
Natural units are indexed by a map e : Fin d → Fin n. The independence hypothesis is $\det D_e\ne0$ .
There is no trivializing reading: the goal fixes the number of $\pi$ -groups to $n-\operatorname{rank}D$ and requires them to be linearly independent and dimensionless, so the identity map does not qualify.

Contributions such as general lemmas on Real.rpow products and on annihilators of matrix column spaces are reusable beyond this mission.

Selected references

J.-P. Uzan, Varying Constants, Gravitation and Cosmology, Living Rev. Relativity 14 (2011) 2. http://www.livingreviews.org/lrr-2011-2 (§2.1, pp. 9–17).
E. Buckingham, On physically similar systems; illustrations of the use of dimensional equations, Phys. Rev. 4 (1914) 345. https://doi.org/10.1103/PhysRev.4.345

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Convex Optimization Linear Optimization Operations Research+1·Captain: mikedeng1

Linear Programming: Foundations and Extensions VI: The Homogeneous Self-Dual Predictor–Corrector MethodTextbook

Motivation

Interior-point methods are the standard polynomial-time algorithms for linear programming, and the path-following method that practitioners implement (Chapter 18 of Vanderbei's Linear Programming: Foundations and Extensions) comes without a complete convergence proof. Chapter 22 of the same book presents a closely related algorithm for which a complete analysis can be written down: the homogeneous self-dual predictor–corrector method. It combines two ideas. The first is the self-dual embedding of Ye, Todd and Mizuno (1994), which folds a linear program and its dual into one auxiliary problem that always has feasible solutions, so no feasible starting point is needed. The second is the predictor–corrector scheme of Mizuno, Todd and Ye (1993), which alternates an affine-scaling step with a centering step while keeping the iterates in a neighbourhood of the central path, and reduces the duality measure by a factor $1 - 1/(2\sqrt n)$ every two iterations. The result is an $O(\sqrt n\,L)$ iteration bound, the best known for interior-point methods on linear programs.

Setting

Let $A$ be a real $n \times n$ matrix with $n \ge 2$ that is skew symmetric, $A = -A^T$ . The homogeneous self-dual problem (22.4) is

\text{maximize } 0 \quad \text{subject to } Ax + z = 0,\quad x, z \ge 0.

For $x, z \in \mathbb{R}^n$ write $X, Z$ for the diagonal matrices with the entries of $x, z$ on the diagonal and $e$ for the vector of ones. The infeasibility is $\rho(x, z) = Ax + z$ and the noncomplementarity is $\mu(x, z) = \frac1n x^T z$ . For a centering parameter $0 \le \delta \le 1$ , step directions $(\Delta x, \Delta z)$ solve the linear system

A\Delta x + \Delta z = -(1 - \delta)\rho(x, z), \qquad Z\Delta x + X\Delta z = \delta\mu(x, z)e - XZe. \qquad (22.5)\text{–}(22.6)

For $0 \le \beta \le 1$ the neighbourhood is

\mathcal N(\beta) = \{(x, z) > 0 : \|XZe - \mu(x, z)e\| \le \beta\mu(x, z)\},

with $\|\cdot\|$ the Euclidean norm and $(x, z) > 0$ meaning that every component is strictly positive. The algorithm starts at $x^{(0)} = z^{(0)} = e$ and alternates two steps. A predictor step starts from $(x, z) \in \mathcal N(1/4)$ , uses $\delta = 0$ , and takes the step length (22.10) $\theta = \max\{t : (x + t\Delta x, z + t\Delta z) \in \mathcal N(1/2)\}$ . A corrector step starts from $(x, z) \in \mathcal N(1/2)$ , uses $\delta = 1$ and $\theta = 1$ .

A general linear program (22.1), $\max c^Tx$ subject to $Ax \le b$ , $x \ge 0$ with $A$ now $m \times n$ , and its dual (22.2), $\min b^Ty$ subject to $A^Ty \ge c$ , $y \ge 0$ , are embedded in the homogeneous self-dual problem (22.21):

-A^Ty + c\phi + z = 0,\quad Ax - b\phi + w = 0,\quad -c^Tx + b^Ty + \psi = 0,\quad x, y, \phi, z, w, \psi \ge 0.

A feasible solution of (22.21) is strictly complementary if $x_j + z_j > 0$ , $y_i + w_i > 0$ and $\phi + \psi > 0$ for all $i, j$ .

Formalization targets

Goal: Theorem 22.5 (p. 330)

In each predictor step, starting from $(x, z) \in \mathcal N(1/4)$ with any solution $(\Delta x, \Delta z)$ of (22.5)–(22.6) at $\delta = 0$ ,

\theta \ge \frac{1}{2\sqrt n}.

The formal statement asserts that every $t \in [0, 1/(2\sqrt n)]$ keeps $(x + t\Delta x, z + t\Delta z)$ in $\mathcal N(1/2)$ , and that the supremum of the admissible step lengths is at least $1/(2\sqrt n)$ .

Milestones

Theorem 22.1: (22.4) is feasible, every feasible point is optimal, and $z^Tx = 0$ on the feasible set.
Theorem 22.2: $\Delta z^T\Delta x = 0$ , $\bar\rho = (1 - \theta + \theta\delta)\rho$ , $\bar\mu = (1 - \theta + \theta\delta)\mu$ , and $\bar X\bar Ze - \bar\mu e = (1 - \theta)(XZe - \mu e) + \theta^2\Delta X\Delta Ze$ .
Lemma 22.4: $\|PQe\| \le \frac12\|r\|^2$ for the scaled directions $p = X^{-1/2}Z^{1/2}\Delta x$ , $q = X^{1/2}Z^{-1/2}\Delta z$ , $r = p + q$ ; $\|r\|^2 = n\mu$ when $\delta = 0$ ; $\|r\|^2 \le \beta^2\mu/(1 - \beta)$ when $\delta = 1$ and $(x, z) \in \mathcal N(\beta)$ .
Theorem 22.3: a predictor step lands in $\mathcal N(1/2)$ with $\bar\mu = (1 - \theta)\mu$ ; a corrector step lands in $\mathcal N(1/4)$ with $\bar\mu = \mu$ .
Theorem 22.7: there are constants $c_j > 0$ with $x_j + z_j \ge c_j$ for every iterate $(x, z) \in \mathcal N(\beta)$ .
Theorem 22.8: a strictly complementary solution of (22.21) with $\bar\phi > 0$ yields optimal solutions $\bar x/\bar\phi$ , $\bar y/\bar\phi$ of (22.1)–(22.2); with $\bar\phi = 0$ it certifies that the primal or the dual is infeasible.

Significance

Theorem 22.5 is the quantitative core of the method. Combined with Theorem 22.3 it gives $\mu^{(2k)} \le (1 - \frac{1}{2\sqrt n})^k$ along the iterates, and therefore at most $4L\sqrt n$ iterations to bring $\mu$ below $2^{-L}$ (§22.2.4). Since the infeasibility tracks the noncomplementarity, $\rho^{(k)} = \mu^{(k)}\rho^{(0)}$ , both go to zero at that rate. Theorem 22.8 then converts the output into an answer for the original linear program: optimal primal and dual solutions, or a certificate that one of them is infeasible. Theorem 22.7 is the mechanism behind strict complementarity of the limit (Theorem 22.6, stated in the book without proof).

All results are classical and proved in the book. The mission formalizes those proofs. To the best of the curator's knowledge there is no machine-checked convergence analysis of an interior-point method for linear programming in Mathlib or on this platform; the existing platform material on interior-point methods covers a different, short-step path-following method in equality form.

Difficulty

The algebra of Theorem 22.2 is the first obstacle: the orthogonality $\Delta z^T\Delta x = 0$ is not a consequence of (22.5) alone but of skew symmetry combined with both step equations and the definition of $\mu$ , and parts (3)–(4) depend on it. The second is that the book's step length (22.10) is a maximum that need not exist, so a statement about $\theta$ must be phrased about the admissible set of step lengths, and membership in $\mathcal N(1/2)$ requires strict positivity of every component along the whole segment, not only the norm bound at its end. The norm bound alone does not control positivity; an argument that ignores this proves membership in a larger set than $\mathcal N(1/2)$ . Theorem 22.7 needs a strictly complementary feasible solution of (22.4), whose existence (Theorem 10.6 in the book, the Goldman–Tucker theorem) is itself a substantial result not available in Mathlib.

Formalization scope

Vectors are functions Fin n → ℝ (and Fin m → ℝ), matrices are Matrix (Fin m) (Fin n) ℝ, and all declarations sit in the namespace VanderbeiLP.SelfDual. The committed conventions are:

The Euclidean norm is defined explicitly (euclidNorm); Mathlib's default norm on Fin n → ℝ is the sup norm and is not used.
$\mu(x, z) = \frac1n x^Tz$ with $n \ge 2$ , the standing assumption of §22.2, carried as a hypothesis by every theorem about (22.4) together with $A^T = -A$ .
Step directions are any solution of (22.5)–(22.6); existence and uniqueness of the solution are neither assumed nor claimed.
The predictor step length is the supremum of $\{t \in \mathbb{R} : (x + t\Delta x, z + t\Delta z) \in \mathcal N(1/2)\}$ . This set contains $0$ and is bounded above by $1$ along a predictor direction, so the supremum is never a default value. Theorem 22.3(1) is stated under the hypothesis that the maximum exists, as (22.10) presumes.
Theorem 22.7 is stated for points of $\mathcal N(\beta)$ with $0 \le \beta < 1$ satisfying $\rho(x, z) = \mu(x, z)\rho(e, e)$ , the relation all iterates satisfy. The constants $c_j$ are quantified before $(x, z)$ and depend only on $A$ and $\beta$ . The existence of a strictly complementary solution of (22.4) is not a hypothesis.
Theorem 22.8 is for arbitrary $m, n$ and data $(A, b, c)$ ; "optimal" means feasible and attaining the best objective value among feasible points.
No explicit constants beyond those printed in the statements ( $1/4$ , $1/2$ , $1/(2\sqrt n)$ , $\beta^2/(1-\beta)$ ) occur; the book leaves no constant implicit in the formalized results.

A trivializing formalization is ruled out: the step length is not a default-valued supremum, $\mathcal N(\beta)$ requires strict positivity and uses the Euclidean norm, and the goal is also stated as the segment property its proof establishes.

Theorem 22.6 (convergence of the iterates to a strictly complementary solution) is stated without proof in the book and is not part of this mission; the $4L\sqrt n$ iteration count of §22.2.4 is an unnumbered corollary. Both are welcome as follow-up work, as is a proof of Theorem 10.6 for skew-symmetric systems, which Theorem 22.7 needs.

Selected references

R. J. Vanderbei, Linear Programming: Foundations and Extensions, 4th ed., International Series in Operations Research & Management Science 196, Springer, 2014, Chapter 22. https://doi.org/10.1007/978-1-4614-7630-6
S. Mizuno, M. J. Todd, Y. Ye, On adaptive-step primal–dual interior-point algorithms for linear programming, Mathematics of Operations Research 18(4), 964–981, 1993. https://doi.org/10.1287/moor.18.4.964
Y. Ye, M. J. Todd, S. Mizuno, An $O(\sqrt n L)$ -iteration homogeneous and self-dual linear programming algorithm, Mathematics of Operations Research 19(1), 53–67, 1994. https://doi.org/10.1287/moor.19.1.53
A. J. Goldman, A. W. Tucker, Theory of linear programming, in Linear Inequalities and Related Systems, Annals of Mathematics Studies 38, Princeton University Press, 1956, 53–97. https://doi.org/10.1515/9781400881987-005

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Numerical Analysis Operations Research Optimization·Captain: mikedeng1

Minimization Methods for Non-Differentiable Functions VIII: Quadratic Convergence of the Gradient Orthogonalization Method for Nonlinear EquationsTextbook

Motivation

Solving a square system of nonlinear equations $\psi_i(x) = 0$ , $i = 1, \dots, n$ , $x \in \mathbb{R}^n$ , is one of the oldest tasks of numerical analysis. It appears whenever optimality conditions, equilibrium conditions or discretized differential equations have to be solved. N. Z. Shor's Minimization Methods for Non-Differentiable Functions (Springer 1985, translated by K. C. Kiwiel and A. Ruszczyński) treats such a system as the nonsmooth minimization problem

\min_{x \in E_n} f(x), \qquad f(x) = \max_{1 \le i \le n} |\psi_i(x)|,

whose optimal value is $0$ exactly when the system is consistent. Section 3.5 of the book applies the subgradient method with space dilation to this problem. At each iteration that method needs the gradient of one of the $\psi_i$ , not all $n$ of them. Taking the dilation coefficient to its extreme value ( $\beta = 0$ ) turns the method into a gradient orthogonalization method. The book shows that this method converges at a quadratic rate near a regular solution, like Newton's method, although it never forms or factorizes the Jacobian.

This mission is the eighth in a series formalizing Shor's book. It covers Sections 3.5–3.6 (printed pp. 62–71).

Setting

$E_n$ is the $n$ -dimensional Euclidean space with inner product $(x, y)$ and norm $\|x\|$ . The data are functions $\psi_1, \dots, \psi_n : E_n \to \mathbb{R}$ with gradients $g_{\psi_i}(x)$ , and the max-residual function is $f(x) = \max_i |\psi_i(x)|$ (3.24). The Jacobian is the matrix $J(x) = \{\partial\psi_i/\partial t_j\}_{i,j=1}^n$ of partial derivatives with respect to the coordinates $x = \{t_1, \dots, t_n\}$ .

An almost-gradient of a function $f$ at $x_0$ is an accumulation point of a sequence of gradients $\nabla f(x_k)$ , where $x_k \to x_0$ and $f$ is differentiable at every $x_k$ (p. 18).

The regular case (p. 63) is the situation in which $f^* = \min f = 0$ is attained at a point $x^*$ , the $\psi_i$ are continuously differentiable near $x^*$ , and $J(x^*)$ is nonsingular.

The gradient orthogonalization method (3.27) runs in stages of $n$ steps. A stage starting at $x_0$ produces $x_1, \dots, x_n$ as follows. At step $k+1$ , $0 \le k < n$ , the gradient $g_{\psi_{k+1}}(x_k)$ is projected on the subspace orthogonal to the gradients $g_{\psi_1}(x_0), \dots, g_{\psi_k}(x_{k-1})$ used earlier in the stage. Call the result $\varphi_{k+1}$ . The step is

x_{k+1} = x_k - \frac{\psi_{k+1}(x_k)}{\|\varphi_{k+1}\|^2}\, \varphi_{k+1}.

For $k = 0$ there is nothing to project, so $\varphi_1 = g_{\psi_1}(x_0)$ . The next stage starts at $x_n$ .

Formalization targets

Goal: Theorem 3.9, one stage squares the error

Let $x^* = 0$ solve the system, let the $\psi_i$ be continuously differentiable with gradients Lipschitz with constant $L$ on a ball $S_\delta = \{x : \|x\| < \delta\}$ (3.28), and let $g_{\psi_1}(0), \dots, g_{\psi_n}(0)$ be linearly independent. Then there exist $\varepsilon > 0$ and $c > 0$ such that every stage with $\|x_0\| \le \varepsilon$ satisfies

\|x_n\| \le c\, \|x_0\|^2 .

Milestones inside the proof

The proof of Theorem 3.9 passes through these displayed results, stated here in attack order:

Eq. (3.29): near the solution, the orthogonalized gradients satisfy $\|\varphi_{k+1}\| > b > 0$ , with $b$ uniform over stages.
Eq. (3.33): after step $k+1$ , $|\psi_{k+1}(x_{k+1})| \le (L/b^2)\, \psi_{k+1}^2(x_k)$ .
Lemma 3.1: a step of length $h$ in a unit direction orthogonal to $g_{\psi_i}(x_1)$ changes $\psi_i$ by at most $L h (h + \|x_1 - x_2\|)$ .
Eqs. (3.36)–(3.37): within a stage, $\max_k \|x_k\| \le c_1 \|x_0\|$ , and at the end point $f(x_n) \ge c_2 \|x_n\|$ .

Companion result: Theorem 3.8

In the regular case, for every $\delta > 0$ there is a neighborhood of $x^*$ in which every almost-gradient satisfies

(1-\delta) f(x) \le (g_f(x), x - x^*) \le (1+\delta) f(x).

Significance

Theorem 3.9 says that a method which evaluates one equation and one gradient per step, and never solves a linear system, has the local quadratic rate of Newton's method on a regular system. The stages combine: once one stage starts close enough to the solution, the errors of later stages satisfy $\|x_0^{(r+1)}\| \le c\|x_0^{(r)}\|^2$ . Theorem 3.8 describes the geometry of $f$ near a regular solution. The ratio of $(g_f(x), x - x^*)$ to $f(x)$ tends to $1$ , which is what allows the dilation parameters of the book's Theorem 3.3 to approach their extreme values near the solution. Both results connect the nonsmooth space-dilation methods of Chapter 3 to classical Newton-type theory.

All results of this mission are proved in the book. None of them has, to our knowledge, a machine-checked proof: the platform has local quadratic convergence results for cubic-regularized Newton, proximal Newton and semismooth Newton, but not for this method. The work is formalizing Shor's proof. This includes building the Gram-determinant argument behind (3.29) and the second-order estimates, which are reusable for other projection-based solvers of Kaczmarz type.

Difficulty

A naive argument analyses each step as a Newton step for one equation and stops there. That shows each equation $\psi_{k+1}$ is small right after its own step, but the theorem needs all equations to be small at the end of the stage. Later steps move the point and can undo the progress on earlier equations. The step directions are orthogonal to the earlier gradients, but those gradients were taken at earlier points, not at the current one. Controlling this drift needs Lemma 3.1 together with the bound (3.36), which says that the whole stage stays within a multiple of $\|x_0\|$ .

A second difficulty is that the method is only defined near the solution. The step divides by $\|\varphi_{k+1}\|^2$ , and a uniform positive lower bound on these norms comes from the nonvanishing of a Gram determinant evaluated at $n$ different points. The final passage from small residuals to a small error needs the growth bound (3.37), which uses the nonsingularity of the Jacobian once more.

Formalization scope

Representation. $E_n$ is EuclideanSpace ℝ (Fin n). The equations are a family ψ : Fin n → E_n → ℝ indexed from $0$ , so ψ k is the book's $\psi_{k+1}$ . Gradients are Mathlib's gradient, and $S_\delta$ is the open ball Metric.ball 0 δ. Continuous differentiability is ContDiff ℝ 1; in Theorem 3.8 it is ContDiffOn ℝ 1 on a neighborhood of $x^*$ .
The stage. A stage is a predicate IsOrthStage ψ x on a sequence $x : \mathbb{N} \to E_n$ , which fixes $x_1, \dots, x_n$ from $x_0$ . The projection is starProjection onto the orthogonal complement of the span of the earlier gradients.
Corrected misprint. The printed Step 1, (3.27a), divides by $\|g_{\psi_1}(x_0)\|$ rather than its square. The formalization uses the square, which is what (3.27b) and the proof's (3.31)–(3.33) require at $k = 0$ .
Division by zero. The book gives no rule for $\varphi_{k+1} = 0$ ; Lean's $t/0 = 0$ would leave the point unchanged. All statements apply the method only near the solution, where (3.29) bounds $\|\varphi_{k+1}\|$ away from zero.
Quantifier order. In Theorem 3.9 and in (3.29), (3.36)–(3.37), the constants ( $\varepsilon$ , $c$ , $\delta'$ , $b$ , $\varepsilon_0$ , $c_1$ , $c_2$ ) are existentials placed before the stage. They depend only on the data $\psi, \delta, L$ . A statement with $c$ chosen after $x_0$ would be trivial, with $c = \|x_n\|/\|x_0\|^2$ , and is ruled out.
Nonsingularity. In Theorem 3.9 it is the linear independence of $g_{\psi_i}(0)$ , the book's primary hypothesis. In Theorem 3.8 it is $\det J(x^*) \ne 0$ , with $J$ defined by coordinate partial derivatives.
Almost-gradients. They are quantified universally: Theorem 3.8 holds for every almost-gradient, not for one chosen one.

Not formalized: Theorem 3.10 (pp. 70–71), the $n$ -step quadratic convergence of the $\beta = 0$ r-algorithm with resetting. Its directional minimization rule is not pinned down on the page. The algorithm says "determined by minimizing", while the proof uses "the smallest positive root" of a stationarity equation. A contribution that fixes this rule and states the theorem is welcome as a follow-up.

Contributions of any of the five milestones are welcome independently. The Gram-determinant lower bound (3.29) and Lemma 3.1 are general facts about orthogonalization and Lipschitz gradients, and are useful beyond this mission.

Selected references

N. Z. Shor, Minimization Methods for Non-Differentiable Functions, Springer Series in Computational Mathematics 3, Springer, 1985, §§3.5–3.6, pp. 62–71. https://doi.org/10.1007/978-3-642-82118-9
J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, 1970. https://doi.org/10.1137/1.9780898719468

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Convex Optimization Operations Research Optimization·Captain: mikedeng1

Minimization Methods for Non-Differentiable Functions IV: Linear Convergence of the Subgradient Method under Level-Set Shape ConditionsTextbook

Motivation

The subgradient method minimizes a convex function $f$ on $\mathbb{R}^n$ that need not be differentiable, by stepping against an arbitrary subgradient. With stepsizes $h_k \to 0$ , $\sum h_k = \infty$ it converges (Shor, Theorem 2.2), but in general only slowly: no stepsize rule that ignores the structure of $f$ gives a geometric rate. Section 2.3 of N. Z. Shor's Minimization Methods for Non-Differentiable Functions (Springer 1985) identifies geometric conditions on $f$ under which a simple geometric stepsize rule does give linear convergence — convergence with the speed of a geometric progression — and computes the rate explicitly.

The results are the origin of what is now studied as sharpness or error-bound conditions for nonsmooth optimization. Their practical content is that nonsmooth problems whose level sets are not too elongated near the minimum (piecewise-linear functions, maxima of finitely many well-conditioned pieces, positive definite quadratics) can be solved by the subgradient method at a linear rate, with a stepsize rule that needs only one or two scalar parameters.

Timeline. Shor proposed the subgradient method in 1962. The book presents the geometric stepsize rule under an angle condition (Theorem 2.7) and its level-surface form (Theorem 2.8), and attributes the block-halving rule of Theorem 2.9 to its reference [94]. Goffin (Math. Programming 13, 1977) gave the sharp rate in terms of a condition number of the level sets. The book compares the quadratic case with L. V. Kantorovich's rate for steepest descent.

Setting

Let $E_n$ be the $n$ -dimensional Euclidean space with inner product $(x, y)$ and $f : E_n \to \mathbb{R}$ convex. A vector $g$ is a subgradient of $f$ at $x$ if $f(y) - f(x) \ge (g, y - x)$ for all $y$ ; $g_f(x)$ denotes an arbitrary subgradient at $x$ , chosen once for each $x$ . Let $M^*$ be the set of minimum points of $f$ , assumed nonempty; for $x \in E_n$ , $x^*(x)$ is the point of $M^*$ nearest to $x$ .

Given a starting point $x_0$ and positive stepsizes $h_1, h_2, \dots$ , the normalized subgradient method is

x_{k+1} = x_k - h_{k+1}\frac{g_f(x_k)}{\|g_f(x_k)\|}, \qquad k = 0, 1, 2, \dots,

stopped when $g_f(x_k) = 0$ (then $x_k \in M^*$ ).

Two shape conditions are used. The angle condition (2.12) with angle $0 \le \varphi < \pi/2$ asks that every subgradient make an angle at most $\varphi$ with the direction to the nearest minimum point:

(g_f(x), x - x^*(x)) \ge \cos\varphi\,\|g_f(x)\|\,\|x - x^*(x)\|.

The level-surface ratio condition (2.20), for a function with unique minimum point $x^*$ , asks that on a ball $Y$ around $x^*$ any two points $x, z$ on a common level surface $f(x) = f(z) \ne f(x^*)$ satisfy $\|x - x^*\| \le \sigma\|z - x^*\|$ .

Formalization targets

Goal: Theorem 2.8 (p. 32)

If $f$ has a unique minimum point $x^*$ , $\sigma \ge \sqrt2$ , $h_1 \ge \|x_0 - x^*\|/\sigma$ , and (2.20) holds on $Y = \{y : \|y - x^*\| \le \sigma h_1\}$ , then with $h_{k+1} = h_k\sqrt{\sigma^2-1}/\sigma$

\|x_k - x^*\| \le h_{k+1}\,\sigma, \qquad k = 0, 1, 2, \dots

Milestones

Theorem 2.7 (pp. 30–31): under (2.12) and the geometric rule $h_{k+1} = h_k r(\varphi)$ with $r(\varphi) = \sin\varphi$ for $\varphi \ge \pi/4$ and $r(\varphi) = 1/(2\cos\varphi)$ for $\varphi < \pi/4$ ,

\|x_k - x^*(x_k)\| \le h_{k+1}/\cos\varphi \quad\text{resp.}\quad 2h_{k+1}\cos\varphi .

Remark after Theorem 2.7 (p. 32): the same conclusion when (2.12) holds only at the iterates.
Inequality (2.22) (p. 33): (2.20) on $Y$ implies $(g, x - x^*) \ge \sigma^{-1}\|g\|\,\|x - x^*\|$ for every $x \in Y$ and every subgradient $g$ at $x$ .
Example (p. 33): for $A$ symmetric positive definite with extreme eigenvalues $\lambda \le \mu$ ,

\min_{x \ne 0}\frac{(Ax, x)}{\|Ax\|\,\|x\|} = \frac{2\sqrt{\lambda\mu}}{\lambda + \mu},

attained at $x = \sqrt{\mu/(\lambda+\mu)}\,s_1 + \sqrt{\lambda/(\lambda+\mu)}\,s_2$ . 5. Theorem 2.9 (p. 34): under the assumptions of Theorem 2.8 with $\sigma \ge 2$ , the rule $h_{k+1} = h_0 2^{-[(k+1)/N]}$ with $N \ge 3\sigma^2 + 1$ gives $\|x_k - x^*\| \le 2\sigma h_{k+1}$ .

Significance

The results. Theorem 2.7 shows that the subgradient method, often dismissed as sublinear, converges linearly once the geometry of $f$ is controlled and the stepsizes decrease geometrically at the right ratio; the rate $r(\varphi)$ depends only on the angle. Theorem 2.8 restates the hypothesis in terms of the shape of level surfaces, a condition that can be checked for concrete functions, and gives rate $\sqrt{\sigma^2-1}/\sigma$ . The Example computes the angle for positive definite quadratics, yielding rate $(\varrho - 1)/(\varrho + 1)$ with $\varrho = \mu/\lambda$ the condition number — the same rate as steepest descent with exact line search in Kantorovich's analysis, obtained with less storage. Theorem 2.9 removes the need to know $\sigma$ exactly in the stepsize ratio.

Formalizing them. All five results are proved in the book; none is formalized, and no linear-rate result for a nonsmooth first-order method is on the platform. The formalization pins the constants (2.13)–(2.19), the stepsize indexing, and the treatment of the stopped iteration; the Example is a Kantorovich-type inequality for symmetric operators that is reusable beyond this mission.

Difficulty

The obvious one-step estimate $\|x_{k+1} - x^*\|^2 = \|x_k - x^*\|^2 - 2h_{k+1}(g, x_k - x^*)/\|g\| + h_{k+1}^2$ alone does not contract: the step length is fixed in advance and does not shrink with the distance, so a step may overshoot the minimum. The rate argument has to track the ratio between the current distance and the current stepsize, and the admissible ratio of stepsizes is dictated by the worst case of this quadratic in the distance; in the two regimes $\varphi \ge \pi/4$ and $\varphi < \pi/4$ the worst case sits at different ends. For Theorem 2.8, the shape condition is only assumed on the ball $Y$ , so the iterates must be shown to stay in $Y$ , and the passage from level surfaces to subgradients needs the distance from $x^*$ to a level surface, which is not a quantity the iteration computes. Theorem 2.9's constant stepsize blocks are not monotone in distance at all, and the count $3\sigma^2 + 1$ must be matched against a worst-case phase.

Formalization scope

The space $E_n$ is EuclideanSpace ℝ (Fin n); $f$ is real-valued and ConvexOn ℝ Set.univ. A subgradient selection is an arbitrary function g with g x a subgradient at every x, quantified universally. Stepsizes are h : ℕ → ℝ with h (k+1) used at step k; the recursions $h_{k+1} = h_k r$ are imposed for $k \ge 1$ , $h_1$ being the chosen initial step (the book's " $k = 0, 1, 2, \dots$ " in Theorem 2.8 is read this way). The iteration stops at $g_f(x_k) = 0$ by an explicit branch that repeats $x_k$ , never through $x/0 = 0$ ; the bounds are asserted for every $k$ , which implies the book's "either the method stops or …" form. $x^*(x)$ is a definition (the nearest point of the set of minima), and the set of minima is assumed nonempty. Unique minimum is stated as $M^* = \{x^*\}$ . The ball $Y$ is closed, and (2.20) is assumed only for pairs in $Y$ with a common value different from $f(x^*)$ . In (2.16) the ratio is $1/(2\cos\varphi)$ , as the proof requires, where the page prints "1/2 cos φ". In Theorem 2.9, "the assumptions of Theorem 2.8" are taken with $h_1 = h_0$ , which fixes $h_0 \ge \|x_0 - x^*\|/\sigma$ and $Y$ of radius $\sigma h_0$ . In the Example, the extreme eigenvalues are pinned by $\lambda\|x\|^2 \le (Ax,x) \le \mu\|x\|^2$ together with unit eigenvectors, and the minimum is stated with IsLeast.

A statement in which the stepsizes or the bound constants could be chosen after the iterates, or in which the shape condition quantified over an empty set of pairs, would be trivially true; here every constant is fixed by the hypotheses before the sequence is generated, and the conditions are the book's.

Needed infrastructure: the subgradient inequality, continuity of convex functions on $\mathbb{R}^n$ , nearest points of closed convex sets, and elementary trigonometry. The nearest-point map and the stepsize ratio are defined within the mission; the subgradient inequality, the set of minima and the iteration come from the series' shared definitions. Contributions welcome: proofs of the milestones, and a reusable Kantorovich-type cosine bound for symmetric positive definite operators.

Selected references

N. Z. Shor, Minimization Methods for Non-Differentiable Functions, Springer Series in Computational Mathematics 3, Springer 1985, §2.3, pp. 30–36. https://doi.org/10.1007/978-3-642-82118-9
J.-L. Goffin, On convergence rates of subgradient optimization methods, Mathematical Programming 13 (1977) 329–347. https://doi.org/10.1007/BF01584346
L. V. Kantorovich, Functional analysis and applied mathematics, Uspekhi Mat. Nauk 3 (1948) 89–185 (steepest descent rate for quadratics, cited by Shor as [45]).

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Operations Research Probability Statistics+1·Captain: mikedeng1

Elements of Queueing Theory V: Strassen's Theorems and the Stochastic Ordering of QueuesTextbook

Strassen's Theorems and the Stochastic Ordering of Queues

Background

Chapters 1–3 of Baccelli and Brémaud's Elements of Queueing Theory compute exact quantities: Palm identities, stability criteria, PASTA, Pollaczek–Khinchin. Chapter 4 asks a different question. When you cannot compute a queue, can you at least say it is better than another one?

That requires an order on distributions. The chapter builds a family of them — integral orders — by choosing a class ℒ of test functions and declaring F ≤_ℒ G when ∫f dF ≤ ∫f dG for all f ∈ ℒ. Three matter: {i} the non-decreasing functions, giving the strong (stochastic) order; {cx} the convex functions, giving the convex order; and their intersection {icx}.

The goal

An integral order compares two distributions that need not live on the same probability space, and that is both its convenience and its difficulty. Strassen's theorems say each of these orders is secretly a statement about a coupling.

Theorem 4.2.2 (p.278), Strassen's ≤_cx theorem:

F ≤_cx G   ⟺   ∃ X ~ F, Y ~ G on one space with  E[Y | X] = X  a.s.
F ≤_icx G  ⟺   the same with  E[Y | X] ≥ X  a.s.

The convex order holds exactly when G is a martingale dilation of F — obtained by spreading each point out without moving its conditional mean. That is what makes the order usable: comparison results for queues become induction arguments on a coupling instead of analytic manipulations of convolutions of c.d.f.'s.

Its companion Theorem 4.2.1 is the ≤_st version, where the coupling is the simpler X ≤ Y a.s. In dimension one both are explicit — take X = F⁻¹(U), Y = G⁻¹(U) for a uniform U. In dimension n there is no such formula, and that is why these are Strassen's theorems. The book attributes both to Strassen (1965) and proves neither.

Why FIFO is optimal

§4.1 is a different kind of comparison: not between two queues, but between two service disciplines for the same queue. The order there is majorization ≺, which compares how spread out two vectors of the same total are.

The answer is that FIFO minimizes E⁰[f(V)] for every convex f (Property 4.1.3), and the proof is an interchange argument. Under any non-preemptive discipline that uses no information on the service times, customer k effectively receives service σ_{γ(k)} for some permutation γ; Lemma 4.1.3 shows the same queue is produced by FIFO fed with that reordered input, and that the reordering does not change the law of the input. Lemma 4.1.4 passes to the limit, which needs ρ < 1. Lemmas 4.1.1 and 4.1.2 then do the combinatorics: undoing one inversion of γ makes the waiting-time vector less spread out, so the identity permutation — FIFO — is extremal.

Feller's paradox, and what survives it

§4.4 compares time-stationary queues, and opens with a warning. T_n[P⁰] ≤_i T̃_n[P̃⁰] for every n does not imply T_n[P] ≤_i T̃_n[P̃]: Example 4.4.1, "Feller's paradox revisited", exhibits a Poisson process and a renewal process where the Palm order holds and the stationary one fails. The order does not pass from the Palm probability to the stationary one.

For ≤_cx it does. Lemma 4.4.1 is why: it expands E_P[f(N[0,x))] as a series of second differences of f against Palm expectations, and a convex f makes every coefficient non-negative. Lemma 4.4.2 handles the S-orders, built by dividing Palm integrals by the mean cycle length, and shows that the normalisation does not hide the comparison it normalises by.

Formalization scope

Orders. ≤_i, ≤_cx, ≤_icx on distributions on ℝⁿ are integral orders over the book's test classes (§4.2.1), with the page's qualification that only test functions with well-defined integrals count. Majorization ≺ is (4.1.2) with increasing reorderings of both vectors.
Strassen. Both theorems are stated as equivalences, with the coupling existential over the probability space. Theorem 4.2.2 carries both clauses — E[Y | X] = X for ≤_cx, E[Y | X] ≥ X for ≤_icx, as conditional expectations given σ(X) — and assumes both distributions integrable; Theorem 4.2.1 has no integrability hypothesis. A one-directional statement (the Jensen half) is not the theorem.
The queue of §4.1.3 is constructed: a GI/GI input (i.i.d. inter-arrival and service times, independent), a single work-conserving server started empty, and any non-preemptive discipline whose choices are measurable in the information the book's σ-field 𝒢_t carries (arrivals, service times of customers already started) plus external randomisation. FIFO is one such discipline. The interchange permutations γ_n and their limit γ are built from the schedule as on pp.268–270; Lemma 4.1.4 assumes ρ = E[σ₀]/E[τ₀] < 1.
Lemma 4.4.1 is stated with the exact second-difference series and assumes that series converges absolutely; the page states it for all f, which fails for heavy-tailed counts and sparse f.
The S-orders test against {I-ℒ} — primitives ∫_0^t f(u, x) du of test functions — and apply only to distributions whose first coordinate is a.s. positive with a finite mean.

What this mission provides

None of it exists. Mathlib has no stochastic order, no convex order, no increasing-convex order, no majorization, no Schur-convexity and no Strassen theorem; the platform returns zero hits for q=stochastic ordering. Everything in this chapter is new substrate — and §§4.1–4.2 need nothing from Palm calculus, so this mission can be read on its own.

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Convex Optimization Linear Optimization Operations Research+1·Captain: mikedeng1

Linear Programming: Foundations and Extensions IV: Existence of the Central PathTextbook

Motivation

Interior-point methods solve linear programs by moving through the interior of the feasible region instead of along its edges, as the simplex method does. The methods used in practice are path-following methods: they track a curve, the central path, that runs through the interior of the feasible region and ends at an optimal solution. Before any such method can be analysed, the curve has to exist. This mission formalizes Chapter 17 of R. J. Vanderbei, Linear Programming: Foundations and Extensions (4th ed., Springer 2014), which defines the central path through the logarithmic barrier problem and proves that it exists exactly when the primal and the dual problem both have strictly positive feasible points.

The chapter's results have a short history. Barrier methods for nonlinear programming go back to Fiacco and McCormick (1968). Interest in interior-point methods for linear programming began with Karmarkar (1984), whose projective algorithm does not mention a central path; the connection between Karmarkar's method and the primal–dual central path was found by Megiddo (1989), with central points traced back to Huard (1967) and an extended study of the path by Bayer and Lagarias (1989). The chapter is the textbook entry point to this line of work and the foundation for the path-following algorithm of Chapter 18.

Setting

Let $A$ be a real $m \times n$ matrix, $b \in \mathbb{R}^m$ and $c \in \mathbb{R}^n$ . The primal linear program is to maximize $c^T x$ subject to $Ax \le b$ , $x \ge 0$ ; its dual is to minimize $b^T y$ subject to $A^T y \ge c$ , $y \ge 0$ . With slack variables $w \in \mathbb{R}^m$ and $z \in \mathbb{R}^n$ they read (17.1)

Ax + w = b,\ x, w \ge 0 \qquad\text{and}\qquad A^T y - z = c,\ y, z \ge 0.

For a vector $\xi$ , $\xi > 0$ means that every component is strictly positive. The primal feasible region has nonempty interior when some $(\bar x, \bar w)$ satisfies $A\bar x + \bar w = b$ with $\bar x > 0$ , $\bar w > 0$ ; the dual feasible region has nonempty interior when some $(\bar y, \bar z)$ satisfies $A^T \bar y - \bar z = c$ with $\bar y > 0$ , $\bar z > 0$ .

For a parameter $\mu > 0$ , the barrier function (17.7) is

f(x, w) = c^T x + \mu \sum_{j=1}^n \log x_j + \mu \sum_{i=1}^m \log w_i ,

and the barrier problem (17.2) is to maximize $f(x, w)$ subject to $Ax + w = b$ , over the domain $x > 0$ , $w > 0$ where the logarithms are finite. A solution of the barrier problem is a point of that domain at which $f$ attains its maximum over the domain.

Writing $X, Z, Y, W$ for the diagonal matrices carrying $x, z, y, w$ and $e$ for the all-ones vector, the primal–dual central-path system (17.6) is

Ax + w = b, \qquad A^T y - z = c, \qquad XZe = \mu e, \qquad YWe = \mu e,

with $x, w, y, z > 0$ . The last two equations say $x_j z_j = \mu$ and $y_i w_i = \mu$ for all $j$ and $i$ . The set of its solutions $(x_\mu, w_\mu, y_\mu, z_\mu)$ , $\mu > 0$ , is the primal–dual central path.

The chapter also uses one fact from nonlinear programming: for the problem "maximize $f(x)$ subject to $g_i(x) = 0$ , $i = 1, \dots, m$ ", a critical point is a feasible $x^*$ with $\nabla f(x^*) = \sum_i y_i \nabla g_i(x^*)$ for some Lagrange multipliers $y_i$ (17.3), and $H_f(x^*)$ is the Hessian of $f$ at $x^*$ .

Formalization targets

Goal: Theorem 17.2 (p. 265)

For each fixed $\mu > 0$ ,

\exists\, (x, w) \text{ solving the barrier problem} \iff \big(\exists\, \bar x, \bar w > 0 : A\bar x + \bar w = b\big) \wedge \big(\exists\, \bar y, \bar z > 0 : A^T\bar y - \bar z = c\big).

Both directions are part of the goal. The statement fixes no constants and no rate; it asserts only when the barrier problem is solvable.

Milestones

Theorem 17.1 (p. 261), second-order sufficiency under linear constraints: if the constraints are linear, a critical point $x^*$ with $\xi^T H_f(x^*)\xi < 0$ for every $\xi \ne 0$ satisfying $\xi^T \nabla g_i(x^*) = 0$ for all $i$ is a local maximum on the feasible set.
Exercise 10.7 (p. 150): if the primal is feasible and its feasible set $\{x : Ax \le b,\ x \ge 0\}$ is bounded, then there are $y > 0$ , $z > 0$ with $A^T y - z = c$ .
Corollary 17.3 (p. 266): if the primal feasible set (or the dual feasible set) has nonempty interior and is bounded, then for each $\mu > 0$ the system (17.6) has exactly one solution with $x, w, y, z > 0$ .

The corollary is stronger than the goal in one direction (it adds uniqueness and the dual variables) and weaker in another (it assumes boundedness).

Significance

The result itself. Theorem 17.2 gives an exact criterion for the barrier problem to be solvable for a fixed $\mu$ , and Corollary 17.3 turns it into the statement that the central path is a well-defined curve $\mu \mapsto (x_\mu, w_\mu, y_\mu, z_\mu)$ for all $\mu > 0$ . Every path-following method, including the one analysed in Chapter 18 of the same book, targets points on this curve; without existence and uniqueness the "target" of an iteration is undefined. The system (17.6) is also the starting point of the primal–dual Newton step.

Formalizing it. These results are classical and proved in the book; none is formalized in the $Ax \le b$ , $x \ge 0$ form used here. The platform has the converse fact in the standard form $Ax = b$ , $x \ge 0$ (a solution of the central-path conditions minimizes the barrier, Introduction to Linear Optimization), but not existence. A formal proof of Theorem 17.2 and Corollary 17.3 produces a reusable existence theorem for the central path that downstream missions on path-following and self-dual methods can import.

Difficulty

The "if" direction of Theorem 17.2 is an existence claim on a set that is neither closed nor bounded: the domain $x > 0$ , $w > 0$ is open, and the feasible region itself may be unbounded, so the obvious appeal to "a continuous function on a compact set attains its maximum" does not apply directly. The example "maximize $0$ subject to $x \ge 0$ " (p. 264), whose barrier $\mu \log x$ has no maximum, shows that the dual hypothesis cannot be dropped. The "only if" direction, which the book calls trivial and does not prove, needs first-order conditions at a maximizer over a relatively open set.

Theorem 17.1 needs a second-order Taylor expansion with a remainder that is $o(\|\xi\|^2)$ uniformly along the constraint subspace, not along individual lines. Exercise 10.7 is a theorem of the alternative and is not a consequence of weak duality alone. Uniqueness in Corollary 17.3 requires the positivity of the solution: the equations $x_j z_j = \mu$ , $y_i w_i = \mu$ admit sign-flipped solutions.

Formalization scope

Vectors are Fin n → ℝ and Fin m → ℝ; $A$ is a Matrix (Fin m) (Fin n) ℝ. The book's primal–dual pair in $Ax \le b$ , $x \ge 0$ form with slacks is used throughout; there are no explicit constants in this chapter.

Conventions committed to:

"Nonempty interior" means a feasible point with every component strictly positive, as the proof of Theorem 17.2 says. The topological interior of $\{(x, w) : Ax + w = b,\ x, w \ge 0\}$ in $\mathbb{R}^{n+m}$ is empty whenever $m \ge 1$ ; reading the theorem that way would make its right-hand side always false for $m \ge 1$ , and that reading is ruled out.
The barrier problem is posed over $x > 0$ , $w > 0$ explicitly; Real.log returns $0$ at nonpositive arguments and is never evaluated there.
Solutions of (17.6) are required to be strictly positive, as in Exercise 17.3 (p. 267).
"Bounded" is Bornology.IsBounded of the feasible set in $\mathbb{R}^n$ (resp. $\mathbb{R}^m$ ).
In Theorem 17.1 the constraints are $Gx = \beta$ ; $f$ is differentiable near $x^*$ with derivative differentiable at $x^*$ , and $\xi^T H_f(x^*) \xi$ is the second Fréchet derivative applied to $(\xi, \xi)$ . The local maximum is relative to the feasible set.

Infrastructure a complete development needs: attainment of maxima on compact sets (IsCompact.exists_isMaxOn in Mathlib), first-order conditions on relatively open sets, a theorem of the alternative for Exercise 10.7, and concavity facts about the logarithm. A second-order sufficient condition under affine constraints is not in Mathlib and is reusable beyond linear programming. Proofs of any milestone, including the "only if" half of the goal separately, are welcome contributions.

Selected references

R. J. Vanderbei, Linear Programming: Foundations and Extensions, 4th ed., International Series in Operations Research & Management Science 196, Springer, 2014, Chapter 17 and Exercise 10.7. https://doi.org/10.1007/978-1-4614-7630-6
A. V. Fiacco and G. P. McCormick, Nonlinear Programming: Sequential Unconstrained Minimization Techniques, Wiley, 1968. https://doi.org/10.1137/1.9781611971316
N. Karmarkar, A new polynomial-time algorithm for linear programming, Combinatorica 4 (1984), 373–395. https://doi.org/10.1007/BF02579150
N. Megiddo, Pathways to the optimal set in linear programming, in Progress in Mathematical Programming, Springer, 1989, 131–158. https://doi.org/10.1007/978-1-4613-9617-8_8
D. A. Bayer and J. C. Lagarias, The nonlinear geometry of linear programming I, II, Transactions of the AMS 314 (1989), 499–526 and 527–581. https://doi.org/10.1090/S0002-9947-1989-1005525-6

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Linear Programming: Foundations and Extensions II: Farkas' Lemma and Strict Complementary SlacknessTextbook

Motivation

Every linear program comes with a second linear program, its dual, and most of what is known about linear programming is a statement about how the two interact. Weak duality gives certificates of optimality; strong duality says those certificates always exist; complementary slackness turns optimality into a system of equations. These three facts are the core of any first course in optimization and of every correctness argument for the simplex method.

Strict complementarity is the sharpest statement of the same kind. Complementary slackness says that in each pair (a primal variable and its dual slack, a dual variable and its primal slack) at least one member vanishes at optimality. Strict complementarity says that some optimal pair can be chosen so that exactly one member vanishes in each pair. The result is due to Goldman and Tucker (1956). It is what identifies the optimal face of a linear program and its partition of the variables into those that can be positive at an optimum and those that cannot, and it is a standing ingredient in the analysis of interior-point methods, which approach this strictly complementary optimum rather than a vertex.

This mission formalizes the chain from duality to strict complementarity as it is developed in Chapters 5 and 10 of Vanderbei, Linear Programming: Foundations and Extensions (4th ed., Springer 2014, doi:10.1007/978-1-4614-7630-6). It is the second mission of a series on that book.

Timeline:

1902 — Farkas publishes the lemma on the solvability of linear inequality systems.
1947–1951 — von Neumann, and Gale, Kuhn and Tucker, establish linear programming duality.
1956 — Goldman and Tucker prove the existence of strictly complementary optimal solutions (in Linear Inequalities and Related Systems, Annals of Mathematics Studies 38).

Setting

Fix integers $m, n \ge 0$ , a real $m \times n$ matrix $A = (a_{ij})$ , a vector $b \in \mathbb{R}^m$ and a vector $c \in \mathbb{R}^n$ . The primal problem is

\text{maximize } c^T x \quad\text{subject to}\quad Ax + w = b,\quad x \ge 0,\ w \ge 0, \qquad (10.9)

where $w = b - Ax$ is the primal slack. The dual problem is

\text{minimize } b^T y \quad\text{subject to}\quad A^T y - z = c,\quad y \ge 0,\ z \ge 0, \qquad (10.10)

where $z = A^T y - c$ is the dual slack. A vector $x$ is primal feasible if $x \ge 0$ and $w \ge 0$ ; it is primal optimal if it is feasible and $c^T x' \le c^T x$ for every feasible $x'$ . Dual feasibility and dual optimality are defined in the same way, with minimization. Inequalities between vectors are componentwise, and $\xi > 0$ means that every component of $\xi$ is strictly positive.

A halfspace of $\mathbb{R}^n$ is a set $\{x : a^T x \le \beta\}$ with $a \ne 0$ ; a polyhedron is a set $\{x : Ax \le b\}$ for some $m$ , $A$ and $b$ .

In the Lean development these objects live in the namespace VanderbeiLP.StrictComp: primalSlack A b x, dualSlack A c y, PrimalFeasible, DualFeasible, PrimalOptimal, DualOptimal, IsHalfspace, IsPolyhedron.

Formalization targets

Goal: Strict Complementary Slackness (Theorem 10.7)

If the primal (10.9) has an optimal solution, then there exist a primal optimal $x^*$ and a dual optimal $y^*$ , with slacks $w^* = b - Ax^*$ and $z^* = A^T y^* - c$ , such that

x^* + z^* > 0 \qquad\text{and}\qquad y^* + w^* > 0.

The only hypothesis is primal optimality; the existence of a dual optimum is part of the conclusion.

Milestones

Theorem 5.1 (Weak Duality). Primal feasible $x$ and dual feasible $y$ satisfy $c^T x \le b^T y$ .
Theorem 5.2 (Strong Duality). If the primal has an optimal $x^*$ , the dual has an optimal $y^*$ with $c^T x^* = b^T y^*$ .
Theorem 5.3 (Complementary Slackness). Feasible $x$ , $y$ are both optimal if and only if $x_j z_j = 0$ for all $j$ and $w_i y_i = 0$ for all $i$ .
Lemma 10.5 (Farkas' Lemma). $Ax \le b$ has no solution if and only if some $y$ satisfies $A^T y = 0$ , $y \ge 0$ , $b^T y < 0$ .
Theorem 10.4 (Separation of polyhedra). Two disjoint nonempty polyhedra lie in two disjoint halfspaces.
Theorem 10.6. If both problems are feasible, there are feasible $\bar x$ , $\bar y$ with $\bar x + \bar z > 0$ and $\bar y + \bar w > 0$ .

Theorem 10.6 is the feasible-solution version of the goal; Theorem 10.4 is a further consequence of Farkas' Lemma in the same chapter.

Significance

Strict complementarity determines the optimal partition: the set of indices $j$ for which some optimal $x^*$ has $x^*_j > 0$ is exactly the complement of the set for which some optimal dual slack $z^*_j$ is positive. This partition describes the optimal faces of both problems, is the object that interior-point methods recover in the limit, and is the starting point of sensitivity analysis beyond a single optimal basis. Farkas' Lemma and the separation theorem are the linear-algebraic form of convex separation and are reused across optimization, game theory and polyhedral combinatorics.

All results of this mission are classical and proved in the literature. What the mission adds is a machine-checked version of them in one fixed linear-programming form, the inequality form with explicit slacks used throughout Vanderbei's book. Weak duality, strong duality and complementary slackness are already formalized on this platform for other forms (Bertsimas–Tsitsiklis's general form, a minimization, and a covering pair with the roles of primal and dual exchanged). Those statements are equivalent to the ones here only after a transformation (negating the objective, swapping primal and dual), so they are not the same theorems. The Farkas variant for inequality systems, the separation theorem for two polyhedra, and both strict complementarity theorems have no formal counterpart on the platform.

Difficulty

Weak duality and the converse direction of complementary slackness are short computations. The substance lies elsewhere. Strong duality and Farkas' Lemma require a genuine existence argument; the book obtains them from the simplex method, whose termination is itself a nontrivial fact, and any other route needs an independent theorem of the alternative.

For strict complementarity the obvious attempt fails. Complementary slackness gives, for each optimal pair, only that one member of each complementary pair vanishes; nothing in a single optimal basic solution forces the other member to be positive, and in degenerate problems every basic optimal pair can fail strictness. A strictly complementary pair is in general not a vertex of either optimal face, so it cannot be found by inspecting basic solutions. The goal also asks for more than Theorem 10.6: the positivity must be achieved within the optimal sets, which are faces cut out by an additional objective-level constraint, so the feasible-solution argument does not transfer verbatim.

Formalization scope

Vectors are Fin n → ℝ and Fin m → ℝ; the constraint matrix is Matrix (Fin m) (Fin n) ℝ; m and n are arbitrary natural numbers, including zero. The slacks are functions of the solution (primalSlack A b x = b - A *ᵥ x, dualSlack A c y = Aᵀ *ᵥ y - c), never free variables, so a "solution $(x, w)$ " of the book is the vector $x$ with the slack it determines. Optimality is attainment of the maximum (minimum) over the feasible set; no supremum, value function or extended reals are involved. The strict inequality $\xi > 0$ is written componentwise as ∀ j, 0 < x j + dualSlack A c y j and ∀ i, 0 < y i + primalSlack A b x i.

A halfspace carries a nonzero normal vector. Without that requirement the empty set would be a halfspace and the separation theorem would be trivial; the formal definition rules this out.

The book states every result in this mission with its hypotheses explicit, and none of them asserts the existence of an unspecified constant, so no explicit-constant instantiation was needed. The remark after Theorem 10.7 refers to "the complementary slackness theorem (Theorem 5.1)"; the complementary slackness theorem is Theorem 5.3, and the milestones follow the theorem numbering.

A complete development needs a theorem of the alternative for real linear inequality systems (Mathlib has Farkas-type results for cones and the geometric Hahn–Banach theorem, but no ready-made matrix version of Lemma 10.5) and elementary convex-combination arguments on feasible sets. The definitions of this mission are self-contained and reusable for any later chapter that works in Vanderbei's inequality form. Proofs of any milestone are welcome, as are proofs that avoid the simplex method.

Selected references

R. J. Vanderbei, Linear Programming: Foundations and Extensions, 4th ed., International Series in Operations Research & Management Science 196, Springer, 2014. doi:10.1007/978-1-4614-7630-6
J. Farkas, "Theorie der einfachen Ungleichungen", Journal für die reine und angewandte Mathematik 124 (1902), 1–27. doi:10.1515/crll.1902.124.1
A. J. Goldman and A. W. Tucker, "Theory of linear programming", in Linear Inequalities and Related Systems, Annals of Mathematics Studies 38, Princeton University Press, 1956, 53–97.
D. Gale, H. W. Kuhn and A. W. Tucker, "Linear programming and the theory of games", in Activity Analysis of Production and Allocation, Wiley, 1951, 317–329.

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Minimization Methods for Non-Differentiable Functions XI: Convexity and Subgradients of the Value Function in Decomposition with Respect to VariablesTextbook

Motivation

Large convex programs often have a block structure: a small set of "complicating" variables $x$ couples otherwise separate subproblems in the remaining variables $y$ . Decomposition with respect to variables fixes $x$ , solves the subproblem in $y$ , and treats the optimal subproblem value as a function of $x$ alone. The outer problem in $x$ is then small but nonsmooth, because the optimal value of a constrained program is generally not differentiable in its parameters. Shor's Chapter 4 (Shor 1985, Ch. 4) presents this reduction as a principal application of subgradient methods: once a subgradient of the outer function can be read off from the subproblem, the methods of Chapters 2–3 apply directly. The same construction underlies Benders decomposition (Benders 1962) and its convex generalization (Geoffrion 1972), and parametric decomposition schemes for linear programs.

The chapter's other numbered results serve the same programme from the dual side: the Lagrangian dual function of a program over a compact set gives a lower bound usable in branch and bound (Theorem 4.3), exact nonsmooth penalty functions turn a constrained convex program into one unconstrained nonsmooth minimization (Theorem 4.2; nonsmooth penalties were first studied systematically by I. I. Eremin, 1967), and a stochastic transportation model is shown to be a convex program before being solved through its dual (Lemma 4.3).

Setting

The variables split into $x \in E^x_l$ and $y \in E^y_m$ (Euclidean spaces, inner product $(\cdot,\cdot)$ ). The problem is

\min_{x,y} f_0(x,y) \quad \text{s.t.} \quad f_i(x,y) \le 0,\quad i = 1,\dots,n, \qquad (4.1)\text{–}(4.2)

with $f_0, f_1, \dots, f_n$ convex functions of $z = (x,y)$ (jointly convex), finite everywhere. For a fixed $\bar x$ , the subproblem (4.3)–(4.4) is $\min_{y \in D(\bar x)} f_0(\bar x, y)$ with $D(\bar x) = \{y : f_i(\bar x,y) \le 0\}$ . Where it has a solution $y(\bar x)$ , the value function is

\Phi(\bar x) = \min_{y \in D(\bar x)} f_0(\bar x,y). \qquad (4.5)

The Slater condition at $\bar x$ asks for a $y$ with $f_i(\bar x,y) < 0$ for all $i$ . The Lagrange function is $L_U(x,y) = f_0(x,y) + \sum_i U_i f_i(x,y)$ , and $U \ge 0$ are Kuhn–Tucker multipliers at $\bar x$ when $\Phi(\bar x) = \min_y L_U(\bar x,y)$ . A subgradient of a function of $(x,y)$ is written through its projections $(g^x, g^y)$ on the two blocks; a subgradient of $\Phi$ at $\bar x$ is a $g$ with $\Phi(x) - \Phi(\bar x) \ge (x - \bar x, g)$ .

Formalization targets

Goal: Theorem 4.1 (p. 94)

If $W$ is a convex set of $x$ -values at which the subproblem has a solution, then $\Phi$ is convex on $W$ ; and if $\bar x \in W$ satisfies the Slater condition, then for every optimal $y(\bar x)$ , multipliers $U$ exist, $L_U$ has a subgradient at $(\bar x, y(\bar x))$ with vanishing $y$ -projection, and the $x$ -projection of any such subgradient satisfies

g_\Phi(\bar x) = g^x_{L_U}(\bar x, y(\bar x)) \in \partial \Phi(\bar x). \qquad (4.6)

Milestones for the goal

Convexity of $\Phi$ on $W$ (Theorem 4.1, first assertion).
Existence of Kuhn–Tucker multipliers for the subproblem under Slater (p. 95, display).
Existence of a subgradient of $L_U$ with vanishing $y$ -projection (p. 95).
Formula (4.6) for a given multiplier vector and such a subgradient (p. 95, final display).

Further results of the chapter

Corollary (4.7): if each $f_\alpha(x,\cdot)$ is continuously differentiable in $y$ , then $g^x_{f_0} + \sum_i U_i g^x_{f_i}$ , built from arbitrary subgradients of the $f_\alpha$ , is a subgradient of $\Phi$ .
Lemma 4.3: the stochastic transportation problem (4.163)–(4.165) is a convex program.
Theorem 4.2: with nonsmooth penalties $p_i$ whose slopes $c_i = \lim_{t\to0+} p_i(t)/t$ exceed a Lagrange multiplier vector $\bar y$ , the minimizers of $S = f_0 + \sum p_i \circ f_i$ are exactly the solutions of the constrained program; and if a minimizer of $S$ solves the program, some multiplier vector satisfies $\bar y \le c$ .
Theorem 4.3: for $\Phi(u) = \min_{x\in X}[f_0 + \sum u_i f_i]$ over a compact $X$ , $Q = \max_{u\ge0}\Phi(u) \le f^*$ .

Significance

The result itself. Theorem 4.1 is what makes decomposition with respect to variables an instance of convex nonsmooth minimization: the outer problem is convex, and one subproblem solve returns both $\Phi(\bar x)$ and a subgradient. The algorithm on p. 96 — solve the subproblem at $x_k$ , form $g_\Phi(x_k)$ by (4.6) or (4.7), take a subgradient step — is exactly this, and the step-size theory of Chapter 2 then gives convergence. The Corollary is the version used in practice for linear and quadratic subproblems, where multipliers and partial subgradients are computed directly. Theorem 4.2 justifies replacing constraints by nonsmooth penalties of finite slope, and Theorem 4.3 is the weak-duality bound behind Lagrangian relaxation in branch and bound.

Formalizing it. All results are classical and proved in the book; none is formalized as stated here. The platform already has related statements with different shapes: convexity of the perturbation value function in the constraint right-hand side (VectorSpaceOpt.perturbationValue_convex, Luenberger), Slater strong duality over the whole space (ConvexOptimization.slater_strong_duality), weak duality with an unconstrained domain (ConvexOptimization.weak_duality), weak Lagrangean duality for integer programs (LinearOptimization.integer_program_weak_lagrangean_duality), and the LP special case of convexity of the optimal cost (LinearOptimization.lp_optimal_cost_convex_in_rhs). This mission adds the partial-minimization form in which one block of variables is minimized out under joint convexity, its subgradient calculus, exact nonsmooth penalties, and weak duality over a compact domain.

Difficulty

Convexity of $\Phi$ is elementary once the minimum is attained. The subgradient formula is where the obvious argument fails: an arbitrary subgradient of $L_U$ at $(\bar x, y(\bar x))$ does not project to a subgradient of $\Phi$ , because its $y$ -projection contributes a term $(y(x) - y(\bar x), g^y)$ of unknown sign. The theorem needs a subgradient whose $y$ -projection vanishes, and its existence is a separate fact about partial minimization of a jointly convex, everywhere-finite function. The multipliers come from the Kuhn–Tucker theorem for the subproblem, which requires the Slater condition. In the Corollary, the difficulty is to show that differentiability in $y$ forces the $y$ -projection of any combination $g_{f_0} + \sum U_i g_{f_i}$ to vanish. In Theorem 4.2 the necessity part needs a subdifferential chain rule for $p_i \circ f_i$ .

Formalization scope

$E^x_l$ , $E^y_m$ and $E_N$ are EuclideanSpace ℝ (Fin l), EuclideanSpace ℝ (Fin m), EuclideanSpace ℝ (Fin N); constraints are indexed by Fin n (or Fin m). All functions are real-valued and finite everywhere; joint convexity is convexity on the product $E^x_l \times E^y_m$ .
$\Phi$ is a real infimum over $D(x)$ . It is the book's minimum wherever the minimum is attained, and every statement assumes attainment at each point of $W$ . A statement about $\Phi$ at points where the subproblem has no solution would be about Lean's default value $0$ , and is ruled out by these hypotheses.
"Convex on some convex subset $W$ of $E_n$ " is read as convexity on every convex $W$ on which $\Phi$ is defined. A formalization quantifying over a single unspecified $W$ (e.g. a singleton) would be trivially true.
Formula (4.6) is stated for subgradients of $L_U$ whose $y$ -projection is zero, as the book's proof uses it; the subgradient inequality for $\Phi$ is stated on all of $W$ .
Kuhn–Tucker multipliers relative to an optimal $\bar y$ : $U \ge 0$ , $U_i f_i(\bar x,\bar y) = 0$ , and $\bar y$ minimizes $L_U(\bar x,\cdot)$ over all $y$ .
Theorem 4.2: a Lagrange multiplier vector is $\bar y \ge 0$ with $f_0 + \sum\bar y_i f_i \ge f^*$ everywhere, $f^*$ the finite optimal value; the necessity clause is read as "for some multiplier vector" (for all multiplier vectors it is false); the limits $c_i$ are given as hypotheses.
Theorem 4.3: the minimum (4.187) attained for every $u \ge 0$ (the book's "min"; no continuity assumed); $f^*$ attained; $Q$ attained as the book's "max" presupposes.
Lemma 4.3: demands with densities and finite mean, penalty coefficients $r_j \ge 0$ .

Useful infrastructure beyond this mission: partial minimization of jointly convex functions, subgradients on product spaces, and a Kuhn–Tucker saddle-point theorem for convex programs with inequality constraints. Proofs of any milestone, and reusable lemmas for these, are welcome.

Selected references

N. Z. Shor, Minimization Methods for Non-Differentiable Functions, Springer Series in Computational Mathematics 3, Springer, 1985, Ch. 4, pp. 93–96, 131–133, 146–148. https://doi.org/10.1007/978-3-642-82118-9
J. F. Benders, Partitioning procedures for solving mixed-variables programming problems, Numerische Mathematik 4, 1962, 238–252. https://doi.org/10.1007/BF01386316
A. M. Geoffrion, Generalized Benders decomposition, Journal of Optimization Theory and Applications 10, 1972, 237–260. https://doi.org/10.1007/BF00934810
I. I. Eremin, The penalty method in convex programming, Soviet Mathematics Doklady 8, 1967, 459–462 (Shor's reference [22]).
R. T. Rockafellar, Convex Analysis, Princeton University Press, 1970, §29 (partial minimization and perturbation functions). https://doi.org/10.1515/9781400873173

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Minimization Methods for Non-Differentiable Functions X: The Space-Dilation Ellipsoid Method Localizes the Solution in Ellipsoids Shrinking by the Ratio q_nTextbook

Motivation

The ellipsoid method is the algorithm that settled the polynomial-time solvability of linear programming (Khachiyan, 1979) and that underlies the equivalence of separation and optimization in combinatorial optimization (Grötschel, Lovász and Schrijver, 1981). Its origin is in nonsmooth convex optimization. In 1976 Yudin and Nemirovskii proposed a modified method of centered sections that localizes an optimum inside a sequence of ellipsoids, and in 1977 N. Z. Shor observed independently that the same scheme is a subgradient method with space dilation along the gradient, the family of methods he had developed since 1969. Section 3.8 of Shor's monograph Minimization Methods for Non-Differentiable Functions (Springer 1985) presents the method in this second form and proves its basic localization property.

Timeline:

1965. A. Yu. Levin proposes the method of centered sections (cuts through the center of gravity of a polyhedron); each cut removes at least a fixed fraction of the volume, but computing centers of gravity is impractical for $n > 3$ .
1969–1972. Shor introduces subgradient methods with space dilation along the gradient (SDG methods).
1976. Yudin and Nemirovskii replace the polyhedron by a minimal ellipsoid containing a half-ellipsoid, obtaining a geometric volume decrease depending only on the dimension ([Yudin–Nemirovskii 1976]).
1977. Shor shows that the same method is an SDG algorithm with coefficient $\beta = \sqrt{(n-1)/(n+1)}$ ([Shor 1977]).
1979. Khachiyan applies the method to linear inequalities with integer data, obtaining the first polynomial-time algorithm for linear programming ([Khachiyan 1979]).

Setting

Let $E_n$ be $n$ -dimensional Euclidean space with inner product $(x, y)$ , and $n > 1$ . For a unit vector $\xi$ and a number $\alpha$ , the operator of space dilation along $\xi$ with coefficient $\alpha$ is $R_\alpha(\xi) = I + (\alpha - 1)\xi\xi^T$ : it multiplies the component of a vector along $\xi$ by $\alpha$ and leaves the orthogonal component unchanged.

Let $g : E_n \to E_n$ be a vector field, not necessarily continuous. The problem is to find a point $x^*$ with

(g(x), x - x^*) \ge 0 \quad \text{for all } x \in E_n,

where it is known that such an $x^*$ exists in the closed ball $S(x_0, R)$ of radius $R > 0$ about a given point $x_0$ . Put $\beta = \sqrt{(n-1)/(n+1)}$ and $r = n/\sqrt{n^2-1}$ . The algorithm (3.57)–(3.60) starts from $x_0$ , $B_0 = I_n$ , $h_0 = R/(n+1)$ , and at iteration $k+1$ stops if $g(x_k) = 0$ , and otherwise sets

\xi_k = \frac{B_k^T g(x_k)}{\|B_k^T g(x_k)\|}, \quad x_{k+1} = x_k - h_k B_k \xi_k, \quad B_{k+1} = B_k R_\beta(\xi_k), \quad h_{k+1} = r h_k .

With $A_k = B_k^{-1}$ , the localizing ellipsoid is $\Phi_k = \{x : \|A_k(x - x_k)\| \le (n+1)h_k\}$ , and the dimension-dependent ratio is

q_n = \sqrt{\frac{n-1}{n+1}}\left(\frac{n}{\sqrt{n^2-1}}\right)^n < 1 .

Three problems produce such a field: minimizing a convex $f$ on a ball (the field (3.62), a subgradient inside the ball and the outward radial direction outside); the convex program $\min f_0$ s.t. $f_i \le 0$ (the field (3.65), a subgradient of the objective at feasible points and of a most violated constraint otherwise); and a convex–concave saddle point problem (the field $\{g_f^x, -g_f^y\}$ ).

Formalization targets

Goal: Theorem 3.14 (p. 86)

For every $k$ ,

\|A_k(x_k - x^*)\| \le h_k (n+1), \tag{3.61}

that is, $x^* \in \Phi_k$ . The statement holds for every field $g$ satisfying the monotonicity condition at $x^*$ ; nothing about continuity or convexity is assumed.

Milestones

Eq. (3.4): $\|R_\alpha(\xi)x\| = \sqrt{\|x\|^2 + (\alpha^2-1)(x,\xi)^2}$ for unit $\xi$ .
Volume of $\Phi_k$ (p. 87): $(n+1)h_k = R r^k$ and $v(\Phi_k) = v_0 R^n r^{nk}/\det A_k$ , $v_0$ the volume of the unit ball.
Volume ratio (p. 87–88): $v(\Phi_{k+1}) = q_n\, v(\Phi_k)$ with $q_n < 1$ , the volumes being positive and finite.
Eq. (3.62): the ball field satisfies $(g(x), x - x^*) \ge 0$ .
Eq. (3.65): the convex-programming field satisfies $(g(x), x - x^*) \ge 0$ .
Saddle point field (p. 90): $(g(z), z - z^*) \ge 0$ .

Significance

The result. Theorem 3.14 with the volume identity says that after $k$ steps the solution is confined to an ellipsoid of volume $q_n^k$ times that of the initial ball, for any field of the above kind. Milestones 4–6 turn this into localization guarantees for constrained convex minimization, general convex programming and convex–concave saddle points, with a rate that depends only on the dimension. The same localization underlies the complexity bounds of the ellipsoid method for linear programming and the polynomial equivalence of separation and optimization.

Formalizing it. The results are classical and proved in the book. No machine-checked proof of the space-dilation form of the method is known to exist. The platform already contains a proved version of the Bertsimas–Tsitsiklis form (LinearOptimization.ellipsoid_update_halfspace_subset, LinearOptimization.ellipsoid_update_volume_lt: a half-ellipsoid $E(z, D) \cap \{a^Tx \ge a^Tz\}$ is covered by an updated ellipsoid whose volume is smaller by a factor below $e^{-1/(2(n+1))}$ ), and Khachiyan's feasibility algorithm (SmaleNinth.khachiyan_ellipsoid_decides). Those statements are about a center/shape-matrix update and give a volume inequality; this mission is about the iterates of Shor's matrix recursion $B_{k+1} = B_k R_\beta(\xi_k)$ and the exact ratio $q_n$ . Relating the two parametrizations ( $D_k = (n+1)^2 h_k^2 B_k B_k^T$ ) is a welcome side result.

Difficulty

The obvious approach, tracking the ellipsoid through the center and shape matrix and invoking a minimum-volume covering argument, is not what the algorithm computes: here the iterate is updated through the factor $B_k$ and the stepsize $h_k$ is fixed in advance, independent of the field, so the induction must be carried out in the transformed coordinates $z_k = A_k(x_k - x^*)$ in which the ellipsoid is a ball. The difficulty is that the monotonicity condition gives only the sign of one inner product, $(z_k, \xi_k) \ge 0$ , while the norm of $z_{k+1}$ depends on both $(z_k,\xi_k)$ and $\|z_k\|$ ; the constants $\beta$ and $r$ are exactly those for which the resulting quadratic estimate closes. For the volume identity, the main technical step is computing $\det R_\beta(\xi) = \beta$ and the Lebesgue measure of a linear image of a ball in EuclideanSpace.

Formalization scope

$E_n$ is EuclideanSpace ℝ (Fin n); matrices act through Matrix.toEuclideanLin; $B_k^*$ is the transpose. $R_\alpha(\xi)$ is the matrix $I + (\alpha-1)\xi\xi^T$ (the book's property 10); for unit $\xi$ this is the operator of the book's definition.
The algorithm is the definition ellipsoidMethod g R x₀ : ℕ → EllState n, with state $(x_k, B_k, h_k)$ , $B_0 = I$ , $h_0 = R/(n+1)$ . If $g(x_k) = 0$ the state is repeated from then on (the book stops); the normalization in (3.57) is only performed when $g(x_k) \ne 0$ . $A_k$ is the matrix inverse of $B_k$ , which is nonsingular.
Theorem 3.14 is stated for $n > 1$ , $R > 0$ , $x^*$ with $\|x_0 - x^*\| \le R$ and $(g(x), x - x^*) \ge 0$ for all $x$ , for every $k$ . The book's additional assumption that $g(x) \ne 0$ for $x \ne x^*$ is not used by its proof and is omitted. The iterates are those of the recursion; a statement about an arbitrary ellipsoid containing $x^*$ , or about an arbitrary invertible matrix in place of $B_k$ , would not be this theorem and is ruled out by the definitions.
Volumes are Lebesgue measure in ℝ≥0∞. The ellipsoid is given for an arbitrary center (the book writes $x^*$ in one place and $x_k$ in another; the volume is the same). The volume formula requires the first $k$ iterations to have been performed ( $g(x_j) \ne 0$ , $j < k$ ); the ratio requires iteration $k+1$ to be performed.
The printed chain on p. 87 has misprints (exponents $2$ and $1$ on $n/\sqrt{n^2-1}$ where $n$ is meant, and $(n-1)/(n+1)$ for $\sqrt{(n-1)/(n+1)}$ ); the statement follows the value of $q_n$ given on p. 88. The estimate for $x \notin S(x_0,R)$ before (3.62) has a sign misprint; only the conclusion is stated.
Needed infrastructure: determinant of a rank-one perturbation of the identity ( $\det(I + c\,\xi\xi^T) = 1 + c\|\xi\|^2$ , available in Mathlib as the matrix determinant lemma), the measure of a linear image (MeasureTheory.Measure.addHaar_image_linearMap), and elementary real inequalities for $q_n < 1$ . The space-dilation lemmas are reusable in the other Shor missions on SDG methods and the $r$ -algorithm.

Selected references

N. Z. Shor, Minimization Methods for Non-Differentiable Functions, Springer Series in Computational Mathematics 3, Springer, 1985, §3.8. https://doi.org/10.1007/978-3-642-82118-9
D. B. Yudin and A. S. Nemirovskii, Informational complexity and efficient methods for the solution of convex extremal problems, Ekonomika i Matematicheskie Metody 12 (1976), 357–369 (English translation: Matekon 13 (1977), 25–45).
N. Z. Shor, Cut-off method with space extension in convex programming problems, Cybernetics 13 (1977), 94–96.
L. G. Khachiyan, A polynomial algorithm in linear programming, Soviet Mathematics Doklady 20 (1979), 191–194.
R. G. Bland, D. Goldfarb and M. J. Todd, The ellipsoid method: a survey, Operations Research 29 (1981), 1039–1091. https://doi.org/10.1287/opre.29.6.1039
M. Grötschel, L. Lovász and A. Schrijver, Geometric Algorithms and Combinatorial Optimization, Springer, 1988. https://doi.org/10.1007/978-3-642-97881-4

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Convex Optimization Operations Research Optimization·Captain: mikedeng1

Minimization Methods for Non-Differentiable Functions III: Convergence of the Normalized Subgradient Method with Divergent-Series StepsizesTextbook

Motivation

Many optimization problems of operations research have objectives that are convex but not differentiable: Lagrangian duals of integer and combinatorial programs, maxima of finitely many affine or smooth functions, penalty functions for systems of inequalities, and the value functions produced by decomposition. For such functions the gradient method and steepest descent fail. Constant steps cannot work because the subgradients need not tend to zero at a nondifferentiable minimum, and exact line search along the negative gradient can converge to a point that is not a minimizer (the example on pp. 22–23 of the source).

The subgradient method replaces the gradient by an arbitrary subgradient and gives up monotone decrease of the objective. Its convergence theory is the foundation of nondifferentiable optimization and of Lagrangian relaxation in integer programming.

Timeline. N. Z. Shor proposed the method with normalized steps in 1962 (Kiev). Yu. M. Ermoliev proved convergence in finite dimensions with divergent-series stepsizes (Kibernetika, 1966), and B. T. Polyak proved it for constrained problems in Hilbert space (Doklady Akad. Nauk SSSR, 1967). Held, Wolfe and Crowder (Mathematical Programming, 1974) brought the method to large combinatorial problems through Lagrangian relaxation. This mission formalizes the exposition of Section 2.1–2.2 of Shor's monograph (Springer, 1985), which gives self-contained proofs of these results.

Setting

Let $E_n$ be the $n$ -dimensional Euclidean space with inner product $(x, y)$ and norm $\|x\|$ . Let $f : E_n \to \mathbb{R}$ be a convex function finite everywhere. A vector $g$ is a subgradient of $f$ at $x_0$ if

f(x) - f(x_0) \ge (g, x - x_0) \quad \text{for all } x \in E_n.

Every convex $f$ has at least one subgradient at every point. Let $M^* = \{x : f(x) \le f(y) \ \forall y\}$ be the set of minimum points and, when it is nonempty, $f^* = \min f$ .

A subgradient selection $g_f$ assigns to each $x$ some subgradient $g_f(x)$ of $f$ at $x$ . No particular choice is made: every result holds for every selection. Given stepsizes $h_1, h_2, \dots > 0$ and a starting point $x_0$ , the normalized subgradient method is

x_{k+1} = x_k - h_{k+1}\, \frac{g_f(x_k)}{\|g_f(x_k)\|}, \qquad k = 0, 1, \dots \tag{2.4}

If $g_f(x_k) = 0$ , then $x_k$ is a minimizer and the computation stops. The unnormalized method is $x_{k+1} = x_k - h_{k+1} g_f(x_k)$ (2.5), and the method with restarts takes that step when $h_{k+1}\|g_f(x_k)\| \le c$ and returns to $x_0$ otherwise.

Formalization targets

Goal: Theorem 2.2 (p. 25)

If $M^*$ is nonempty and bounded, $h_k > 0$ , $h_k \to 0$ and $\sum_{k \ge 1} h_k = +\infty$ , then for every $x_0$ and every subgradient selection, the method (2.4) either reaches $M^*$ at some index $\bar k$ or

\lim_{k \to \infty} \min_{y \in M^*} \|x_k - y\| = 0, \qquad \lim_{k \to \infty} f(x_k) = f^*.

Milestones

Eq. (2.3), the one-step inequality $\|x_{k+1} - x^*\|^2 \le \|x_k - x^*\|^2 + h^2 - 2h\,\rho(x^*, U_k)$ , where $U_k = \{x : f(x) = f(x_k)\}$ .
Theorem 2.1: with constant step length $h$ , some level surface $\{f = f(x_{k^*})\}$ passes within $h(1+\varepsilon)/2$ of any $x^* \in M^*$ .
Corollaries 1 and 2: a suitable constant step length yields a subsequence with $f(x_{k_i}) - f^* < \delta$ . If $M^*$ contains a ball of radius $r > h/2$ , the method terminates in $M^*$ .
Theorem 2.5: if $M^*$ contains a ball of radius $r$ , $\sum h_k = \infty$ and $\limsup h_k < 2r$ , then (2.4) terminates in $M^*$ .
Theorem 2.3: for the unnormalized method (2.5), bounded subgradients along the trajectory imply convergence, and unbounded subgradients rule it out.
Theorem 2.4: the method with restarts converges for every $c > 0$ .

Significance

Theorem 2.2 is the basic convergence guarantee for first-order methods on general nonsmooth convex functions. It needs no Lipschitz constant, no bound on the subgradients and no smoothness: normalizing the step makes the step length independent of the size of the subgradient. The divergent-series rule $h_k \to 0$ , $\sum h_k = \infty$ is the standard stepsize condition of stochastic approximation and of Lagrangian relaxation codes. Theorems 2.3–2.5 mark its boundaries. The unnormalized method needs bounded subgradients (Theorem 2.3), restarts remove that need (Theorem 2.4), and a solution set with nonempty interior gives finite termination (Theorem 2.5). The last result is the basis of the finite methods for systems of convex inequalities and for the dual of an assignment problem with a unique solution (pp. 28–29).

All of these results are classical and proved in the source. None of them is formalized in Lean's Mathlib. The platform has neighbouring results that are not the same statements: Poljak's divergent-series theorem for concave piecewise-linear maximization (in Validation of Subgradient Optimization I), and rate bounds for Lipschitz objectives (First-Order and Stochastic Optimization Methods for ML II, Understanding Machine Learning X). This mission adds the general convex case with normalized steps, the dichotomy for unnormalized steps, and finite termination.

Difficulty

The standard rate analysis of the subgradient method bounds $\|x_{k+1} - x^*\|^2 - \|x_k - x^*\|^2$ by $-2h_{k+1}(f(x_k) - f^*)/\|g_f(x_k)\| + h_{k+1}^2$ . It then needs a uniform bound on $\|g_f(x_k)\|$ , which is exactly what is not assumed here. Nothing a priori keeps the iterates in a bounded set, and the subgradients of a general convex function (for instance $f(x) = x^4$ , the source's example on p. 26) grow without bound away from $M^*$ ; with unnormalized steps this makes the method diverge. Even with normalized steps, the distance to a minimizer decreases only outside a neighbourhood of $M^*$ whose size is of the order of the current step, so a monotone decrease argument gives at best a subsequence with small function values. Convergence of the whole sequence $\min_{y \in M^*}\|x_k - y\|$ to zero is a stronger statement, and boundedness of $M^*$ is essential to it.

Formalization scope

$E_n$ is EuclideanSpace ℝ (Fin n); $f$ is real-valued (finite everywhere) with ConvexOn ℝ Set.univ f.
The subgradient selection g is arbitrary, with the hypothesis ∀ x, IsSubgradient f x (g x). The starting point is arbitrary.
The iterations are defined recursively (normalizedIter, plainIter, resetIter); the stepsize sequence is h : ℕ → ℝ with h (k+1) used at step $k$ . In (2.4), a zero subgradient is handled by an explicit branch that repeats the current iterate (which is then in $M^*$ ). No statement relies on Lean's convention $x/0 = 0$ .
$M^*$ is required to be nonempty wherever the book writes $\min_{y \in M^*}$ or $f^* = \min f$ . $\min_{y \in M^*}\|x_k - y\|$ is Metric.infDist, and $f^*$ is ⨅ y, f y.
$\sum_{k \ge 1} h_k = +\infty$ is Tendsto (fun N => ∑ k ∈ Finset.range N, h (k+1)) atTop atTop. $\limsup h_k < 2r$ is "for some $q < 2r$ , eventually $h_k \le q$ ", so it cannot hold vacuously for an unbounded sequence.
Corollary 1's step length $h_\delta$ is quantified before the selection and the starting point: it depends only on $f$ and $\delta$ .
Theorem 2.3 is stated as two implications, (bounded subgradients ⇒ convergence) and (unbounded ⇒ no convergence), not as a disjunction that one case could satisfy trivially.
A formalization of Theorem 2.2 that assumes bounded subgradients, a Lipschitz $f$ , or a specific subgradient choice (such as the minimal-norm one) proves a different and weaker theorem, and does not close the goal.

Needed infrastructure: continuity of convex functions on $E_n$ (in Mathlib), compactness of sublevel sets when $M^*$ is bounded, and the geometry of level surfaces relative to supporting hyperplanes. The one-step inequality (2.3) and the level-set compactness lemma are reusable by the later missions of this series (linear rate, Polyak's stepsize, stochastic subgradient). Contributions that prove Eq. (2.3) or Theorem 2.1 first are welcome.

Selected references

N. Z. Shor, Minimization Methods for Non-Differentiable Functions, Springer Series in Computational Mathematics 3, Springer, 1985, Chapter 2, pp. 22–30. https://doi.org/10.1007/978-3-642-82118-9
B. T. Polyak, A general method for solving extremal problems, Doklady Akademii Nauk SSSR 174 (1967), 33–36 (the source's reference [64]).
Yu. M. Ermoliev, Methods for solving nonlinear extremal problems, Kibernetika (Kiev), no. 4 (1966), 1–17 (the source's reference [24]).
M. Held, P. Wolfe, H. P. Crowder, Validation of subgradient optimization, Mathematical Programming 6 (1974), 62–88. https://doi.org/10.1007/BF01580223

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Analysis Convex Optimization Optimization·Captain: mikedeng1

Minimization Methods for Non-Differentiable Functions II: Convex Functions Are Almost Differentiable and Their Almost-Gradients Are SubgradientsTextbook

Motivation

Gradient methods assume a continuous gradient; subgradient methods assume convexity. Many objective functions met in practice satisfy neither. Shor's example is economic planning, where components of the objective are piecewise-smooth, not necessarily convex functions of a parameter describing the productivity of a unit, and where minimax formulations produce kinks as a rule (Shor, Minimization Methods for Non-Differentiable Functions, Springer 1985, §1.4, p. 17, DOI 10.1007/978-3-642-82118-9). Such problems need a class of functions wide enough to contain piecewise-smooth and minimax functions and narrow enough to carry a usable replacement for the gradient.

Shor's answer is the class of almost differentiable functions, introduced in his 1972 work and presented in §1.4 of the book, together with the almost-gradient, a limit of gradients taken at nearby points of differentiability. The capstone of the section, Theorem 1.15, connects this class with convex analysis: every convex function on $E_n$ is almost differentiable, and its almost-gradients are subgradients. This is what lets the later chapters treat convex minimization and almost-differentiable minimization with one set of tools. Clarke's generalized gradient of locally Lipschitz functions (Clarke 1975) is the closest relative; Shor's definition differs in requiring the gradient to be continuous on its domain (p. 19).

Setting

Let $E_n$ denote $n$ -dimensional Euclidean space with inner product $(x, y)$ and norm $\|x\|$ . For $f : E_n \to \mathbb{R}$ , write $M = \{x \in E_n : f \text{ is differentiable at } x\}$ and $\nabla f(x)$ for the gradient at $x \in M$ .

A function $f$ is almost differentiable if

on every bounded set $S$ it is Lipschitz: $|f(x) - f(y)| \le L_S \|x - y\|$ for $x, y \in S$ , with a constant $L_S$ depending on $S$ ;
it is differentiable at Lebesgue-almost every point of $E_n$ ;
the map $x \mapsto \nabla f(x)$ , restricted to $M$ , is continuous.

An almost-gradient of $f$ at $x_0$ is a vector $g$ that is an accumulation point of a sequence $\nabla f(x_1), \nabla f(x_2), \dots$ with $x_k \in M$ and $x_k \to x_0$ . The set of almost-gradients is $G(x_0)$ . A generalized almost-gradient is a point of the closure of the convex hull of $G(x_0)$ .

A vector $g$ is a subgradient of $f$ at $x_0$ if $f(x) - f(x_0) \ge (g, x - x_0)$ for all $x \in E_n$ ; for convex $f$ the set of subgradients is the subdifferential $G_f(x_0)$ .

In Lean these objects are AlmostDifferentiable f, almostGradients f x₀ and IsSubgradient f x₀ g in the namespace ShorNonsmooth.AlmostDiff, with $E_n$ = EuclideanSpace ℝ (Fin n).

Formalization targets

Goal: Theorem 1.15 (p. 18)

For convex $f : E_n \to \mathbb{R}$ ,

f \text{ is almost differentiable} \quad\text{and}\quad G(x_0) \subseteq G_f(x_0) \ \text{ for every } x_0 \in E_n .

The printed statement says the almost-gradients "coincide with" the subgradients. As a set equality this is false (for $f(x) = |x|$ on $E_1$ , $G(0) = \{-1, 1\}$ while $G_f(0) = [-1, 1]$ ), and the book's proof establishes the inclusion. The goal is the inclusion.

Milestones

Proof of Theorem 1.15, display (p. 18). For convex $f$ differentiable at $x_k$ : $f(x) - f(x_k) \ge (\nabla f(x_k), x - x_k)$ for all $x$ .
Proof of Theorem 1.15 (p. 18). For convex $f$ and bounded $S$ there is $C$ with $|f'_v(x)| \le C\|v\|$ for all $x \in S$ and all $v$ , the one-sided directional derivatives existing.
Proof of Theorem 1.15 (p. 18). A convex $f$ is differentiable almost everywhere and $\nabla f$ is continuous on $M$ .
Theorem 1.14 (p. 18). For almost differentiable $f$ , $G(x)$ is nonempty, bounded and closed at every $x$ .
p. 19. For almost differentiable $f$ , $\overline{\operatorname{conv}}\, G(x)$ is convex, bounded and closed.

Significance

The result. Theorem 1.15 embeds convex functions into the almost differentiable class and identifies each almost-gradient of a convex function as a subgradient. Consequently any method that only needs almost-gradients (limits of gradients at nearby differentiable points, which is what a numerical procedure can actually compute) produces valid subgradients when applied to a convex function. Theorem 1.14 supplies the compactness that makes $G(x)$ usable as a set-valued substitute for the gradient.

Formalizing it. All results here are classical and proved on paper. Mathlib contains Rademacher's theorem for Lipschitz functions (LipschitzWith.ae_differentiableAt) and local Lipschitz continuity of convex functions on open sets; it does not contain the continuity of the gradient of a convex function on its domain of differentiability, nor any notion of almost-gradient. The mission produces those, together with a formal record that the printed "coincide" is an inclusion. A formal proof of the true equality $G_f(x_0) = \overline{\operatorname{conv}}\, G(x_0)$ would be a welcome further contribution.

Difficulty

Two parts of the goal carry real content. The first is condition (c): the gradient of a convex function, restricted to the set where it exists, is continuous. The book cites this from the literature; it is not a consequence of Rademacher's theorem, which gives differentiability almost everywhere and says nothing about how gradients at nearby points relate. The second is condition (b) in a form Lean accepts: Rademacher's theorem in Mathlib is stated for globally Lipschitz functions, while a convex function on $E_n$ is only Lipschitz on bounded sets, so the almost-everywhere statement has to be assembled from local pieces. The passage from gradients to subgradients in the second conjunct is comparatively routine.

The analogous closure properties claimed on the same pages for sums, products and maxima of almost differentiable functions (Theorems 1.16 and 1.17) are false as printed and are not targets; see the formalization scope.

Formalization scope

$E_n$ is EuclideanSpace ℝ (Fin n); $n = 0$ is allowed and harmless. Functions are total and real-valued; convexity is ConvexOn ℝ Set.univ f.
The gradient is Mathlib's gradient f x; condition (c) is ContinuousOn (gradient f) {x | DifferentiableAt ℝ f x}, continuity of the restriction in the subspace topology.
Condition (a) is quantified over every bounded set with a set-dependent constant: ∀ S, Bornology.IsBounded S → ∃ L, LipschitzOnWith L f S. Condition (b) is ∀ᵐ x ∂volume, DifferentiableAt ℝ f x.
An almost-gradient is a cluster point (MapClusterPt) of the gradient sequence, not its limit; the points $x_k$ may equal $x_0$ .
Subgradients are taken relative to the whole space, the domain of every function in this mission.
The goal states the inclusion $G(x_0) \subseteq G_f(x_0)$ proved in the book. Stating the printed set equality would make the goal false; weakening the first conjunct to "locally Lipschitz and differentiable almost everywhere" would drop condition (c) and with it the substance of the theorem. Neither is acceptable.
Theorem 1.16 (sums, differences, products) and Theorem 1.17 (maxima) are omitted: both are false for the class as defined. With $h(x) = x^2 \sin(1/x)$ , the functions $h + |x|$ and $-|x|$ are almost differentiable but their sum $h$ is differentiable everywhere with a derivative discontinuous at $0$ ; and $\max(h - |x|,\, h - |x| + 2x) = h + x$ . Theorem 1.18 (Mifflin's superposition theorem for semismooth functions) is cited from the literature without proof and rests on Clarke's generalized gradient; it is outside this mission.

Infrastructure that is reusable beyond this mission: continuity of the gradient of a convex function on its domain; Rademacher's theorem for locally Lipschitz functions on $E_n$ ; the almost-gradient set and its compactness. Contributions that prove these as standalone lemmas are welcome.

Selected references

N. Z. Shor, Minimization Methods for Non-Differentiable Functions, Springer Series in Computational Mathematics 3, Springer, 1985. https://doi.org/10.1007/978-3-642-82118-9
R. T. Rockafellar, Convex Analysis, Princeton University Press, 1970, Theorem 25.5 (continuity of the gradient of a convex function). https://doi.org/10.1515/9781400873173
F. H. Clarke, Generalized gradients and applications, Transactions of the American Mathematical Society 205 (1975), 247–262. https://doi.org/10.1090/S0002-9947-1975-0367131-6
H. Rademacher, Über partielle und totale Differenzierbarkeit von Funktionen mehrerer Variabeln und über die Transformation der Doppelintegrale, Mathematische Annalen 79 (1919), 340–359. https://doi.org/10.1007/BF01498415

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Markov Chain Operations Research Probability+1·Captain: mikedeng1

Stochastic Dynamic Programming and the Control of Queueing Systems XIII: Lyapunov Criteria and z Standard Markov Chains with CostsTextbook

Motivation

Average cost control of queues rests on a small amount of Markov chain theory: when does a chain with costs have a well defined long-run average cost, and how can that be checked for a concrete model with an unbounded state space? Appendix C of L. I. Sennott, Stochastic Dynamic Programming and the Control of Queueing Systems (Wiley, 1999, doi:10.1002/9780470317037) collects this material for countable state spaces and packages it in one hypothesis, the $z$ standard chain. Chapters 7–10 of the book verify this hypothesis for the Markov chains induced by stationary policies in admission, routing and service-rate control models, and use its consequences to prove existence of average cost optimal policies.

The tools are Lyapunov functions in the sense of Foster (1953): a nonnegative function on the states whose expected one-step change is negative away from a finite set. Foster's criterion for positive recurrence, and its refinements bounding expected first passage times and costs, are the standard way to verify stability of queueing networks (Meyn and Tweedie, Markov Chains and Stochastic Stability, 1993/2009).

Setting

A Markov chain $\Gamma$ on a countable set $S$ is given by transition probabilities $P_{ij}\ge 0$ with $\sum_j P_{ij}=1$ . $X_t$ is the state at time $t$ and $P^{(t)}_{ij}$ the $t$ -step transition probability ( $P^{(0)}_{ij}=\delta_{ij}$ ). State $i$ leads to $j$ if $P^{(t)}_{ij}>0$ for some $t\ge0$ ; states that lead to each other communicate, which partitions $S$ into communicating classes.

For a nonempty $G\subseteq S$ the first passage time from $i$ is $T_{iG}=\min\{t\ge1: X_t\in G\}$ given $X_0=i$ , and $m_{iG}=E[T_{iG}]\in[0,\infty]$ ; $m_{ij}$ is the case $G=\{j\}$ and $m_{ii}$ the expected return time. The taboo probability $_G P^{(t)}_{ik}$ is the probability of going from $i$ to $k$ in $t$ steps without visiting $G$ at the intermediate times, and $_G u_{ik}$ is the expected number of visits to $k$ at times $0\le t<T_{iG}$ . A state is transient if $P(T_{ii}<\infty)<1$ and positive recurrent if $m_{ii}<\infty$ ; a positive recurrent class is a communicating class of positive recurrent states. The steady state probability is $\pi_j=(m_{jj})^{-1}$ (zero when $m_{jj}=\infty$ ).

Each state carries a finite cost $C(i)\ge0$ . The expected average cost over $[0,n-1]$ from $i$ is

J^{(n)}_i=\frac1n\,E\Big[\sum_{t=0}^{n-1}C(X_t)\,\Big|\,X_0=i\Big]=\frac1n\sum_{t=0}^{n-1}\sum_j P^{(t)}_{ij}C(j),

$c_{iG}$ is the expected cost $E[\sum_{t=0}^{T_{iG}-1}C(X_t)\mid X_0=i]$ of a first passage (defined when $m_{iG}<\infty$ ), and $J_R=\sum_{j\in R}\pi_jC(j)$ is the average cost on a positive recurrent class $R$ . The chain is $z$ standard (Definition C.2.5) if for a distinguished state $z$

m_{iz}<\infty\quad\text{and}\quad c_{iz}<\infty\qquad\text{for all } i\in S.

Formalization targets

Goal: Proposition C.2.6

If $\Gamma$ is $z$ standard, then $S$ is the union of a positive recurrent class $R\ni z$ and a set of transient states, $J_R<\infty$ , and

\lim_{n\to\infty}J^{(n)}_i=J_R\qquad\text{for every } i\in S.

The statement fixes no constants: it asserts that the average cost exists, is finite, and does not depend on the initial state.

Milestones

Proposition C.1.2: $\pi$ is the unique stationary distribution of a positive recurrent class, and $\pi_j=e_{ij}/m_{ii}=\pi_ie_{ij}$ .
Proposition C.1.4: the first-step equations (C.2)–(C.4) for taboo probabilities, visit counts and $m_{iG}$ ; $\sum_{i\in G}\pi_im_{iG}=1$ for $G$ inside a positive recurrent class; $m_{ij}<\infty$ within such a class.
Proposition C.1.5: if $\sum_jP_{ij}[y(j)-y(i)]\le-\epsilon$ off $G$ , then $m_{iG}\le y(i)/\epsilon$ .
Corollary C.1.6: the same with $G=\{z\}$ and $\sum_jP_{zj}y(j)<\infty$ makes $z$ positive recurrent.
Proposition C.2.1: on a positive recurrent class, $J^{(n)}_i\to J_R=c_{ii}/m_{ii}$ .
Proposition C.2.2: $c_{iG}=\sum_kC(k)\,{}_Gu_{ik}$ , the first-step equation (C.13), and $J_R=\sum_{i\in G}\pi_ic_{iG}$ .
Proposition C.2.3 and Corollary C.2.4: the cost drift condition $\sum_jP_{ij}[r(j)-r(i)]\le-C(i)$ off a finite set bounds $c_{iG}\le r(i)+Fm_{iG}$ , and gives $c_{zz}<\infty$ .
Remark C.2.7: the hypotheses of C.1.6 and C.2.4 together imply the chain is $z$ standard; so do irreducibility, positive recurrence and finite average cost.

Significance

Proposition C.2.6 is what makes the $z$ standard hypothesis useful: an average cost criterion that is a genuine limit, finite, and independent of the initial state, even for chains with transient states and unbounded state spaces. Every average cost optimality result of the book that works with a stationary policy's induced chain (the (SEN) and (BOR) assumption sets, the approximating-sequence method, the continuous-time chapter) calls on this proposition or on the Lyapunov criteria of Remark C.2.7 to establish its hypotheses for queueing models.

All results of the mission are classical and proved in the literature; parts are stated in the book without proof and referred to Chung (1967), Grassmann et al. (1985) and renewal theory. None of them has been machine-checked in this form as far as the platform and Mathlib show: Mathlib has kernels and Ionescu-Tulcea trajectories but no countable-state Markov chain classification, no first passage calculus, and no Foster–Lyapunov criterion. Existing platform results on countable chains (the Levin–Peres–Wilmer series) treat irreducible chains without costs. A complete development here produces a reusable library of first passage identities, Foster–Lyapunov bounds for times and costs, and average cost limits on reducible chains.

Difficulty

The Lyapunov bounds (C.1.5, C.2.3) are telescoping arguments, but they require a clean handling of truncated passages and of sums that may be infinite: (C.7) is an inequality between possibly divergent series, and the step "iterate $n$ times and let $n\to\infty$ " must be made rigorous for $[0,\infty]$ -valued expectations.

The central difficulty is part (iii) of the goal for transient initial states. On the class $R$ , the limit of $J^{(n)}_i$ is a renewal reward theorem over successive returns to $z$ ; from a transient state the first cycle has a different law, so a delayed renewal reward argument is needed, and it has to cover the case where costs are unbounded. The obvious approach, bounding $J^{(n)}_i$ between $J_R$ and the average over the first $n$ steps of the chain started in $z$ , fails because $P^{(t)}_{ij}$ need not converge (periodic classes) and because finite $c_{iz}$ does not bound individual cost terms. Proposition C.1.2's uniqueness and the Kac-type identity of C.1.4(iv) likewise need the full cycle decomposition of a positive recurrent class.

Formalization scope

The chain is a structure MC S with P : S → S → ℝ≥0∞ and ∑' j, P i j = 1, over a countable type S; costs are C : S → ℝ≥0. Probabilities and expectations are ℝ≥0∞-valued sums over finite paths Fin (t+1) → S, so every quantity is defined without summability side conditions and may be $\infty$ . The first passage time is $T_{iG}\ge1$ ; $m_{iG}$ is the expectation of $T_{iG}$ from its law (and $\infty$ when $P(T_{iG}<\infty)<1$ ), not defined by the recursion (C.4), so that (C.4) is a theorem. $_Gu_{ik}$ counts visits at times $0\le t<T_{iG}$ . $c_{iG}$ is computed over first passage paths and is used only when $m_{iG}<\infty$ , as in the book. $\pi_j$ is $(m_{jj})^{-1}$ , which the book states equals the Cesàro limit $\lim_nQ^{(n)}_{jj}$ . $J^{(n)}_i$ is meaningful for $n\ge1$ , and limits are taken in $[0,\infty]$ . The drift conditions $\sum_jP_{ij}[y(j)-y(i)]\le-\epsilon$ and $\sum_jP_{ij}[r(j)-r(i)]\le-C(i)$ are written in the equivalent additive form $\sum_jP_{ij}y(j)+\epsilon\le y(i)$ , which is equivalent for finite $y$ and makes the case $\sum_jP_{ij}y(j)=\infty$ fail, as it does in the book.

A trivializing formalization, such as defining $m_{iG}$ or $c_{iG}$ by the equations (C.4) or (C.13), defining $J_R$ as the limit of $J^{(n)}_i$ , or allowing a $z$ standard chain whose return time or return cost to $z$ is infinite, is ruled out: $z$ standard requires $m_{iz}<\infty$ and $c_{iz}<\infty$ for every $i$ including $z$ , and each quantity is defined from path probabilities.

Needed infrastructure: path-sum manipulation in $[0,\infty]$ (first-step and last-step decompositions), the ratio limit / renewal reward theorem for a positive recurrent class, and the delayed version for transient starts. The first passage calculus and the Lyapunov bounds are reusable by the book's other chapters on average cost, which state the $z$ standard property for policy-induced chains. Contributions of lemmas on path sums and of an independent renewal reward library are welcome.

Selected references

L. I. Sennott, Stochastic Dynamic Programming and the Control of Queueing Systems, Wiley, 1999, Appendix C, pp. 292–302. doi:10.1002/9780470317037
K. L. Chung, Markov Chains with Stationary Transition Probabilities, 2nd ed., Springer, 1967. doi:10.1007/978-3-642-62015-7
F. G. Foster, On the stochastic matrices associated with certain queuing processes, Annals of Mathematical Statistics 24 (1953), 355–360. doi:10.1214/aoms/1177728976
S. P. Meyn and R. L. Tweedie, Markov Chains and Stochastic Stability, 2nd ed., Cambridge University Press, 2009. doi:10.1017/CBO9780511626630
D. P. Heyman and M. J. Sobel, Stochastic Models in Operations Research, Vol. I, McGraw-Hill, 1982.

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Analysis Probability·Captain: mikedeng1

Stochastic Dynamic Programming and the Control of Queueing Systems XII: A Tauberian Theorem Linking Abel and Cesàro Means of Nonnegative SeriesTextbook

Motivation

In the theory of Markov decision chains two cost criteria dominate: the discounted cost, in which a cost incurred at time $n$ is weighted by $\alpha^n$ for a discount factor $\alpha \in (0,1)$ , and the average cost, the long-run cost per unit time. A standard route to average cost optimal policies is to solve the discounted problem and let $\alpha \to 1^-$ . That route needs a precise link between the two criteria for a single sequence of expected costs $u_0, u_1, u_2, \dots \ge 0$ : the discounted cost multiplied by $1-\alpha$ on one side, the running average on the other. Appendix A.4 of Sennott, Stochastic Dynamic Programming and the Control of Queueing Systems (Wiley, 1999) supplies this link as Theorem A.4.2, and the book invokes it whenever it passes from discounted to average cost (for instance in Chapter 6, where it is applied to $u_n = E_\theta[C(X_n, A_n)]$ ).

The result belongs to classical summability theory.

Abelian direction. Convergence of averages implies convergence of the power-series (Abel) means: a consequence of Abel's and Frobenius's theorems on power series (19th century).
Tauber (1897). The first converse, under a growth condition on the terms.
Hardy and Littlewood (1914). The converse for Cesàro means of nonnegative terms, the "Hardy–Littlewood Tauberian theorem".
Karamata (1930). A short proof of that converse by polynomial approximation, reproduced in Titchmarsh, The Theory of Functions (1939, pp. 227–229). The book follows this proof.
Widder (1941). The continuous-time (Laplace transform) version.
Sennott (1986b). A proof of the discrete statement, cited by the book as its source for Theorem A.4.2.

Setting

Let $u_0, u_1, u_2, \dots$ be nonnegative terms $u_n \in [0, \infty]$ with $u_0 < \infty$ . For $\alpha \in [0, \infty)$ the power series

U(\alpha) = \sum_{n=0}^{\infty} \alpha^n u_n \in [0, \infty]

always has a sum, possibly $+\infty$ , because its partial sums increase. Its radius of convergence is $R = (\limsup_{n} u_n^{1/n})^{-1} \in [0,\infty]$ . The partial sums are $w_n = \sum_{k=0}^{n-1} u_k$ for $n \ge 1$ . Two ways of averaging the sequence are compared:

the Cesàro means $w_n / n$ , as $n \to \infty$ ;
the Abel means $(1-\alpha) U(\alpha)$ , as $\alpha \to 1^-$ (from below).

For the constant sequence $u_n \equiv B$ both equal $B$ . In the Lean development these are U, radius, w, cesaroMean and abelMean in the namespace SennottDP.Tauberian, all valued in ℝ≥0∞. The proof also uses the function $r$ of Fig. A.1, equal to $0$ on $(0, e^{-1})$ and to $1/x$ on $[e^{-1}, 1)$ .

Formalization targets

Goal: Theorem A.4.2

\liminf_{n\to\infty} \frac{w_n}{n} \le \liminf_{\alpha\to 1^-} (1-\alpha)U(\alpha) \le \limsup_{\alpha\to 1^-} (1-\alpha)U(\alpha) \le \limsup_{n\to\infty} \frac{w_n}{n}, \tag{A.28}

and the following are equivalent: (i) all terms in (A.28) are equal and finite; (ii) $\lim_n w_n/n$ exists and is finite; (iii) $\lim_{\alpha \to 1^-} (1-\alpha)U(\alpha)$ exists and is finite.

No growth or boundedness condition on $u_n$ is assumed: nonnegativity is the only Tauberian condition.

Milestones

Remark A.3.3: the geometric series. $R = 1$ , $U(\alpha) = B/(1-\alpha)$ and $B\sum_n n\alpha^n = B\alpha/(1-\alpha)^2$ .
Remark A.3.1: inside the radius of convergence, $U$ is differentiable term by term, and the derived series has the same radius.
Eq. (A.31): $\big(\sum \alpha^n\big) U(\alpha) = \sum_n \alpha^n w_{n+1} = U(\alpha)/(1-\alpha)$ .
Eq. (A.28): the Abelian inequalities on their own.
Eq. (A.26): $\int_0^1 r = 1$ .
Lemma A.4.1: continuous $s^* \le r \le s$ with integrals within $\varepsilon$ of $1$ .
Eqs. (A.35)–(A.36): if the Abel means tend to $L < \infty$ , then $(1-\alpha)\sum_n \alpha^n u_n f(\alpha^n) \to L\int_0^1 f$ for $f(x) = x^k$ .
The same statement for every $f$ continuous on $[0,1]$ .
The same statement for $f = r$ .
Example A.5.1: a $0/1$ block sequence with $\liminf w_n/n = 1/2$ and $\limsup w_n/n = 2/3$ (Choice One) or $1$ (Choice Two), for which the middle inequality of (A.28) is strict.

Significance

The result itself. Theorem A.4.2 lets every statement about $\lim_{\alpha\to1^-}(1-\alpha)V_\alpha$ for a discounted value be read as a statement about long-run average cost, and conversely. The strict cases of Example A.5.1 show why the book must work with $\liminf$ and $\limsup$ rather than limits: average costs of policies need not exist as limits, even for bounded costs.

Formalizing it. Mathlib contains Abel's theorem for convergent series (Mathlib/Analysis/Complex/AbelLimit.lean) and Cesàro convergence of convergent sequences, but no Tauberian theorem. The pinned checkout has no file mentioning "Tauberian", and neither does the platform. The mathematics has been proved since 1914. Formalizing it here adds:

a machine-checked Hardy–Littlewood–Karamata theorem for nonnegative $[0,\infty]$ -valued sequences;
the $\liminf$ / $\limsup$ comparison (A.28) with infinite values allowed;
a reusable component for the average cost chapters of this series.

Difficulty

The Abelian inequalities (A.28) follow from rearranging the series. The converse (iii) ⇒ (ii) does not: the first idea, recovering $w_n/n$ from $(1-\alpha)U(\alpha)$ at $\alpha = 1 - 1/n$ , fails, because knowing $U$ near $1$ controls only weighted averages of all the $u_k$ , not a sharp truncation. Some positivity argument is unavoidable, since the converse fails for signed sequences (for example $u_n = (-1)^n n$ ). The sharp truncation at $n \approx (-\ln\alpha)^{-1}$ corresponds to the discontinuous weight $r$ . Passing from polynomial weights to the jump function $r$ requires uniform approximation together with the nonnegativity of the $u_n$ at every step.

The infinite values add bookkeeping. A single $u_{n_0} = \infty$ makes every term of (A.28) infinite, and $R < 1$ forces $U(\alpha) = \infty$ on $(R,1)$ .

Formalization scope

Conventions the statements commit to:

Values. Terms $u : \mathbb{N} \to [0,\infty]$ (ℝ≥0∞) with $u_0 \ne \infty$ . $U$ , the Abel means, $w_n$ and the Cesàro means are ℝ≥0∞-valued, with ℝ≥0∞ sums (no summability side conditions).
Limits. $\liminf$ and $\limsup$ are those of the complete lattice $[0,\infty]$ , never real liminf. "Exists and is finite" means convergence in $[0,\infty]$ to some $L \ne \infty$ .
Discount factor. $\alpha \in \mathbb{R}_{\ge 0}$ , and $\alpha \to 1^-$ is the filter 𝓝[<] 1.
Index $n = 0$ . $n \to \infty$ is atTop on $\mathbb{N}$ ; the value $w_0/0 = 0$ is irrelevant.
The equivalence is List.TFAE.
The function $r$ . Its value at the jump is $r(e^{-1}) = e$ (Fig. A.1).
Continuity. In Lemma A.4.1 and in the continuous-weight step, continuity is required on the closed interval $[0,1]$ , which is what the Weierstrass approximation step uses. The book writes " $(0,1)$ ".
Finite terms. Remark A.3.1 is stated for finite terms, which the book's discussion of the radius presumes.

A trivializing formalization is ruled out. Real-valued $\liminf$ / $\limsup$ would make (A.28) hold by junk values on unbounded sequences; hypotheses such as boundedness of $u_n$ would replace the theorem by an easier special case; and a definition of "exists and is finite" allowing $L = \infty$ would make (ii) hold for $u_n = n$ . None of these is used. As sanity checks: for $u_n \equiv 1$ all four terms of (A.28) equal $1$ , and for $u_n = n$ all equal $\infty$ and (i)–(iii) all fail.

A complete development needs:

Cauchy products of $[0,\infty]$ -valued power series;
the Weierstrass approximation theorem, which Mathlib has (polynomialFunctions.topologicalClosure);
interval integrals of step-like functions;
$\liminf$ / $\limsup$ manipulation along 𝓝[<] 1.

Reusable beyond this mission: the ℝ≥0∞ power-series toolkit and the Karamata argument for general continuous weights (milestone 8). Contributions welcome: proofs of any milestone, and alternative proofs of (iii) ⇒ (ii).

Selected references

L. I. Sennott, Stochastic Dynamic Programming and the Control of Queueing Systems, Wiley, 1999, Appendix A.3–A.5, pp. 279–287. https://doi.org/10.1002/9780470317037
E. C. Titchmarsh, The Theory of Functions, 2nd ed., Oxford University Press, 1939, pp. 227–229 (Karamata's proof).
J. Karamata, "Über die Hardy–Littlewoodschen Umkehrungen des Abelschen Stetigkeitssatzes", Mathematische Zeitschrift 32 (1930), 319–320.
G. H. Hardy and J. E. Littlewood, "Tauberian theorems concerning power series and Dirichlet's series whose coefficients are positive", Proc. London Math. Soc. (2) 13 (1914), 174–191.
D. V. Widder, The Laplace Transform, Princeton University Press, 1941.
L. I. Sennott, "A new condition for the existence of optimum stationary policies in average cost Markov decision processes — unbounded cost case", Proc. 25th IEEE Conference on Decision and Control, Athens, 1986, pp. 1719–1721 (cited by the book as Sennott 1986b).
T. M. Liggett and S. A. Lippman, "Short notes: Stochastic games with perfect information and time average payoff", SIAM Review 11 (1969), 604–607. https://doi.org/10.1137/1011093

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Analysis Probability·Captain: mikedeng1

Stochastic Dynamic Programming and the Control of Queueing Systems XI: The Generalized Dominated Convergence Theorem for Approximating DistributionsTextbook

Motivation

Linn Sennott's Stochastic Dynamic Programming and the Control of Queueing Systems (Wiley, 1999) studies Markov decision chains with countably many states, finite action sets and unbounded nonnegative costs. Its main computational device, the approximating sequence method (Definition 2.5.1, p. 28), replaces the countable state space $S$ by an increasing sequence of state spaces $S_N$ and the transition law out of each state by distributions $P_j(N)$ on $S_N$ that converge pointwise to the original law. Every convergence argument of that method has the same shape: a sum $\sum_{j\in S_N} P_j(N)\,u(j,N)$ , an expected cost or value under the $N$ -th approximating model, must be shown to converge to $\sum_{j\in S} P_j\,u(j)$ , or at least to satisfy a one-sided bound in the limit.

Appendix A, Sections A.1–A.2 (pp. 270–278), collects the analysis results used for this. The author proves them for sums over countable sets rather than general integrals, so they need no measure theory, and the proofs of the discounted-cost and average-cost approximation theorems (Chapters 4 and 8) cite them. This mission formalizes that appendix section as a self-contained unit.

Setting

Let $S$ be a countable set and $(P_j)_{j\in S}$ a probability distribution on $S$ : $P_j\ge 0$ and $\sum_j P_j=1$ . Values of functions are extended reals in $[-\infty,\infty]$ , with the convention $0\cdot\infty=0$ . For $u:S\to[-\infty,\infty]$ , the weighted sum is

\sum_{j\in S}P_j\,u(j)=\sum_{j\in S}P_j\,u^+(j)-\sum_{j\in S}P_j\,u^-(j),

which is well defined, in $(-\infty,\infty]$ , whenever the negative parts have finite weighted sum. In Lean this is wsum P u.

A family of approximating distributions for $(P_j)$ consists of

an increasing sequence of subsets $S_0\subseteq S_1\subseteq\cdots$ of $S$ with $\bigcup_N S_N=S$ ;
for each $N$ , a probability distribution $(P_j(N))_{j\in S_N}$ on $S_N$ ;
pointwise convergence $\lim_{N\to\infty}P_j(N)=P_j$ for every $j\in S$ .

Since the sets increase to $S$ , a fixed state $j$ lies in $S_N$ for every large $N$ , so $\liminf_N u(j,N)$ and $\lim_N u(j,N)$ make sense for a function $u(j,N)$ defined only for $j\in S_N$ . In Lean these three conditions are the structure ApproxDist P SN Q, with Q N j $=P_j(N)$ .

Formalization targets

Goal: Theorem A.2.6 (Generalized Dominated Convergence Theorem), p. 278

Under the approximating-distribution hypotheses, let $u(j,N)$ and $w(j,N)$ be finite with $|u(j,N)|\le w(j,N)$ for $j\in S_N$ , let $u(j,N)\to u(j)$ and $w(j,N)\to w(j)$ for every $j$ , and assume

\lim_{N\to\infty}\sum_{j\in S_N}P_j(N)\,w(j,N)=\sum_{j\in S}P_j\,w(j)<\infty.

Then

\lim_{N\to\infty}\sum_{j\in S_N}P_j(N)\,u(j,N)=\sum_{j\in S}P_j\,u(j).

Milestones, in the book's order

Proposition A.1.3: $\liminf$ commutes with a minimum over a finite nonempty set; $\lim$ does too when all limits exist.
Proposition A.1.5: $\liminf_N\sum_{j\in G}u(j,N)\ge\sum_{j\in G}\liminf_N u(j,N)$ for finite $G$ and $u>-\infty$ , when the right side has no indeterminate form.
Proposition A.1.7: the same inequality for countable sums of $[0,\infty]$ -valued terms.
Proposition A.1.8: the same, with the left sum over $S_N$ increasing to $S$ .
Proposition A.1.10: $\liminf u_n\le\liminf w_n/n\le\limsup w_n/n\le\limsup u_n$ for $w_n=\sum_{k<n}u_k$ .
Proposition A.2.1 (Fatou's lemma): $\liminf_N\sum_jP_j u(j,N)\ge\sum_jP_j\liminf_N u(j,N)$ for $u\ge -L$ .
Theorem A.2.3 (dominated convergence, with a convergent dominating sequence $w(j,N)$ ) and Corollary A.2.4 (fixed dominating function).
Proposition A.2.5 (generalized Fatou's lemma):

\liminf_{N\to\infty}\sum_{j\in S_N}P_j(N)\,u(j,N)\ \ge\ \sum_{j\in S}P_j\,\liminf_{N\to\infty}u(j,N)\qquad(u\ge -L\text{ on }S_N).

Corollary A.2.7: bounded convergence under approximating distributions.

Significance

The results. Proposition A.2.5 and Theorem A.2.6 are the two limit interchanges the approximating sequence method needs. The generalized Fatou inequality gives the lower bound on limits of value functions of the approximating models, and the generalized dominated convergence theorem identifies the limit of expected costs when a dominating function is available. Corollary A.2.7 gives the bounded case, and Proposition A.1.3 moves limits inside the minimization of an optimality equation. Proposition A.1.10 compares Cesàro averages with the sequence itself, which the average-cost chapters use.

Formalizing them. All statements are classical and proved in the book. A.1.7, A.2.1, A.2.3 and A.2.4 are, after translation, instances of Mathlib's Fatou lemma (MeasureTheory.lintegral_liminf_le) and dominated convergence theorem for a discrete measure. The mission states them in the book's discrete, extended-real form so later chapters can cite them by number. The versions in which the distribution and its support change with $N$ (A.2.5–A.2.7) are not in Mathlib in this form; the generalized dominated convergence theorem appears in the measure-theoretic literature (Royden, Real Analysis, Ch. 4) but has no Lean formalization. None of these statements has been formalized for this book.

Difficulty

For the generalized results, Mathlib's Fatou lemma and dominated convergence theorem do not apply directly: they fix one measure, while here both the weights $P_j(N)$ and the support $S_N$ change with $N$ . The dominating function $w(j,N)$ also changes with $N$ and is not integrable uniformly in $N$ ; only the convergence of its expectations is assumed. For small $N$ , $\sum_{j\in S_N}P_j(N)w(j,N)$ may even be infinite, and then $\sum_{j\in S_N} P_j(N)u(j,N)$ has no value.

A second difficulty is bookkeeping in the extended reals. Lower limits may be $\pm\infty$ , products use $0\cdot\infty=0$ , and a sum with a $+\infty$ term and a $-\infty$ term is undefined. The inequality of Proposition A.1.5 holds only when that indeterminate form is excluded. The bound $u\ge -L$ in A.2.1 and A.2.5 cannot be dropped: Examples A.1.9 and A.2.2 of the book show the inequalities fail without it.

Formalization scope

Types. $S$ is any type with [Countable S]. Probabilities are ℝ≥0∞-valued with ∑' j, P j = 1, and $P_j(N)$ is Q N j. Extended-real values are EReal, and nonnegative extended values (A.1.7, A.1.8) are ℝ≥0∞. Limits and lower and upper limits are along atTop in ℕ. The book's $N\ge N_0$ convention (Remark A.1.2) needs nothing extra, since only large $N$ matter.
Sums over $S_N$ are sums over $S$ of (SN N).indicator (Q N) times the function. Values of $u(j,N)$ and $P_j(N)$ for $j\notin S_N$ never enter any statement, and the hypotheses on $u$ ( $u\ge -L$ , $|u|\le w$ ) are imposed only on $S_N$ , as in the book.
Weighted sums are wsum, the difference of the weighted positive and negative parts, each in [0,∞]. When both are infinite the book's sum is undefined and wsum returns $-\infty$ . Every sum used in a hypothesis or conclusion is defined, except possibly for finitely many $N$ in limit statements, where it does not matter.
Limits of $u(j,N)$ and $w(j,N)$ are taken in EReal, since the book allows extended-real limits (p. 271). The hypotheses force them to be finite wherever $P_j>0$ .
Ruled out. Statements in which wsum returns its junk value $-\infty$ could make an inequality trivial. Every left-hand side here is a lower limit of sums whose negative parts are bounded, or a limit of eventually defined sums, so no statement holds only because of that junk value. Real-valued tsum with its junk value $0$ is not used anywhere.
Welcome contributions. A proof of A.1.7 from lintegral_liminf_le for the counting measure, and lemmas relating wsum to Mathlib's tsum and to integrals against Measure.sum (fun j => P j • dirac j), can be reused by the other missions of this series.

Selected references

L. I. Sennott, Stochastic Dynamic Programming and the Control of Queueing Systems, Wiley Series in Probability and Statistics, John Wiley & Sons, 1999, Appendix A, pp. 270–278. https://doi.org/10.1002/9780470317037
H. L. Royden, Real Analysis, 3rd ed., Macmillan, 1988, Chapter 4 (Fatou's lemma, generalized Lebesgue convergence theorem).
The Mathlib Community, The Lean Mathematical Library, CPP 2020, https://doi.org/10.1145/3372885.3373824 (MeasureTheory.lintegral_liminf_le, MeasureTheory.tendsto_integral_of_dominated_convergence).

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Dynamic Programming Operations Research Probability+1·Captain: mikedeng1

Stochastic Dynamic Programming and the Control of Queueing Systems X: Average Cost Optimization of Continuous Time Markov Decision ChainsTextbook

Motivation

Many queueing systems evolve in continuous time: customers arrive according to a Poisson process, services take exponentially distributed times, and a controller may change the service rate, admit or reject customers, or route them whenever the state changes. Minimizing the long-run average cost of such a system is a standard problem in the control of queues (Lippman 1975; Puterman 1994, Ch. 11; Sennott 1999, Ch. 10). The continuous time model does not fit directly into the discrete time theory of Markov decision chains developed in the earlier chapters of Sennott's book, because time spent in a state now matters and the natural average cost is a ratio of expected cost to expected elapsed time.

This mission formalizes Sections 10.1–10.4 of L. I. Sennott, Stochastic Dynamic Programming and the Control of Queueing Systems (Wiley, 1999): the elementary properties of the exponential distribution, the continuous time Markov decision chain and its average cost, a reduction of the continuous time problem to an auxiliary discrete time Markov decision chain, and the theorem stating that finite state approximating sequences of the auxiliary chain compute optimal average costs and optimal stationary policies of the continuous time chain. The chapter closes with an explicit average cost computation for the M/M/1 queue with service rate control.

Setting

A random variable $X$ has the exponential distribution with rate $\mu>0$ if $P(X\le t)=1-e^{-\mu t}$ for $t\ge0$ . A function $r(\delta)$ is $o(\delta)$ if $r(\delta)/\delta\to0$ as $\delta\to0^+$ .

A continuous time Markov decision chain (CTMDC) $\Psi$ has a countable state space $S$ and, for each $i\in S$ , a finite nonempty action set $A_i$ . Choosing $a\in A_i$ in state $i$ incurs an instantaneous cost $G(i,a)\ge0$ and a cost rate $g(i,a)\ge0$ in effect until the next transition. The time until the next transition is exponential with rate $\nu(i,a)>0$ , so its mean is $\tau(i,a)=1/\nu(i,a)$ ; the next state is $j$ with probability $P_{ij}(a)$ , where $P_{ii}(a)=0$ . A policy $\theta$ chooses, at each transition, an action (possibly at random) from the history of past states, actions and sojourn times; a stationary policy $e$ chooses $e(i)$ in state $i$ . With $C_n$ the cost and $T_n$ the time of the first $n$ transition periods, the average cost and the minimum average cost are

J^\Psi_\theta(i)=\limsup_{n\to\infty}\frac{E_\theta[C_n\mid X_0=i]}{E_\theta[T_n\mid X_0=i]},\qquad J^\Psi(i)=\inf_\theta J^\Psi_\theta(i).

Assumption (CTB) requires constants $\tau$ and $B$ with $0<\tau<\inf_{i,a}\tau(i,a)\le\sup_{i,a}\tau(i,a)\le B<\infty$ . The auxiliary MDC $\Delta$ has the same states and actions, costs $C(i,a)=G(i,a)\nu(i,a)+g(i,a)$ , and transition probabilities $P^*_{ij}(a)=\tau\nu(i,a)P_{ij}(a)$ for $j\ne i$ , $P^*_{ii}(a)=1-\tau\nu(i,a)$ . Its average cost $J^\Delta_\theta(i)=\limsup_n n^{-1}\sum_{t<n}E_\theta[C(X_t,Y_t)]$ and minimum average cost $J^\Delta(i)$ are those of Chapter 2. Assumption (CTAC) is $J^\Delta(\cdot)\le J^\Psi(\cdot)$ .

An approximating sequence $(\Delta_N)_{N\ge N_0}$ for $\Delta$ uses finite state spaces $S_N$ increasing to $S$ and transition probabilities $P^*_{ij}(a;N)$ on $S_N$ converging to $P^*_{ij}(a)$ . The (AC) assumptions ask for constants $J^N$ and functions $r^N$ on $S_N$ solving

J^N+r^N(i)=\min_{a\in A_i}\Big\{C(i,a)+\sum_{j\in S_N}P^*_{ij}(a;N)\,r^N(j)\Big\},\qquad i\in S_N,\ N\ge N_0,\tag{10.21}

with $\limsup_N r^N(i)<\infty$ , $\liminf_N r^N(i)\ge-Q$ for a constant $Q\ge0$ , and $\limsup_N J^N=:J^*<\infty$ , $J^*\le J^\Delta(i)$ .

Formalization targets

Goal: Theorem 10.3.3

Under (CTB), (CTAC) and the (AC) assumptions for an approximating sequence of $\Delta$ :

J^*=\lim_{N\to\infty}J^N\ \text{exists and}\ J^\Delta(i)=J^\Psi(i)=J^*\quad(i\in S),

and every limit point $e^*$ of a sequence $e^N$ of stationary policies realizing the minimum in (10.21) satisfies $J^\Delta_{e^*}=J^\Delta$ and $J^\Psi_{e^*}=J^\Psi$ . The goal leaves the chain, the approximating sequence and the constants of (CTB) arbitrary.

Milestones

Proposition 10.1.2: $P(X>x+y\mid X>y)=P(X>x)$ for $x,y>0$ , and $P(X\le\delta)=\mu\delta+o(\delta)$ .
Proposition 10.1.3: for independent exponentials, $P(X_1\le\delta,X_2\le\delta)=o(\delta)$ , $P(X_1<X_2)=\mu_1/(\mu_1+\mu_2)$ , and $\min(X_1,X_2)$ is exponential with rate $\mu_1+\mu_2$ .
Lemma 10.3.1: if $z$ is bounded below and $Z\tau(i,e)+z(i)\ge G(i,e)+g(i,e)\tau(i,e)+\sum_jP_{ij}(e)z(j)$ for all $i$ (10.15), then $J^\Psi_e\le Z$ .
Lemma 10.3.2: $(Z,w)$ satisfies $Z+w(i)\ge C(i,e)+\sum_jP^*_{ij}(e)w(j)$ (10.20) if and only if $(Z,\tau w)$ satisfies (10.15).
Proposition 10.4.1: in the M/M/1 queue with arrival rate $\lambda$ , holding cost $H(i)=Hi$ and service cost rate $c(a)$ , the policy that always serves at rate $a>\lambda$ has average cost $\rho_ac(a)+H\rho_a/(1-\rho_a)$ , $\rho_a=\lambda/a$ .

Significance

The goal theorem turns the average cost control of a continuous time chain on an infinite state space into a finite computation: solve the optimality equation (10.21) of a finite truncation of the auxiliary chain, let the truncation grow, and read off the optimal average cost and an optimal stationary policy of the original continuous time chain. The auxiliary chain is the book's form of uniformization, and the result is what licenses the numerical study of the M/M/1 service rate control problem in Section 10.4 and of the M/M/K and polling models in Sections 10.5–10.6. Proposition 10.4.1 gives the closed-form benchmark against which the computed optimal policy is compared.

The results are proved in the book, some with details left to the reader (Lemma 10.3.2(ii), Problem 10.10), and the goal rests on Theorem 8.1.1 and Lemma 7.2.1 of the same book. None of them has, as far as a search of Mathlib and the Prove2Me catalogue shows, a machine-checked proof: Mathlib provides the exponential law (ProbabilityTheory.expMeasure) and its distribution function, but not memorylessness or the minimum of independent exponentials, and no continuous time Markov decision model. A formalization would supply these, together with a checked average cost comparison between a continuous time chain and its discrete time auxiliary chain.

Difficulty

The obvious argument compares the two chains policy by policy, but the policy classes differ: a policy for $\Delta$ may change action in every time slot, including slots where the state does not change, while a policy for $\Psi$ acts only at transitions and may use the observed sojourn times. Only the stationary policies coincide. The lower bound $J^\Psi\ge J^*$ therefore cannot be obtained by transferring policies, and it is exactly what Assumption (CTAC) supplies. The upper bound requires passing from the discrete time inequality (10.20) for the limit point $e^*$ to a bound on a ratio of expected cost to expected time in continuous time, where the denominator depends on the policy; the uniform bounds of (CTB) on the mean sojourn times are what control it. Inside Lemma 10.3.1 the function $z$ is only bounded below, so the telescoping of expectations must be justified without integrability of $z$ from above.

Formalization scope

The state space is a countable type S, actions a type Act, and action sets A i : Finset Act; the CTMDC and MDC structures hold data, and their axioms (nonempty action sets, nonnegative costs, positive rates, stochastic transition rows with $P_{ii}(a)=0$ ) are separate predicates. Transition probabilities are ℝ≥0∞-valued; costs, rates and the functions $z,w,r^N$ are real. Expected costs, expected times and all average costs are ℝ≥0∞-valued, so $+\infty$ is a legitimate value, and they are compared with real constants in EReal; the limits superior and inferior of (AC) are taken in EReal. The expected cost of $n$ transition periods under a general policy is a recursion over the periods in which the sojourn time is integrated against expMeasure ν(i,a) and the next state is drawn independently from $P_{i\cdot}(a)$ ; policies are measurable in the past sojourn times. In (10.15) and (10.20) the convergence of the series is part of the inequality. The strict inequality $\tau<\inf\tau(i,a)$ of (CTB) is kept strict (as a positive margin); weakening it to $\le$ would make $P^*_{ii}(a)$ vanish or turn negative.

The average cost $J^\Psi_\theta$ is a ratio of expectations, not the expectation of a ratio, and the infimum $J^\Psi$ ranges over history dependent randomized policies that may use sojourn times; replacing either by a stationary-only class, or dropping (CTAC), gives a different theorem.

A complete development needs: expected rewards of a chain with exponential holding times, the average cost theory of Chapter 8 for the auxiliary chain (Theorem 8.1.1 and Lemma 7.2.1, restated here as needed), and renewal-reward reasoning for Proposition 10.4.1. The exponential-distribution lemmas are reusable beyond this mission and are welcome as independent contributions.

Selected references

L. I. Sennott, Stochastic Dynamic Programming and the Control of Queueing Systems, Wiley Series in Probability and Statistics, John Wiley & Sons, 1999. https://doi.org/10.1002/9780470317037
M. L. Puterman, Markov Decision Processes: Discrete Stochastic Dynamic Programming, John Wiley & Sons, 1994. https://doi.org/10.1002/9780470316887
S. A. Lippman, Applying a new device in the optimization of exponential queuing systems, Operations Research 23(4), 687–710, 1975. https://doi.org/10.1287/opre.23.4.687
D. Gross and C. M. Harris, Fundamentals of Queueing Theory, 3rd ed., John Wiley & Sons, 1998.

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Operations Research Probability Stochastic Systems·Captain: mikedeng1

Stochastic Dynamic Programming and the Control of Queueing Systems IX: Bounded Mean Residual Lifetimes Imply Finite Moments of All OrdersTextbook

Motivation

In a discrete-time queue the service of a customer lasts a random number $Y$ of slots. When such a system is modelled as a Markov decision chain (Sennott, Stochastic Dynamic Programming and the Control of Queueing Systems, Wiley, 1999, DOI 10.1002/9780470317037, Chapter 9), the state must record how much service is still owed, and the controller only observes that a service has lasted $s$ slots and is not yet finished. The relevant random quantity is then the residual life $Y_s$ : the remaining service time given that $s$ slots have elapsed without completion. Verifying the book's average cost assumptions for such a model requires bounds on expected first passage times and costs, and these reduce to moment bounds on $Y$ and on the residual lives $Y_s$ .

Section 9.2 isolates a single condition that makes those bounds available: the expected remaining service time is bounded uniformly in the elapsed time. The concept of mean residual life comes from reliability theory, where $Y$ is the lifetime of a component and $E[Y_s]$ is its expected remaining lifetime at age $s$ . This mission formalizes Section 9.2 of the book, together with the moment computation for batch arrivals (Lemma 9.5.2) that the same verification uses.

Setting

Let $Y$ be a random variable with values in $\{1,2,3,\dots\}$ and distribution $u_y = P(Y = y)$ , $y \ge 1$ . Write $F(y) = P(Y \le y)$ and $F^*(y) = P(Y > y) = 1 - F(y)$ for $y \ge 0$ , so $F(0) = 0$ and $F^*(0) = 1$ . The $k$ -th moment is

E[Y^k] = \sum_{y \ge 1} y^k u_y \in [0,\infty].

For $s \ge 0$ with $F^*(s) > 0$ , the residual life $Y_s$ has distribution

P(Y_s = y) = P(Y = s + y \mid Y > s) = \frac{u_{s+y}}{F^*(s)}, \qquad y \ge 1,

with $Y_0 = Y$ ; its tail is $F^*_s(y) = F^*(s+y)/F^*(s)$ , and $E[Y_s]$ is the mean residual lifetime.

The distribution of $Y$ has bounded mean residual lifetimes (BMRL- $U$ , Definition 9.2.4) if there is a finite constant $U$ with

E[Y_s] \le U \qquad \text{for every } s \ge 0 \text{ with } F^*(s) > 0,

and it is BMRL if it is BMRL- $U$ for some $U$ .

Three families appear by name: the geometric distribution $\mathrm{geo}(\mu)$ of the number of Bernoulli( $\mu$ ) trials to the first success, $P(Y=y) = \mu(1-\mu)^{y-1}$ ; the negative binomial $\mathrm{neg\,bin}(\mu, r)$ of the number of trials to the $r$ -th success, $P(Y = y) = \binom{y-1}{r-1}\mu^r(1-\mu)^{y-r}$ for $y \ge r$ ; and the truncated Poisson $\mathrm{trun\,Pois}(\lambda)$ , $P(Y=y) = \frac{e^{-\lambda}}{1-e^{-\lambda}}\frac{\lambda^y}{y!}$ for $y \ge 1$ .

For Lemma 9.5.2, batches of customers arrive in each slot; the batch sizes $X_1, X_2, \dots$ are independent with common distribution $p_j$ , mean $\lambda = \sum_j j p_j$ and second moment $\lambda^{(2)} = \sum_j j^2 p_j$ , and $X(s) = X_1 + \dots + X_s$ is the number of arrivals in $s$ slots.

Formalization targets

Goal: Proposition 9.2.5

If the distribution of $Y$ is BMRL, then

E[Y^k] < \infty \qquad \text{for every } k.

The goal fixes no constant: it asserts only that a uniform first-moment bound on the residual lives forces every moment of $Y$ to be finite.

Milestones

Proposition 9.2.1. $E[Y] = \sum_{y=0}^\infty F^*(y)$ and, for $k \ge 2$ ,

E[Y^k] = 1 + \sum_{z=0}^{k-1}\binom{k}{z}\left[\sum_{y=1}^\infty y^z F^*(y)\right]. \tag{9.4}

Remark 9.2.2. For $k \ge 2$ , $E[Y^k] < \infty$ if and only if $\sum_y y^{k-1}F^*(y) < \infty$ .
Proposition 9.2.3. For a positive integer $k$ , $E[Y^k] < \infty$ implies $E[Y_s^k] < \infty$ for all $s \ge 0$ .
Proposition 9.2.6. The geometric ( $0<\mu<1$ ), negative binomial ( $0<\mu<1$ , $r \ge 2$ ) and truncated Poisson ( $\lambda > 0$ ) distributions are BMRL.
Lemma 9.5.2. Under $\lambda^{(2)} < \infty$ ,

E[X(s)] = \lambda s, \qquad E[(X(s))^2] = \lambda^{(2)}s + \lambda^2 s(s-1). \tag{9.25}

Significance

The result itself. Proposition 9.2.5 turns a condition that is easy to check for concrete service distributions, and natural for services (a service whose expected remaining duration grows without bound as it goes on is undesirable), into the moment bounds that the average cost analysis consumes. With Proposition 9.2.6 it shows that the most common unbounded service distributions on $\{1,2,\dots\}$ have finite moments of all orders; with Lemma 9.5.2 it supplies the linear and quadratic growth of expected arrivals and their second moments that the verification of the (WAC) assumptions for the batch-arrival queue of Example 9.3.1 needs (Section 9.5). Every bounded distribution is BMRL as well (the book's Problem 9.3).

Formalizing it. All results here are proved in the book; none has a machine-checked proof on the platform or in Mathlib, which has geometric and Poisson distributions but no residual lives, negative binomial or truncated Poisson laws. A complete development gives a reusable tail-sum calculus for moments of $\mathbb N$ -valued random variables in $[0,\infty]$ , a residual-life construction for discrete distributions, and the BMRL property of three standard families. The platform's mean residual life order (the "Stochastic Orders II" mission, Shaked–Shanthikumar) compares two variables; BMRL is a uniform bound on one variable's residual lives and is not an order, so none of that material states these results.

Difficulty

BMRL controls only first moments, of the conditional laws $Y_s$ ; the goal asks for moments of every order of $Y$ itself. Bounding $E[Y^k]$ by expanding $E[Y_s]$ for each fixed $s$ gives nothing, because each single bound is compatible with a heavy tail: the uniformity in $s$ is essential. The residual lives are also only defined where $P(Y > s) > 0$ , so every argument must handle distributions with bounded support separately. Proposition 9.2.6 requires explicit control of ratios of tail sums for three families; for the negative binomial and truncated Poisson the tails have no closed form.

Formalization scope

$Y$ is represented by its law, a function $u : \mathbb N \to [0,\infty]$ with $\sum_y u_y = 1$ and $u_0 = 0$ (IsDistOnPos). $F^*$ , moments and residual-life moments are ℝ≥0∞-valued series; an infinite moment is $+\infty$ and "finite" means $< \infty$ . No Bochner integral is used, so a finite-moment conclusion cannot hold vacuously through an integrability default.
The residual life $Y_s$ is defined by (9.7) and is used only where $F^*(s) > 0$ ; BMRL- $U$ is required exactly at those $s$ , and $U$ is a finite nonnegative real. A formalization requiring the bound at every $s$ with a junk value of $E[Y_s]$ where $F^*(s) = 0$ is ruled out: the definitions never divide by $F^*(s) = 0$ in a used position, and bounded distributions remain BMRL.
The geometric and negative binomial laws count trials (support starting at $1$ and $r$ ), not failures as Mathlib's geometricPMF does.
Lemma 9.5.2 is stated on a probability space with measurable, mutually independent (iIndepFun) batch sizes of common law $p$ , expectations as lower Lebesgue integrals, and only assumption (BA1), $\lambda^{(2)} < \infty$ , which is the part of the book's (BA) that concerns arrivals.
Welcome contributions: the tail-sum identity (9.4) and its reindexing lemmas, the residual-life tail formula (9.8) and moment formula (9.9), each as a separate lemma; and proofs that the three named families are probability distributions on their supports.

Selected references

Linn I. Sennott, Stochastic Dynamic Programming and the Control of Queueing Systems, Wiley Series in Probability and Statistics, John Wiley & Sons, 1999, Section 9.2 (pp. 202–206) and Section 9.5 (pp. 214–215). DOI 10.1002/9780470317037
Moshe Shaked and J. George Shanthikumar, Stochastic Orders, Springer Series in Statistics, Springer, 2007, Section 2.A (the mean residual life order). DOI 10.1007/978-0-387-34675-5

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Dynamic Programming Markov Chain Operations Research+1·Captain: mikedeng1

Stochastic Dynamic Programming and the Control of Queueing Systems VIII: Computing Average Cost Optimal Policies by Approximating SequencesTextbook

Motivation

Control problems for queueing systems (admission control, service rate control, routing to parallel servers) are naturally modelled as Markov decision chains whose state is a vector of queue lengths. The state space is therefore denumerably infinite, and the performance measure of interest is usually the long-run average cost per slot. For such models the existence theory of average cost optimal stationary policies is well developed (Chapter 7 of Sennott's book), but existence gives no algorithm: an optimal policy is a function on an infinite set, and value iteration cannot be run on an infinite state space.

The approximating sequence method answers this by replacing the infinite model $\Delta$ with a sequence of finite models $\Delta_N$ on truncated state spaces $S_N$ , solving the average cost optimality equation in each, and passing to the limit. Chapter 8 of L. I. Sennott, Stochastic Dynamic Programming and the Control of Queueing Systems (Wiley, 1999) gives a set of conditions, the (AC) assumptions, under which this limit procedure provably produces the minimum average cost and an average cost optimal policy of $\Delta$ .

Timeline. The approximating sequence method for the average cost criterion and the (AC) assumptions were introduced in Sennott (1997a), with further results in Sennott (1997b) (bibliographic notes, p. 194). The book collects these results, adds the four step verification template (Proposition 8.2.1), the finite-set augmentation route based on the (BOR) assumptions (Proposition 8.2.3), and the weakening (WAC) of Section 8.7, which Chapter 9 uses.

Setting

An MDC $\Delta$ has a countable state space $S$ , a finite nonempty action set $A_i$ in each state $i$ , a finite cost $C(i,a)\ge0$ , and transition probabilities $P_{ij}(a)$ . A general policy $\theta$ chooses actions at random using the whole past history. Its average cost is

J_\theta(i)=\limsup_{n\to\infty}\frac1n\sum_{t=0}^{n-1}E_\theta[C(X_t,A_t)\mid X_0=i],

and the minimum average cost is $J(i)=\inf_\theta J_\theta(i)\in[0,\infty]$ . A policy is average cost optimal if $J_\theta\equiv J$ .

An approximating sequence $(\Delta_N)_{N\ge N_0}$ consists of finite sets $S_N$ increasing to $S$ and MDCs $\Delta_N$ on $S_N$ with the same actions and costs and with transition probabilities $P_{ij}(a;N)$ on $S_N$ converging to $P_{ij}(a)$ . Write $v^N_n$ and $V^N_\alpha$ for the $n$ -horizon and discounted value functions of $\Delta_N$ .

The (AC) assumptions are:

(AC1) there are finite constants $J^N$ and finite functions $r^N$ on $S_N$ with

J^N+r^N(i)=\min_a\Big\{C(i,a)+\sum_{j\in S_N}P_{ij}(a;N)\,r^N(j)\Big\},\qquad i\in S_N; \tag{8.1}

(AC2) $\limsup_N r^N(i)<\infty$ ;
(AC3) $\liminf_N r^N(i)\ge -Q$ for a constant $Q\ge0$ ;
(AC4) $J^*:=\limsup_N J^N<\infty$ and $J^*\le J(i)$ for all $i$ .

The (WAC) assumptions of Section 8.7 allow $Q$ to depend on the state, at the price of integrability conditions along every stationary policy.

Formalization targets

Goal: Theorem 8.1.1

Under (AC), the limit $\lim_{N\to\infty}J^N$ exists and

J(i)=\lim_{N\to\infty}J^N\qquad\text{for all } i\in S,

and every limit point $e^*$ of stationary policies $e^N$ realizing the minimum in (8.1) is average cost optimal for $\Delta$ . The statement fixes no constants; it asserts the shape of the conclusion for any model satisfying (AC).

Milestones

Proposition 8.2.1 (the four step template): unichain and aperiodicity of the finite models, an $x$ standard policy at which the approximating sequence is conforming, a comparison of $v^N_n$ (or $V^N_\alpha$ ) with $v_n$ (or $V_\alpha$ ), and a lower bound on $v^N_n - v^N_n(x)$ together imply that the value iteration limits

r^N(i)=\lim_{n\to\infty}\big(v^N_n(i)-v^N_n(x)\big)

exist and satisfy (AC).

Corollary 8.2.2: on $S=\{0,1,2,\dots\}$ with $S_N=\{0,\dots,N\}$ and excess probability sent to $N$ , monotonicity of $v^N_n$ , $v_n$ and of the first passage moments of a $0$ standard policy suffices.
Proposition 8.2.3: an augmentation type approximating sequence that sends excess probability to a finite set of cheap states satisfies the template.
Proposition 8.5.1: in the single-server queue with Bernoulli( $p$ ) arrivals and constant service rate $a>p$ ,

J_{d(a)}=\frac{Hp(1-p)}{a-p}+\frac{pC(a)}{a}.

Proposition 8.7.1: the conclusions of Theorem 8.1.1 hold under (WAC).

Significance

The result itself. Theorem 8.1.1 is what turns the existence theory of Chapter 7 into a computation. It certifies that the minimum average costs of the truncations converge to the minimum average cost of the infinite model, that this cost is constant, and that the policies produced by value iteration on $\Delta_N$ converge, along subsequences, to an optimal policy for $\Delta$ . Propositions 8.2.1–8.2.3 reduce (AC) to properties that can be checked model by model; Section 8.3 checks them for queues with reject option, service rate control, and routing to parallel queues. Proposition 8.7.1 is the version used in Chapter 9 for models whose relative values are not uniformly bounded below. Proposition 8.5.1 gives the closed-form open-loop benchmark used in the numerical study of Section 8.5.

Formalizing it. All results are proved in the book; none has a machine-checked proof. A formalization would give the first verified convergence theorem for truncations of denumerable-state average cost MDPs, and would make the approximating sequence method usable as a certified reduction from infinite to finite models. The template results (8.2.1–8.2.3) additionally require a formal treatment of conformity of approximating Markov chains (Appendix C.4–C.5), which is of independent use.

Difficulty

The naive argument takes limits in (8.1) along $N$ : the minimum over $a$ and the finite sums pass to the limit only in the inequality direction, and only after a Fatou-type lemma for sums against the converging distributions $P_{ij}(a;N)$ with integrands $r^N$ that are neither bounded nor monotone. The lower bound $-Q$ in (AC3) is exactly what makes this possible; without it the limit inequality can fail. The limit inequality then produces only an average cost optimality inequality, and turning it into optimality of the limit policy requires a separate argument that a function bounded below and satisfying the inequality yields an upper bound on the average cost. Existence of $\lim_N J^N$ is not given: (AC4) controls only the limit superior, and the limit must be identified through every subsequence. For the template results, the difficulty is in the Markov chain side: the convergence of first passage times and costs of the truncated chains, which fails for general approximating sequences (Examples C.4.4, C.4.7).

Formalization scope

The Lean development is in the namespace SennottDP.AvgASM. The state space is any countable type; action sets are Finsets, assumed nonempty; costs are finite and nonnegative (ℝ≥0); transition probabilities are ℝ≥0∞-valued, with each row a probability distribution for admissible actions. All value functions and average costs take values in $[0,\infty]$ (ℝ≥0∞), and every infimum over policies ranges over the full class of history-dependent randomized policies. The $J^N$ and $r^N$ of (AC1) are real; the limits superior and inferior over $N$ in (AC2)–(AC4) and (WAC) are taken in EReal, so an unbounded sequence cannot produce a junk finite value. The equality $J(i)=\lim_N J^N$ is stated in EReal, which also asserts that $J(i)$ is finite. Quantities of $\Delta_N$ at states outside $S_N$ are junk values that affect only finitely many $N$ for each state.

A trivializing formalization is ruled out: $J^N$ and $r^N$ are the witnesses of (AC1), not free variables, the policies $e^N$ must realize the minimum in (8.1) for those witnesses, and the minimum average cost $J$ is an infimum over all policies, so the goal cannot be satisfied by choosing $J^*$ or the limit policy.

A complete development needs: the induced process law of a general policy; the average cost optimality inequality argument (Lemma 7.2.1); the finite-state average cost results of Chapter 6 (Propositions 6.4.1, 6.5.1, 6.6.3); a Fatou lemma for converging distributions (Proposition A.2.5); limit points of policy sequences (Proposition B.5); and, for the template results, the theory of $z$ standard chains and conformity (Appendix C.2–C.5). The Markov chain layer and the approximating sequence definitions are reusable beyond this mission. Contributions of any of these intermediate results as separate theorems are welcome.

Selected references

L. I. Sennott, Stochastic Dynamic Programming and the Control of Queueing Systems, Wiley Series in Probability and Statistics, John Wiley & Sons, 1999. https://doi.org/10.1002/9780470317037
L. I. Sennott, "The computation of average optimal policies in denumerable state Markov decision chains", Advances in Applied Probability 29 (1997), 114–137 (cited as Sennott (1997a) in the book).
L. I. Sennott, "On computing average cost optimal policies with application to routing to parallel queues", ZOR — Mathematical Methods of Operations Research 45 (1997), 45–62 (cited as Sennott (1997b) in the book).

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Dynamic Programming Markov Chain Operations Research+1·Captain: mikedeng1

Stochastic Dynamic Programming and the Control of Queueing Systems IV: Average Cost Optimal Stationary Policies Exist for Finite State SpacesTextbook

Why average cost on finite state spaces

Controlled queues, inventories and communication links are run for a long time, and the quantity an operator usually cares about is the long-run average cost per period rather than a discounted total. The average cost criterion is harder to work with than the discounted one: its value is a $\limsup$ of Cesàro means, it is not given by a contraction, and for general (history dependent, randomized) policies the limit need not exist. Chapter 6 of L. I. Sennott, Stochastic Dynamic Programming and the Control of Queueing Systems (Wiley, 1999) treats the case of a finite state space, where the strongest results hold: an average cost optimal policy exists, can be taken stationary, and can be obtained as a limit of discount optimal policies as the discount factor tends to one.

The results go back to D. Blackwell, "Discrete dynamic programming", Ann. Math. Statist. 33 (1962), who showed that for finite states and actions some stationary policy is discount optimal for all discount factors close to one. Such a policy is now called Blackwell optimal. Sennott's Chapter 6 derives average cost optimality of this policy and the multichain average cost optimality equation from it, in the notation used throughout the book.

Setting

A Markov decision chain (MDC) $\Delta$ has a countable state space $S$ , a finite nonempty action set $A_i$ in each state $i$ , nonnegative finite costs $C(i,a)$ , and transition probabilities $P_{ij}(a)$ with $\sum_j P_{ij}(a) = 1$ . A policy $\theta$ chooses the action at time $t$ from a distribution $\theta(\cdot \mid h_t)$ on $A_{i_t}$ that may depend on the whole history $h_t = (i_0,a_0,\ldots,i_t)$ . A stationary policy $f$ always chooses a fixed action $f(i) \in A_i$ in state $i$ .

With $X_t, A_t$ the state and action at time $t$ and $X_0 = i$ , define

the discounted cost $V_{\theta,\alpha}(i) = \sum_{t \ge 0} \alpha^t E_\theta[C(X_t,A_t)]$ for $0<\alpha<1$ , and the discounted value function $V_\alpha(i) = \inf_\theta V_{\theta,\alpha}(i)$ ;
the $n$ horizon cost $v_{\theta,n}(i) = \sum_{t=0}^{n-1} E_\theta[C(X_t,A_t)]$ ;
the average cost $J_\theta(i) = \limsup_n v_{\theta,n}(i)/n$ , its $\liminf$ version $J^*_\theta(i)$ , and the minimum average cost $J(i) = \inf_\theta J_\theta(i)$ .

All infima range over all general policies, and every quantity may equal $+\infty$ . A policy is $\alpha$ discount optimal if $V_{\theta,\alpha} = V_\alpha$ , and average cost optimal if $J_\theta = J$ .

For a stationary policy $f$ on a finite state space, the induced Markov chain splits into positive recurrent classes $R_1,\ldots,R_K$ and transient states. With $p_k(i)$ the probability of reaching $R_k$ from $i$ , distinguished states $z_k \in R_k$ , and $W_\alpha(i) = \sum_k p_k(i) V_\alpha(z_k)$ , the relative value function is $w_\alpha(i) = V_\alpha(i) - W_\alpha(i)$ .

Formalization targets

Goal: Proposition 6.2.3

For an MDC with a finite state space there are $\alpha_0 \in (0,1)$ and one stationary policy $f$ such that $f$ is $\alpha$ discount optimal for every $\alpha \in (\alpha_0,1)$ , $f$ is average cost optimal, and

J(i) = \lim_{\alpha\to 1^-} (1-\alpha) V_\alpha(i) = \lim_{n\to\infty} \frac{v_{f,n}(i)}{n}, \qquad i \in S.

Milestones

Proposition 4.5.3. For finite $S$ and stationary $e$ , $\alpha \mapsto V_{e,\alpha}(i)$ is a finite, continuous, rational function on $(0,1)$ .
Proposition 6.1.1. For every policy on a countable state space,

J^*_\theta(i) \le \liminf_{\alpha\to1^-}(1-\alpha)V_{\theta,\alpha}(i) \le \limsup_{\alpha\to1^-}(1-\alpha)V_{\theta,\alpha}(i) \le J_\theta(i),

with three equivalent conditions for equality. 3. Proposition 6.2.2. For finite $S$ and stationary $e$ , $J_e(i) = \lim_{\alpha\to1^-}(1-\alpha)V_{e,\alpha}(i) = \lim_n v_{e,n}(i)/n$ . 4. Proposition 4.5.1, Proposition 4.5.4, Corollary 4.5.5. The power series structure of $V_{\theta,\alpha}$ in $\alpha$ ; monotonicity and left continuity of $V_\alpha$ ; continuity under bounded costs. 5. Theorem 6.3.1. For the policy $f$ of the goal, $\lim_{\alpha \to 1^-} w_\alpha(i) = w(i)$ exists, and

J(i) + w(i) = C(i,f) + \sum_j P_{ij}(f) w(j) \ \ge\ \min_{a} \Big\{C(i,a) + \sum_j P_{ij}(a) w(j)\Big\},

together with the limit identities (i)–(iii) and the optimality criterion (v). 6. Proposition 6.3.3. $V_\alpha(i) = J(i)/(1-\alpha) + w^*(i) + \varepsilon_\alpha(i)$ with $\varepsilon_\alpha(i) \to 0$ as $\alpha \to 1^-$ .

Significance

The goal says that on a finite state space nothing is gained by randomizing or by remembering the past when minimizing average cost, and that the minimum average cost is the vanishing-discount limit of the discounted value function. This justifies computing average cost optimal policies through discounted problems and value iteration, the route taken in the rest of Chapter 6 and, via approximating sequences, for countable state spaces in Chapters 7 and 8. Theorem 6.3.1 supplies an optimality equation without any unichain or communication assumption. The book's Example 6.3.2 shows that the inequality in that equation can be strict, and that a stationary policy attaining the minimum need not be optimal.

The results are classical and proved in the book. No machine-checked version of them is known to exist. The platform has average-reward results for unichain finite MDPs with Markov policies (the Puterman series) and an average-cost optimality equation under recurrence assumptions (the Bertsekas series). Neither covers existence of a Blackwell optimal policy against the class of all history dependent randomized policies, or the multichain equation. A formal development also yields reusable infrastructure: the law of a controlled process under a general policy, first passage quantities of finite chains, and the Abelian inequality between Abel and Cesàro means of a nonnegative sequence.

Difficulty

The obvious argument picks, for each $\alpha$ , a stationary discount optimal policy $f_\alpha$ and lets $\alpha \to 1$ . Finiteness of the set of stationary policies gives one policy that is optimal along some sequence $\alpha_n \to 1$ , but not on an interval. Excluding infinite switching between two policies requires the analytic structure of $\alpha \mapsto V_{f,\alpha}(i)$ (Proposition 4.5.3), which in turn rests on matrix inversion of $I - \alpha P$ . Passing from the discounted criterion to the average one requires an Abelian inequality for nonnegative series whose terms may be infinite (Proposition 6.1.1), and comparison against general policies rules out any argument that works only within stationary or Markov policies. For Theorem 6.3.1 the difficulty is the multichain structure: the relative value function has to be assembled class by class from first passage times and costs, and its limit must be identified.

Formalization scope

States form a type S; [Countable S] for Section 4.5 and Proposition 6.1.1, [Fintype S] from Section 6.2 on, as in the book. Actions form a type Act with A i : Finset Act nonempty. Costs are in ℝ≥0, transition probabilities in ℝ≥0∞.
A general policy is a function of the list of past state-action pairs (most recent first) and the current state, giving a distribution on A i. Stationary policies embed as degenerate policies. The law of the process is built from this data, and every infimum ranges over all general policies.
$V_{\theta,\alpha}$ , $V_\alpha$ , $v_{\theta,n}$ , $J_\theta$ , $J^*_\theta$ , $J$ are in ℝ≥0∞, so $+\infty$ is represented. $\alpha \to 1^-$ is the filter 𝓝[<] 1. On a finite state space these quantities are finite. The real valued objects of Section 6.3 ( $w_\alpha$ , $w$ , $w^*$ , equation (6.6)) are therefore formed with toReal, and this switch from ℝ≥0∞ to ℝ happens only in Theorem 6.3.1 and Proposition 6.3.3.
The objects of Section 6.3 ( $p_k$ , $m_{i|k}$ , $c_{i|k}$ , $\pi_s$ , $W_\alpha$ ) are defined from $f$ . The distinguished states are a hypothesis quantified over.
A trivializing formalization would take the infimum over stationary policies only, let the optimal policy depend on $\alpha$ , or state rationality as an equation p/q without requiring $q \ne 0$ . Each is excluded here: $J$ and $V_\alpha$ are infima over all general policies, one pair $(\alpha_0,f)$ is quantified before all $\alpha$ , and the denominator is required to be nonzero on $(0,1)$ .

Useful infrastructure includes rational functions of one real variable and their finitely many sign changes, the resolvent $(I-\alpha P)^{-1}$ of a stochastic matrix, the Abelian inequality for $[0,\infty]$ -valued sequences, and renewal-reward identities for finite chains. Contributions of general lemmas on these topics are welcome, as are proofs of individual milestones.

Selected references

L. I. Sennott, Stochastic Dynamic Programming and the Control of Queueing Systems, Wiley, 1999. https://doi.org/10.1002/9780470317037
D. Blackwell, "Discrete dynamic programming", Annals of Mathematical Statistics 33 (1962), 719–726. https://doi.org/10.1214/aoms/1177704593
M. L. Puterman, Markov Decision Processes: Discrete Stochastic Dynamic Programming, Wiley, 1994. https://doi.org/10.1002/9780470316887

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Dynamic Programming Operations Research Probability+1·Captain: mikedeng1

Stochastic Dynamic Programming and the Control of Queueing Systems II: The Discount Optimality EquationTextbook

Motivation

Control problems for queueing systems (admission control, routing, service rate selection, inventory replenishment) are naturally modelled as Markov decision chains with a countable state space, such as the number of customers in a buffer, and with costs that grow without bound in the state, such as holding costs proportional to queue length. The expected discounted cost criterion is the first infinite horizon criterion applied to such models, and it is also the tool through which the average cost criterion is treated later in the same book (Chapters 6–8 of Sennott's text reach average cost optimal policies through limits of discounted problems as the discount factor tends to one).

Classical treatments of discounted dynamic programming assume bounded costs, under which the dynamic programming operator is a contraction and has a unique bounded fixed point. That assumption fails for queueing models. This mission formalizes Chapter 4, Sections 4.1–4.4, of L. I. Sennott, Stochastic Dynamic Programming and the Control of Queueing Systems (Wiley, 1999), which develops the discounted theory for nonnegative, possibly unbounded costs, where value functions may be infinite.

Timeline of the underlying theory:

1965. Blackwell (Ann. Math. Statist. 36) establishes the discounted theory with bounded rewards.
1966. Strauch (Ann. Math. Statist. 37) treats "negative" dynamic programming, the case of nonpositive rewards (equivalently nonnegative costs), with no boundedness assumption.
1977–1978. Bertsekas (SIAM J. Control Optim. 15) and Bertsekas and Shreve (Stochastic Optimal Control: The Discrete-Time Case) give the abstract monotone-mapping framework covering both cases.
1999. Sennott's text states the countable-state, finite-action, nonnegative-cost discounted theory in the form used for queueing control, with general history-dependent randomized policies.

Setting

A Markov decision chain $\Delta$ has a countable state space $S$ ; for each state $i$ a finite nonempty action set $A_i$ ; for each $a \in A_i$ a nonnegative finite cost $C(i,a)$ and a probability distribution $(P_{ij}(a))_{j \in S}$ of the next state. A policy $\theta$ chooses the action at time $n$ at random from a distribution $\theta(\cdot \mid h_n)$ on $A_{i_n}$ that may depend on the entire history $h_n = (i_0, a_0, \dots, a_{n-1}, i_n)$ . A stationary policy $f$ always chooses $f(i) \in A_i$ in state $i$ ; for it one writes $C(i,f) = C(i,f(i))$ and $P_{ij}(f) = P_{ij}(f(i))$ .

Fix a discount factor $\alpha \in (0,1)$ . For an initial state $i$ and a policy $\theta$ , the $n$ -horizon cost with terminal cost zero and the infinite horizon discounted cost are

v_{\theta,\alpha,n}(i) = \sum_{t=0}^{n-1} \alpha^t E_\theta[C(X_t,A_t) \mid X_0 = i], \qquad V_{\theta,\alpha}(i) = \sum_{t=0}^{\infty} \alpha^t E_\theta[C(X_t,A_t) \mid X_0 = i],

and the value functions are $v_{\alpha,n}(i) = \inf_\theta v_{\theta,\alpha,n}(i)$ and $V_\alpha(i) = \inf_\theta V_{\theta,\alpha}(i)$ , infima over all policies. All of these lie in $[0,\infty]$ . A policy is discount optimal if $V_{\theta,\alpha} = V_\alpha$ . The discount optimality equation is

W(i) = \min_{a \in A_i} \Big\{ C(i,a) + \alpha \sum_j P_{ij}(a) W(j) \Big\}, \qquad i \in S. \tag{4.9}

With $W = V_\alpha$ , $B_i(\alpha)$ denotes the set of actions attaining the minimum at $i$ .

Formalization targets

Goal: Theorem 4.1.4

$V_\alpha$ solves (4.9); every $W : S \to [0,\infty]$ solving (4.9) satisfies $V_\alpha \le W$ ; and every stationary policy $f_\alpha$ with

C(i,f_\alpha) + \alpha \sum_j P_{ij}(f_\alpha) V_\alpha(j) = \min_a \Big\{ C(i,a) + \alpha \sum_j P_{ij}(a) V_\alpha(j) \Big\} \quad \text{for all } i

is discount optimal. No boundedness of costs and no finiteness of $V_\alpha$ is assumed.

Milestones

In attack order: Lemma 4.1.1 ( $v_{\theta,\alpha,n} \uparrow V_{\theta,\alpha}$ ); Proposition 4.1.2 (a supersolution of the one-policy equation dominates $v_{e,\alpha,n} + \alpha^n E_e[W(X_n)]$ and $V_{e,\alpha}$ ); Corollary 4.1.3 (a supersolution of the optimality inequality dominates $V_{f,\alpha} \ge V_\alpha$ ); then, beyond the goal, Corollary 4.1.5 ( $\alpha^n E_{f_\alpha}[V_\alpha(X_n) \mid X_0 = i] \to 0$ where $V_\alpha(i) < \infty$ ), Proposition 4.2.2 and Corollary 4.2.4 (conditions under which a solution of (4.9) equals $V_\alpha$ ), Proposition 4.3.1 ( $v_{\alpha,n} \uparrow V_\alpha$ , and limit points of finite horizon optimal stationary policies are discount optimal) and Proposition 4.4.1 (optimal policies are exactly those concentrated on the sets $B_{i}(\alpha)$ along histories of positive probability).

Significance

Theorem 4.1.4 is the foundation for everything in the book that concerns discounted costs: it produces an optimal stationary deterministic policy, identifies $V_\alpha$ among the many solutions of (4.9) (Example 4.2.1 of the book gives a one-parameter family of finite solutions), and underlies value iteration (Proposition 4.3.1) and the approximating-sequence method of Sections 4.6–4.7. The average cost results of Chapters 6–8 are proved from it by letting $\alpha \to 1$ . Proposition 4.4.1 describes the full set of optimal policies, including randomized and history-dependent ones.

These are known results with published proofs. The contribution of this mission is a machine-checked development of the discounted theory for countable state spaces with unbounded costs and infinite values, over the general policy class. Related statements on the platform (the monotone-mapping propositions of Bertsekas 1977 in the MonotoneDP missions, and bounded-cost or finite-state discounted results) use different models and are open; no machine-checked proof of the present statements is known to this mission.

Difficulty

The contraction argument that settles the bounded case is unavailable: with unbounded costs the operator in (4.9) has many fixed points, and $V_\alpha$ can equal $+\infty$ at some states, so neither uniqueness of fixed points nor subtraction of values is available. The optimality equation compares the infimum over all history-dependent randomized policies with a one-step minimum, so the general policy class and the law of the process under it must be handled directly; restricting attention to Markov or stationary policies begs the question. Every limit exchange (monotone limits of finite horizon costs, the passage to limit points of policies in Proposition 4.3.1) takes place in $[0,\infty]$ , where finite-valued arguments do not transfer verbatim.

Formalization scope

The Lean development lives in the namespace SennottDP.Discounted. Conventions:

The state space is a type S with [Countable S]; actions form a type Act and A i : Finset Act is nonempty. Costs are ℝ≥0; transition probabilities are ℝ≥0∞ with ∑' j, P i a j = 1 for a ∈ A i.
A history at time $n$ is a pair Fin (n+1) → S, Fin n → Act; a policy assigns to every history a distribution on the action set of its last state. The probability of a history is the product of the policy and transition probabilities; expectations are ℝ≥0∞ sums over histories, so no integrability conditions arise.
All values ( $v_{\theta,\alpha,n}$ , $V_{\theta,\alpha}$ , $v_{\alpha,n}$ , $V_\alpha$ , and the competing solutions $W$ ) are ℝ≥0∞-valued; $0 \cdot \infty = 0$ . The discount factor is α : ℝ≥0 with 0 < α and α < 1. Terminal costs are zero.
$V_\alpha$ and $v_{\alpha,n}$ are infima over the type of all general policies. Defining them over stationary policies only would make the optimality of $f_\alpha$ a tautology; that formalization is ruled out.
Proposition 4.4.1: the book states the equivalence without a finiteness assumption, but its necessity argument needs $V_\alpha < \infty$ , and necessity fails otherwise. Sufficiency is stated in general and necessity under $V_\alpha < \infty$ everywhere.

Useful infrastructure, reusable by the later missions of this series (approximating sequences, average cost): the shift of a general policy after its first step, the Chapman–Kolmogorov identity for the history law, and the computation $E_f[W(X_{n+1})] = E_f[\sum_j P_{X_n j}(f) W(j)]$ for stationary policies. Contributions of such lemmas, and proofs of any milestone, are welcome.

Selected references

L. I. Sennott, Stochastic Dynamic Programming and the Control of Queueing Systems, Wiley Series in Probability and Statistics, John Wiley & Sons, 1999, Chapter 4. https://doi.org/10.1002/9780470317037
D. Blackwell, Discounted dynamic programming, Annals of Mathematical Statistics 36 (1965), 226–235. https://doi.org/10.1214/aoms/1177700285
R. E. Strauch, Negative dynamic programming, Annals of Mathematical Statistics 37 (1966), 871–890. https://doi.org/10.1214/aoms/1177699369
D. P. Bertsekas, Monotone mappings with application in dynamic programming, SIAM Journal on Control and Optimization 15 (1977), 438–464. https://doi.org/10.1137/0315031
D. P. Bertsekas and S. E. Shreve, Stochastic Optimal Control: The Discrete-Time Case, Academic Press, 1978.
M. L. Puterman, Markov Decision Processes: Discrete Stochastic Dynamic Programming, Wiley, 1994. https://doi.org/10.1002/9780470316887

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