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.

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

1. Theorem 5.1 (Weak Duality). Primal feasible $x$ and dual feasible $y$ satisfy $c^T x \le b^T y$.
2. Theorem 5.2 (Strong Duality). If the primal has an optimal $x^*$, the dual has an optimal $y^*$ with $c^T x^* = b^T y^*$.
3. 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 $i$.
4. Lemma 10.5 (Farkas' Lemma). $Ax \le b$ has no solution if and only if some $y$ satisfies $A^T y = 0$, $y \ge 0$, $b^T y < 0$.
5. Theorem 10.4 (Separation of polyhedra). Two disjoint nonempty polyhedra lie in two disjoint halfspaces.
6. 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.

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

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

Further results of the chapter

5. 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 $\Phi$.
6. Lemma 4.3: the stochastic transportation problem (4.163)–(4.165) is a convex program.
7. 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 $\bar y \le c$.
8. 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

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

1. Eq. (3.4): $\|R_\alpha(\xi)x\| = \sqrt{\|x\|^2 + (\alpha^2-1)(x,\xi)^2}$ for unit $\xi$.
2. 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.
3. Volume ratio (p. 87–88): $v(\Phi_{k+1}) = q_n\, v(\Phi_k)$ with $q_n < 1$, the volumes being positive and finite.
4. Eq. (3.62): the ball field satisfies $(g(x), x - x^*) \ge 0$.
5. Eq. (3.65): the convex-programming field satisfies $(g(x), x - x^*) \ge 0$.
6. 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

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

1. 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 $U_k = \{x : f(x) = f(x_k)\}$.
2. Theorem 2.1: with constant step length $h$, some level surface $\{f = f(x_{k^*})\}$ passes within $h(1+\varepsilon)/2$ of any $x^* \in M^*$.
3. 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 $M^*$.
4. 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 $M^*$.
5. Theorem 2.3: for the unnormalized method (2.5), bounded subgradients along the trajectory imply convergence, and unbounded subgradients rule it out.
6. 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

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

1. Proposition C.1.2: $\pi$ is the unique stationary distribution of a positive recurrent class, and $\pi_j=e_{ij}/m_{ii}=\pi_ie_{ij}$.
2. 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.
3. Proposition C.1.5: if $\sum_jP_{ij}[y(j)-y(i)]\le-\epsilon$ off $G$, then $m_{iG}\le y(i)/\epsilon$.
4. Corollary C.1.6: the same with $G=\{z\}$ and $\sum_jP_{zj}y(j)<\infty$ makes $z$ positive recurrent.
5. Proposition C.2.1: on a positive recurrent class, $J^{(n)}_i\to J_R=c_{ii}/m_{ii}$.
6. Proposition C.2.2: $c_{iG}=\sum_kC(k)\,{}_Gu_{ik}$, the first-step equation (C.13), and $J_R=\sum_{i\in G}\pi_ic_{iG}$.
7. 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 $c_{zz}<\infty$.
8. 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.

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.