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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

Combinatorics Dynamic Programming Operations Research·Captain: mikedeng1

Assortment Optimization Under Consider-then-Choose Choice Models: the dynamic program's state space has at most min(2^N, N·2^K) subproblemsResearch Paper

Motivation

Assortment optimization asks a retailer which subset of products to offer so as to maximize expected revenue, given a model of how customers choose among what is offered. A widely used class of choice models in operations management describes customers by ranked preference lists: each customer type buys the highest-ranked product on offer, or nothing (Farias, Jagabathula & Shah 2013). These nonparametric models fit sales data well, but optimizing over them is hard: even when all customer types share one ranking and differ only in which products they are willing to consider, the problem is NP-hard to approximate within a factor $O(N^{1-\epsilon})$ (Aouad, Farias, Levi & Segev 2018).

Aouad, Farias and Levi (Management Science 2021) study consider-then-choose models, a behavioural refinement from marketing in which a customer first screens products into a consideration set and then picks her favourite among the offered products in that set. Their main algorithmic device is a dynamic program whose number of subproblems they bound by structural properties of the consideration sets. This mission formalizes the first and most general such bound, for a single common ranking, together with the lemmas it rests on. Earlier exact algorithms for special cases of ranking-based models are due to Honhon, Jonnalagedda & Pan (2012).

Setting

There are $N$ products $[N]=\{1,\dots,N\}$ with prices $P_i$ , and $K$ customer types $[K]$ . Type $j$ has a consideration set $C_j\subseteq[N]$ and occurs with probability $\lambda_j$ , where $(\lambda_1,\dots,\lambda_K)$ is a probability vector. All types rank products by the same permutation, and products are numbered in that order: product $1$ is the most preferred. If assortment $A\subseteq[N]$ is offered, type $j$ buys the smallest-index product of $A\cap C_j$ , giving revenue

\mathrm{Rev}(j,A)=P_{\min(A\cap C_j)}\quad\text{if }A\cap C_j\neq\emptyset,\qquad \mathrm{Rev}(j,A)=0\text{ otherwise}.

For $S\subseteq[N]$ and $T\subseteq[K]$ the subproblem value is

J(S,T)=\max_{A\subseteq S}\ \sum_{j\in T}\lambda_j\,\mathrm{Rev}(j,A),

and $J([N],[K])$ is the original problem. The bipartite graph $G$ has the products and types as vertices and an edge $\{i,j\}$ when $i\in C_j$ ; $G[S,T]$ is its subgraph induced on $S\cup T$ , and $(S',T')$ is a connected component of $G[S,T]$ when $S'\cup T'$ induces a connected subgraph and no edge of $G[S,T]$ leaves it. For a product $i$ , $T(i)=\{j\in T:i\in C_j\}$ .

The dynamic program processes products in preference order. Its state space $\mathcal S$ is generated as follows: the components of $G[[N],[K]]$ are states; from a state $(S,T)$ with $i=\min(S)$ , the components of $G[S\setminus\{i\},T\setminus T(i)]$ (product $i$ stocked) and of $G[S\setminus\{i\},T]$ (product $i$ not stocked) are states. Only components with at least one product and at least one type are states. Finally,

\Phi(S,T)=[\min(S)]\cap\bigcup_{u\in T}C_u,\qquad [\min(S)]=\{1,\dots,\min(S)\}.

Formalization targets

Goal: Theorem 1 (p. 14)

|\mathcal S|\ \le\ \min\big(2^N,\ N\cdot 2^K\big),

together with its state-space consequence: if $K\le c\log_2 N$ then $|\mathcal S|\le N^{c+1}$ . The bound holds for every family of consideration sets; prices and type probabilities do not enter.

Milestones, in the paper's order

Lemma 1 (p. 12): $J(S,T)=\sum_{u=1}^{r}J(S_u,T_u)$ over the connected components $(S_u,T_u)$ of $G[S,T]$ .
§3, Eq. (1) (p. 13): for connected $G[S,T]$ and $i=\min(S)$ ,

J(S,T)=\max\Big[P_i\sum_{j\in T(i)}\lambda_j+\sum_{u}J(S^+_u,T^+_u),\ \sum_{u}J(S^-_u,T^-_u)\Big],

over the components of the two residual graphs. 3. Claim 1 (p. 15): if two states have equal $\Phi$ and $u'\in T_2\setminus T_1$ , then $C_{u'}\cap S_1=\emptyset$ . 4. Lemma 2 (Central Lemma) (p. 15): $\Phi$ is injective on $\mathcal S$ , so $|\mathcal S|=|\Phi\langle\mathcal S\rangle|$ .

Significance

The result. A naive dynamic program over all pairs $(S,T)$ has $2^{N+K}$ states. Theorem 1 shows that the states actually generated are at most $N\cdot 2^K$ , so the algorithm, which spends $O(NK)$ operations per state, runs in time quadratic in $N$ when the number of customer types is constant and polynomial when $K=O(\log N)$ . This identifies a tractable regime for a problem that is inapproximable in general. Lemma 2 is the paper's main counting tool: the same injection yields its later polynomial bounds for structured consideration sets (consideration sets of bounded diameter, induced intervals, laminar and $d$ -disjunctive families), which are natural follow-up missions on top of this one.

Formalizing it. The results are proved in the paper; none is formalized. The mission produces a machine-checked model of the consider-then-choose problem under a unique ranking, of its graph decomposition, and of the inductively generated state space, plus proofs of the decomposition lemma, the recursion, and the counting bound. The proof of Lemma 2 in the paper is short and relies on invariants of the generated states stated in passing; a formal proof has to establish them for every state the recursion produces.

Difficulty

The upper bound $2^N$ is not a count of subsets of products: the map $(S,T)\mapsto S$ is not injective on $\mathcal S$ , and different states can share their product sets. The bound $N\cdot 2^K$ likewise is not a count of pairs $(\min S, T)$ for arbitrary connected subgraphs: for the family of all connected pairs $(S,T)$ the count exceeds both bounds (when every $C_j=[N]$ , all $(2^N-1)(2^K-1)$ nonempty pairs are connected). The bounds hold only because of how the recursion generates states, processing products in the common preference order, and the injectivity of $\Phi$ has to be derived from properties of that generation process (which types survive, which products remain) rather than from the definition of a state alone. Claim 1, proved in the paper's appendix, is where this history argument happens.

Formalization scope

Products are Fin N and types Fin K. The order on Fin N is the common ranking, so the paper's product $1$ is 0 : Fin N, and $[\min(S)]$ is the set of indices ≤ S.min' h, including $\min(S)$ itself. Prices and type probabilities are real; the standing assumption that $\lambda$ is a probability vector is defined but not needed by any statement, since Lemma 1 and recursion (1) hold for all real data and the state space depends on the consideration sets only.
$J$ is a finite maximum over the subsets of $S$ . Connectivity is Mathlib's SimpleGraph.Connected of the subgraph of the bipartite graph induced on $S\cup T$ ; a component is a connected, closed vertex set. In Lemma 1 and recursion (1) the sums run over all components, including isolated products and isolated types, which contribute $0$ .
The state space is an inductive predicate with three rules (root components, the stock move, the skip move), and its size is Set.ncard of a subset of a finite type. States are nontrivial components (at least one product and one type). This is the paper's own setting: the proof of Lemma 2 uses that every state has an edge, and on isolated products $\Phi$ is not injective.
Theorem 1's second sentence is about running time; only its state-space consequence is stated. Running time itself is not formalized.
Claim 1 is stated under the hypotheses of the paper (two generated states with equal $\Phi$ , $u'\in T_2\setminus T_1$ ). It is a step of a proof by contradiction: by Lemma 2 these hypotheses never hold together, so the claim records the key step rather than an inhabited situation.
A formalization in which the state space is any family other than the one generated by the recursion (all connected pairs, or a family merely closed under the two moves), in which the processed product is not $\min(S)$ , or in which $\Phi$ uses indices strictly below $\min(S)$ , is a different statement, and the counting bounds are false or empty for it.
Not formalized: Corollary 1 (diameter bound), Theorems 2–6 and EC.1–EC.3 (structured consideration sets, multiple rankings), and the general-ranking dynamic program of §5. A reusable by-product is the component decomposition of an induced bipartite subgraph. Proofs of the milestones in any order are welcome, as are lemmas on the invariants of generated states.

Page numbers are those of the accepted manuscript: page $n$ of the body is PDF page $n+1$ of the deposited file, and page ec $n$ of the e-companion is PDF page $35+n$ .

Selected references

A. Aouad, V. Farias, R. Levi, Assortment Optimization Under Consider-then-Choose Choice Models, Management Science 67(6):3368–3386, 2021 (accepted manuscript, LBS Research Online eprint 1426). https://doi.org/10.1287/mnsc.2020.3681
A. Aouad, V. Farias, R. Levi, D. Segev, The Approximability of Assortment Optimization Under Ranking Preferences, Operations Research 66(6):1661–1669, 2018. https://doi.org/10.1287/opre.2018.1754
V. Farias, S. Jagabathula, D. Shah, A Nonparametric Approach to Modeling Choice with Limited Data, Management Science 59(2):305–322, 2013. https://doi.org/10.1287/mnsc.1120.1610
D. Honhon, S. Jonnalagedda, X. A. Pan, Optimal Algorithms for Assortment Selection Under Ranking-Based Consumer Choice Models, Manufacturing & Service Operations Management 14(2):279–289, 2012. https://doi.org/10.1287/msom.1110.0365

6 thms2 active usersReviewed

Completed

Economics Operations Research Probability·Captain: mikedeng1

Changes in Risky Benefits and in Risky Costs: A Question of the Right Order: An Nth-Degree Risk Increase Lowers Exposure to a Risky Benefit, but to a Risky Cost Only When N Is EvenResearch Paper

Motivation

Many decisions under uncertainty share one structure: a choice $b$ scales the exposure to a random quantity $\tilde X$ . A portfolio share scales the excess return of a risky asset, a coinsurance rate scales an uninsured loss, an output level scales a random price or a random cost. The linear-payoff model of Dionne, Eeckhoudt and Gollier (1993) covers all of these, and the comparative-statics question it poses is old: when the risk in $\tilde X$ increases, does the decision maker take less of it?

For second-degree risk (a mean-preserving spread, Rothschild and Stiglitz 1970) the answer is classical and depends on the decision problem. Menegatti and Peter (2022, doi:10.1287/mnsc.2021.4081) ask the question for risk increases of every order $N$ in the sense of Ekern (1980): downside risk ( $N=3$ ), outer risk ( $N=4$ ) and beyond. Their finding is that the answer depends on whether $\tilde X$ is a risky benefit or a risky cost, and, for a risky cost, on the parity of $N$ .

Timeline:

1970, Rothschild and Stiglitz: increases in risk ( $N=2$ ).
1980, Ekern: $N$ th-degree risk increases and their expected-utility characterization (Lemma 1 of the paper).
2008, Eeckhoudt and Schlesinger: relative $(N+1)$ th-degree risk aversion and the threshold $N$ for precautionary saving.
2016, Nocetti: a general criterion for comparative statics of $N$ th-degree risk increases, the source of the paper's Lemma 3.
2022, Menegatti and Peter: the benefit/cost dichotomy (Proposition 2).

Setting

The payoff of decision $b\in B$ at realization $X$ is

z(X,b)=f(b)\,(X-x_0)+z_0,

where $B\subset\mathbb R$ is compact, $f$ is continuous on $B$ , and $x_0,z_0$ are constants. The random variable $\tilde X\ge 0$ takes values in $[\underline x,\bar x]$ . A decision maker with increasing, sufficiently smooth utility $U$ solves $\max_{b\in B}\mathbb E\,U(z(\tilde X,b))$ , with maximizer set $B^*$ .

Problem classes. The problem is risk taking if $f$ is increasing and risk mitigation if $f$ is decreasing; $\tilde X$ is a risky benefit if $f\ge0$ on $B$ and a risky cost if $f\le0$ on $B$ .

$N$ th-degree risk increase. For a distribution function $F$ on $[\underline x,\bar x]$ put $F^{(1)}=F$ and $F^{(k+1)}(z)=\int_{\underline x}^zF^{(k)}(t)\,dt$ . If $\tilde X_1,\tilde X_2$ have distribution functions $F,G$ , then $\tilde X_2$ is an $N$ th-degree risk increase over $\tilde X_1$ when $F^{(k)}(\bar x)=G^{(k)}(\bar x)$ for $k=1,\dots,N$ and $F^{(N)}\le G^{(N)}$ on $[\underline x,\bar x]$ .

Risk attitudes. The decision maker is $N$ th-degree risk averse if $(-1)^{N+1}U^{(N)}\ge0$ . Where $(-1)^{N+1}U^{(N)}>0$ , the relative $(N+1)$ th-degree risk aversion is $r_{N+1}(z)=-zU^{(N+1)}(z)/U^{(N)}(z)$ .

Strong set order. For $D,D'\subset\mathbb R$ , $D\ge_S D'$ if for all $d\in D$ , $d'\in D'$ : $\max\{d,d'\}\in D$ and $\min\{d,d'\}\in D'$ .

Formalization targets

Goal: Proposition 2

Assume the decision maker is $N$ th- and $(N+1)$ th-degree risk averse with $r_{N+1}<N$ on the relevant domain, and let $\tilde X_2$ be an $N$ th-degree risk increase over $\tilde X_1$ , with maximizer sets $B_1^*,B_2^*$ . Then

\text{risky benefit:}\quad B_2^*\le_S B_1^*\ \text{(risk taking)},\qquad B_2^*\ge_S B_1^*\ \text{(risk mitigation)};

\text{risky cost, }N\text{ even:}\quad B_2^*\le_S B_1^*\ \text{(risk taking)},\qquad B_2^*\ge_S B_1^*\ \text{(risk mitigation)},

and for a risky cost with $N$ odd both orderings are reversed.

Milestones

Lemma 1 (Ekern), direction (i) $\Rightarrow$ (ii): an $N$ th-degree risk increase lowers $\mathbb E\,u$ for every $u$ with $(-1)^{N+1}u^{(N)}\ge0$ .
Lemma 3: if $(-1)^Nf(b)^NU^{(N)}(f(b)(X-x_0)+z_0)$ , display (5), is increasing (decreasing) in $b$ for all $X$ , then $B_2^*\ge_S B_1^*$ ( $\le_S$ ).
Display (7): the derivative of (5), stated in the variable $\varphi=f(b)$ .
Display (8): the bracket of (7) equals $(-1)^{N+1}U^{(N)}\,[r_{N+1}-N]-z_0(-1)^{N+2}U^{(N+1)}$ and is negative.
The monotonicity of (5) in $b$ in each of the four problem classes.
Nonemptiness of $B_i^*$ .

Significance

The result. Proposition 2 shows that the familiar statement "more risk, less risk taking" is a statement about risky benefits. For a risky cost, such as an uninsured loss or a random input price, an odd-order risk increase (for example, more downside risk) leads a prudent decision maker to take more risk. The paper uses this to explain apparently conflicting results in the portfolio, coinsurance, output-choice and hedging literatures, and to show that stochastic-dominance assumptions must be chosen differently for benefits and costs (its §5–§6).

Formalizing it. The result is proved in the paper; none of it is machine-checked. A complete development produces a reusable Lean account of Ekern's $N$ th-degree risk order through iterated distribution functions, the integration-by-parts identity behind Lemma 1, and a template for transferring a distributional order through Topkis's monotonicity theorem to argmax sets in the strong set order. Topkis's theorem is already proved on the platform and is included as a reference item.

Difficulty

The calculus steps (7)–(8) are short; the weight sits elsewhere. Lemma 1 is an $N$ -fold integration by parts of a Lebesgue–Stieltjes integral against a distribution function with possible atoms, with boundary terms that vanish only by Definition 2(i). Lemma 3 must turn a pointwise monotonicity of (5) into increasing differences of $b\mapsto\mathbb E\,U(z(\tilde X_i,b))$ in $(b,i)$ , and then into a strong-set-order comparison of possibly set-valued maximizers. A tempting shortcut, differentiating (5) in $b$ , is unavailable: $f$ is only continuous. The sign conditions are then needed at payoffs lying between those of two decisions, not only at the payoffs themselves, which is why the relevant domain must be an interval.

Formalization scope

Random variables are represented by their laws: probability measures on $\mathbb R$ concentrated on $[\underline x,\bar x]$ . Distribution functions are Mathlib's right-continuous cdf. The iterated CDF is indexed from $0$ : iterCDF μ xl k is the page's $F^{(k+1)}$ .
$U:\mathbb R\to\mathbb R$ is $C^{N+1}$ and increasing. The "relevant domain" is an order-connected set $D$ containing every payoff; the risk attitudes are imposed only on $D$ . A $C^{N+1}$ utility on a neighbourhood of the compact payoff range extends to $\mathbb R$ , so this loses nothing.
$N$ th-degree risk aversion is strict, $(-1)^{N+1}U^{(N)}>0$ on $D$ , because Definition 4 defines $r_{N+1}$ only there; "below $N$ " is strict.
Added hypothesis $z_0\ge0$ . The paper's sign argument for (8) marks $z_0(-1)^{N+2}U^{(N+1)}$ as non-negative, which assumes it; without it the bracket of (7) can be positive.
"Increasing" and "decreasing" are weak. Parity is Mathlib's Even/Odd.
The strong set order is the published Supermodularity.Lattices.InducedSetOrder; $B_2^*\le_S B_1^*$ is InducedSetOrder B₂ B₁.
The goal does not assume that (5) is monotone, nor that $B_i^*$ is nonempty: monotonicity of (5) is a milestone derived from the risk attitudes, and nonemptiness is a separate milestone, so the strong-set-order conclusions are not vacuous on the intended instances. A formalization that put the monotonicity of (5) among the goal's hypotheses would reduce it to Lemma 3 and is excluded.
Not formalized here: Proposition 1 ( $(N+1)$ th-degree risk neutrality), Lemma 2 (mirror images), the $(M/N)$ stochastic-dominance extension (Definitions 5–6, Lemma 4, Theorems 1–2) and the applications (Corollaries 1–5).

Contributions are welcome on any milestone; a proof of Lemma 1 is reusable well beyond this mission, for any statement about Ekern's risk orders.

Selected references

M. Menegatti, R. Peter, Changes in risky benefits and in risky costs: a question of the right order, Management Science 68(5):3625–3634, 2022. https://doi.org/10.1287/mnsc.2021.4081
S. Ekern, Increasing Nth degree risk, Economics Letters 6(4):329–333, 1980. https://doi.org/10.1016/0165-1765(80)90005-1
L. Eeckhoudt, H. Schlesinger, Changes in risk and the demand for saving, Journal of Monetary Economics 55(7):1329–1336, 2008. https://doi.org/10.1016/j.jmoneco.2008.08.004
D. C. Nocetti, Robust comparative statics of risk changes, Management Science 62(5):1381–1392, 2016. https://doi.org/10.1287/mnsc.2015.2190
G. Dionne, L. Eeckhoudt, C. Gollier, Increases in risk and linear payoffs, International Economic Review 34(2):309–319, 1993.
M. Rothschild, J. E. Stiglitz, Increasing risk: I. A definition, Journal of Economic Theory 2(3):225–243, 1970. https://doi.org/10.1016/0022-0531(70)90038-4
D. M. Topkis, Minimizing a submodular function on a lattice, Operations Research 26(2):305–321, 1978. https://doi.org/10.1287/opre.26.2.305

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Combinatorics Graph Theory Operations Research+1·Captain: mikedeng1

Semidefinite Approximations for Bicliques and Bi-Independent Pairs: For Every Bipartite Graph, h(G) ≤ ½√g(G) ≤ h₁(G) ≤ ½√g₁(G) ≤ ¼α(G)Research Paper

Motivation

A biclique in a graph is a pair of disjoint vertex sets $(A,B)$ in which every vertex of $A$ is adjacent to every vertex of $B$ . Maximum-edge bicliques model problems in product manufacturing lines and formal concept analysis; balanced bicliques arise in VLSI design and in the analysis of biological data; and the maximum product $|A|\cdot|B|$ over cross-intersecting set families, an extension of the Erdős–Ko–Rado theorem studied by Pyber and by Suda and Tanaka, is an instance of the same parameter. Deciding whether a bipartite graph has a biclique with at least $k$ edges is NP-complete (Peeters, 2003), so polynomial-time upper bounds are of interest. The ratio parameter $h$ was introduced by Vallentin (2020) to bound product-free sets in finite groups.

In the complement of a bipartite graph a biclique becomes a biindependent pair: no edge between $A$ and $B$ . Laurent, Polak and Vargas (arXiv:2302.08886, Mathematics of Operations Research, 2025) study the product parameter $g(G)$ and the ratio parameter $h(G)$ of biindependent pairs in bipartite graphs, and bound them by semidefinite programs from the Lasserre hierarchy, by analogy with Lovász's theta number for the stability number. Their Proposition 1.2 orders these bounds in a single chain; it is the result this mission formalizes.

Setting

A bipartite graph $G=(V_1\cup V_2,E)$ has finite vertex sets $V_1,V_2$ and edges only between them; $V=V_1\cup V_2$ . A bipartite biindependent pair is a pair $(A,B)$ with $A\subseteq V_1$ , $B\subseteq V_2$ and no edge $\{i,j\}$ with $i\in A$ , $j\in B$ . Over all such pairs define

\alpha(G)=\max(|A|+|B|),\qquad g(G)=\max|A|\cdot|B|,\qquad h(G)=\max\frac{|A|\cdot|B|}{|A|+|B|},

where the trivial pair $(\emptyset,\emptyset)$ contributes $0$ to $h$ . The parameter $\alpha(G)$ is the stability number of $G$ . The balanced versions $\alpha_{\mathrm{bal}}$ , $g_{\mathrm{bal}}$ , $h_{\mathrm{bal}}$ take the same maxima over pairs with $|A|=|B|$ .

Let $J$ be the all-ones matrix and $C=\tfrac12\begin{pmatrix}0&J\\J&0\end{pmatrix}$ , indexed by $V$ , so that $x^{\mathsf T}Cx=(\sum_{i\in V_1}x_i)(\sum_{j\in V_2}x_j)$ . For a matrix $X$ indexed by $V$ write $\langle C,X\rangle=\sum_{i,j}C_{ij}X_{ij}$ . The first-order semidefinite bounds are

g_1(G)=\max\Bigl\{\langle C,X\rangle:\begin{pmatrix}1&\mathrm{diag}(X)^{\mathsf T}\\ \mathrm{diag}(X)&X\end{pmatrix}\succeq0,\ X_{ij}=0\text{ if }\{i,j\}\in E\Bigr\},

h_1(G)=\max\{\langle C,X\rangle:X\succeq0,\ \mathrm{Tr}(X)=1,\ X_{ij}=0\text{ if }\{i,j\}\in E\}.

For an arbitrary graph, replacing the objective by $\mathrm{Tr}(X)$ in the first program gives the Lasserre bound $\mathrm{las}_1(G)$ , and by $\langle J,X\rangle$ in the second gives the theta number $\vartheta(G)$ . With $S_G$ the symmetric matrices supported on the edges of $G$ (zero diagonal), the dual of $h_1$ is $\min\{\lambda:\lambda I+Z-C\succeq0,\ Z\in S_G\}$ .

Formalization targets

Goal: Proposition 1.2 (p. 5)

h(G)\le\tfrac12\sqrt{g(G)}\le h_1(G)\le\tfrac12\sqrt{g_1(G)}\le\tfrac14\alpha(G)\qquad\text{for every bipartite graph }G.

Milestones

Lemma 1.1 (p. 2): $\tfrac14\alpha_{\mathrm{bal}}=\tfrac12\sqrt{g_{\mathrm{bal}}}=h_{\mathrm{bal}}\le h\le\tfrac12\sqrt g\le\tfrac14\alpha$ , with $h=\tfrac14\alpha\iff\tfrac12\sqrt g=\tfrac14\alpha\iff\alpha=\alpha_{\mathrm{bal}}$ .
Eq. (25) (p. 12): $\mathrm{las}_1(G)=\vartheta(G)$ for every finite graph.
p. 12: $\vartheta(G)=\alpha(G)$ for bipartite $G$ .
Lemma 3.3 (p. 13): $h_1(G)=\min\{\lambda:\lambda I+Z-C\succeq0,\ Z\in S_G\}$ , the minimum attained.
$\tfrac12\sqrt{g(G)}\le h_1(G)$ (§3.3, p. 16).
Relations (40)–(41) (p. 16): an optimal $X$ of $h_1$ satisfies $\sum_{j\in V_2}X_{ij}=2h_1(G)X_{ii}$ for $i\in V_1$ , and symmetrically.
$h_1(G)\le\tfrac12\sqrt{g_1(G)}$ (§3.3, pp. 16–17).
$h_1(G)\le\tfrac14\alpha(G)$ (§3.3, p. 18).
$g_1(G)\le\alpha(G)\,h_1(G)$ (§3.3, p. 18).

Milestones 1, 5, 7 and 8–9 together give the goal.

Significance

The chain says that the semidefinite bound $h_1$ , designed for the ratio parameter $h$ , also bounds $\tfrac12\sqrt g$ , and is never weaker than $\tfrac12\sqrt{g_1}$ , the bound one obtains from the semidefinite relaxation of $g$ itself; the inequality $h_1<\tfrac12\sqrt{g_1}$ can be strict (for $K_{n,n}$ minus a perfect matching, $n\ge5$ ). Since $g(G)\le 4h_1(G)^2\le g_1(G)$ , the program $h_1$ gives a polynomial-time bound on the maximum product that is at least as good as $g_1$ . The paper builds on this its eigenvalue bounds for regular bipartite graphs (Proposition 1.3) and a sharpened Gowers bound on product-free sets in groups.

The result is proved in the paper. It has not been formalized; nor, to the knowledge of the mission, has Lovász's theta number, its identity with the first Lasserre bound, or its value on bipartite graphs. A completed mission adds a machine-checked theta number with these two classical facts, and the first formal treatment of semidefinite bounds for bicliques.

Difficulty

The two outer inequalities are elementary counting (Lemma 1.1). The inequality $\tfrac12\sqrt g\le h_1$ needs the dual program (32), hence strong duality for (32)–(33), which is not automatic for semidefinite programs. The inequality $h_1\le\tfrac12\sqrt{g_1}$ is the hard step: an optimal solution of $h_1$ is not feasible for $g_1$ , since the bordered constraint of $g_1$ ties the diagonal of $X$ to a border row, and simple scalings do not preserve the value. The paper calls its argument "a bit technical" and compares it with the classical proof of $\mathrm{las}_1\ge\vartheta$ . The last two inequalities need $\vartheta(G)=\alpha(G)$ for bipartite $G$ , a perfect-graph fact the paper cites rather than proves, and the identity $\mathrm{las}_1=\vartheta$ of two formulations of the theta number, also cited.

Formalization scope

The graph is a relation E : V1 → V2 → Prop between two finite types; the vertex type is V1 ⊕ V2 and bipGraph E is the associated SimpleGraph, with no edges inside a side. Either side may be empty; no further hypothesis is assumed.
$\alpha,g,h$ and the balanced parameters are Finset.sup' maxima, as real numbers, over the finite nonempty set of biindependent pairs; $0/0=0$ at $(\emptyset,\emptyset)$ is the paper's convention that trivial pairs have value $0$ .
$X\succeq0$ is Matrix.PosSemidef (which includes symmetry); the bordered matrix is indexed by Unit ⊕ V. SDP maxima are sSup and the minimum (32) is sInf over real numbers. All feasible sets are bounded, so these are the paper's values (all are $0$ when $V$ is empty).
$g_1$ and $h_1$ are defined by the SDP formulations (8) and (9) that the paper states on p. 5, not by the Lasserre hierarchy (29)–(30). Lemma 3.3 is therefore the equality of programs (32) and (33) with attainment; relations (40)–(41) are stated for the matrix $X$ rather than its Gram vectors.
$\vartheta$ and $\mathrm{las}_1$ are defined for an arbitrary finite SimpleGraph.
A formalization in which $h_1$ or $g_1$ reduces to $h$ or $g$ by unfolding, in which $C$ carries $J$ instead of $\tfrac12J$ , or in which the trace or border constraint is dropped (turning a value into the junk $0$ of an unbounded sSup) is not this theorem; the definitions above rule these out.

Needed infrastructure: positive semidefinite matrices, principal submatrices and Schur complements (in Mathlib), Gram factorisations, a strong duality theorem for semidefinite programs, and the stability number of bipartite graphs. The theta number, its two formulations and $\vartheta=\alpha$ on bipartite graphs are reusable well beyond this mission. Contributions are welcome on any milestone, and on intermediate lemmas such as weak duality for (32)–(33).

Selected references

M. Laurent, S. Polak, L. F. Vargas, Semidefinite approximations for bicliques and biindependent pairs, Mathematics of Operations Research, 2025; arXiv:2302.08886v2. https://arxiv.org/abs/2302.08886
L. Lovász, On the Shannon capacity of a graph, IEEE Transactions on Information Theory 25 (1979), 1–7. https://doi.org/10.1109/TIT.1979.1055985
M. Grötschel, L. Lovász, A. Schrijver, Geometric Algorithms and Combinatorial Optimization, Springer, 1988. https://doi.org/10.1007/978-3-642-97881-4
J. B. Lasserre, An explicit exact SDP relaxation for nonlinear 0-1 programs, IPCO 2001, LNCS 2081, 293–303. https://doi.org/10.1007/3-540-45535-3_23
R. Peeters, The maximum edge biclique problem is NP-complete, Discrete Applied Mathematics 131 (2003), 651–654. https://doi.org/10.1016/S0166-218X(03)00333-0

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Combinatorics Operations Research·Captain: mikedeng1

How to Design a Stable Serial Knockout Competition: Scaling a Degree-Ordered Knockout Seeding on GF(2^k) by Each Nonzero Field Element Yields a Stable SKC on 2^k PlayersResearch Paper

Motivation

A knockout tournament for $n = 2^k$ players is played in $k$ rounds: players are placed at the leaves of a complete binary tree, adjacent players meet in round 1, winners of adjacent matches meet in round 2, and the winner of round $k$ , the final, wins the tournament. Who can meet whom, and in which round, is fixed entirely by the seeding, the order in which the players are placed at the leaves.

A serial knockout competition (SKC) runs $n - 1$ knockout tournaments with the same players and different seedings. The format is used in practice: since 2022 the Professional Darts Corporation's Premier League of Darts, with 8 players, consists of knockout tournaments, two blocks of seven of which have seedings fixed in advance by the organiser. For such a competition to be fair, no pair of players should be favoured or disadvantaged by the seedings: every pair of players should be able to meet equally often in each round. Lambers, Pendavingh and Spieksma call such a competition stable and ask whether stable SKCs exist for every number of players $n = 2^k$ (arXiv:2211.02513, published in Mathematics of Operations Research, 2025). For $n = 8$ they give a stable SKC derived from the Fano plane; for general $n = 2^k$ they give an explicit construction from the finite field $\mathrm{GF}(2^k)$ . This mission formalizes the general construction.

Setting

Fix $k \ge 1$ and a set $P$ of $2^k$ players. A knockout tournament $T$ on $P$ is a seeding: a bijection from the leaf positions $0, 1, \dots, 2^k - 1$ , read left to right, to $P$ .

For players $x, x'$ and a round $i \in \{1, \dots, k\}$ write $v_T(x, x') = i$ if $x$ and $x'$ can meet in round $i$ of $T$ , that is, they meet there if both win all their earlier matches (Definition 1.1). For leaf positions $p, p'$ this means that $p$ and $p'$ lie in the same block of $2^i$ consecutive leaves but in different blocks of $2^{i-1}$ leaves.

A finite family $\mathcal T$ of tournaments on $P$ is stable in round $i$ if there is a number $c_i$ with

\#\{T \in \mathcal T : v_T(x, x') = i\} = c_i

for all pairs of distinct players $x, x'$ , and stable if it is stable in every round $i = 1, \dots, k$ (Definition 1.2). An SKC is a family of $2^k - 1$ knockout tournaments (Definition 1.3). Since a player can meet $2^{i-1}$ opponents in round $i$ of each tournament and has $2^k - 1$ opponents, a stable SKC necessarily has $c_i = 2^{i-1}$ .

The construction uses the field $\mathrm{GF}(2^k) \cong \mathbb Z_2[X]/(q)$ , for an irreducible $q$ of degree $k$ with root $\alpha$ . Every $x \in \mathrm{GF}(2^k)$ is uniquely $x = \sum_{i=0}^{k-1} x_i \alpha^i$ , and its degree is $d(x) = \max\{i : x_i \ne 0\}$ . For a tournament $T$ whose players are the elements of $\mathrm{GF}(2^k)$ and a nonzero $z$ , the tournament $T(z)$ is obtained from $T$ by replacing each player $x$ with $zx$ .

Formalization targets

Goal: Theorem 3.1

Let $T$ be a knockout tournament on the elements of $\mathrm{GF}(2^k)$ with $v_T(x, y) = 1 + d(x - y)$ for all distinct $x, y$ . Then

\mathcal T = \{T(z) : z \in \mathrm{GF}(2^k) \setminus \{0\}\}

is a stable SKC, and for all distinct players $x, x'$ and every round $i = 1, \dots, k$ ,

\#\{T' \in \mathcal T : v_{T'}(x, x') = i\} = 2^{i-1}.

Milestones

Lemma 3.1 (p. 10): there is a knockout tournament $T$ on $\mathrm{GF}(2^k)$ with $v_T(x, y) = 1 + d(x - y)$ for all distinct $x, y$ .
Relabelling (§3.3, p. 11): $v_{T(z)}(x, y) = v_T(z^{-1}x, z^{-1}y)$ .
Translation and the degree formula (§3.3, p. 11): under the hypothesis of the goal, $v_{T(z)}(x, y) = v_{T(z)}(0, x - y)$ and $v_{T(z)}(0, y) = 1 + d(z^{-1}y)$ .
The degree count (proof of Theorem 3.1, p. 11): $\#\{r \in \mathrm{GF}(2^k) : r \ne 0,\ 1 + d(r) = i\} = 2^{i-1}$ for $i = 1, \dots, k$ .

Significance

The theorem settles the existence question for every $n = 2^k$ , with an explicit family: the multiplicative group of $\mathrm{GF}(2^k)$ acting on one degree-ordered seeding. A stable SKC gives every pair of players the same number of potential encounters in every round, so a tournament organiser can schedule a series of knockout tournaments without favouring any pair. The paper uses the construction to tabulate stable SKCs for 16 and 32 players.

The result is proved in the paper; none of it is formalized. The mission produces a machine-checked definition of balanced knockout tournaments and of their meeting rounds, a reusable notion of stability for families of tournaments, and a formal proof of the construction. The counting identity for elements of $\mathrm{GF}(2^k)$ of given degree with respect to a power basis is reusable on its own.

Difficulty

The combinatorics is elementary once the right tournament is in hand; the difficulty is in connecting three descriptions of the same object. The meeting round is a property of leaf positions in a binary tree; the degree $d$ is a property of coordinates in a power basis of a field; and the relabelling $T(z)$ is a multiplicative operation, which does not preserve coordinates. The obvious attempt, a seeding that orders the field elements by their coordinates, gives the identity $v_T(x, y) = 1 + d(x - y)$ for one tournament only. The step that needs care is the passage from this single tournament to the whole family: multiplication by $does not preserve the degree, so the meeting rounds of (z)$ differ from those of $, and stability becomes a statement about how the degree of ^{-1}(x - y)$ is distributed as $ ranges over the nonzero elements.

Formalization scope

A tournament on $2^k$ players of type P is a seeding Seeding k P := Fin (2 ^ k) ≃ P. The meeting round is the predicate CanMeetIn T x x' i (leaf blocks of size $2^i$ and $2^{i-1}$ , with $1 \le i \le k$ ), not a function, so it has no junk value at $x = x'$ .
Families of tournaments are indexed, 𝒯 : ι → Seeding k P with [Fintype ι]; IsStableSKC requires Fintype.card ι = 2 ^ k - 1 and stability in every round $1 \le i \le k$ . The goal's family is indexed by the units Fˣ.
$\mathrm{GF}(2^k)$ with its root $\alpha$ is a field F with a power basis pb : PowerBasis (ZMod 2) F; $k$ is pb.dim, $\alpha$ is pb.gen, and the coordinates $x_i$ are pb.basis.repr x i. deg pb 0 = 0 is a junk value; every statement uses $d$ only at nonzero arguments, and the hypothesis on $T$ is required only for distinct players, since the page leaves $d(0)$ undefined.
$T(z)$ is defined as the relabelling scale T z of the seeding by $x \mapsto zx$ , not by the formula $1 + d(z^{-1}(x - y))$ ; the formula is milestone 3.
The proof of Theorem 3.1 prints the count as $2^i$ ; the formalization states $2^{i-1}$ , in agreement with $c_i = 2^{i-1}$ on p. 4 and with the count of elements of degree $i - 1$ .
No assumption $k \ge 2$ is made.

The goal states the explicit count $2^{i-1}$ and the family size $2^k - 1$ , so it cannot be satisfied by a degenerate meeting predicate or by an empty family; Lemma 3.1 shows that its hypothesis on $T$ is satisfiable for every power basis, so it is not vacuous.

Contributions welcome: the coordinate lemmas for power bases over ZMod 2 (the bijection with binary expansions of leaf positions), the milestones in any order, and the goal.

The $n = 8$ construction from the Fano plane (Theorem 2.1, Lemma 2.1) and the tables of §4 are not part of this mission.

Selected references

R. Lambers, R. Pendavingh, F. Spieksma, How to Design a Stable Serial Knockout Competition, Mathematics of Operations Research, 2025; preprint arXiv:2211.02513v1, 2022. https://arxiv.org/abs/2211.02513
R. Lidl, H. Niederreiter, Finite Fields, Encyclopedia of Mathematics and its Applications 20, Cambridge University Press, 1997. https://doi.org/10.1017/CBO9780511525926

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Operations Research Optimization·Captain: Shuze Chen

Scheduling Parallel Machines On-Line: Removing advance knowledge of release datesResearch Paper

Motivation

A scheduler in a computing system may know how to allocate a fixed collection of jobs efficiently while having no reliable forecast of future arrivals. A guarantee for jobs available at the outset does not immediately answer whether decisions remain effective when new work appears during execution. Unknown release dates capture this distinction: a job becomes available and visible only at its release time. Shmoys, Wein and Williamson separate this information restriction from uncertainty about processing requirements and give a general conversion between the two arrival models. The present target is their Theorem 2.1, rather than the independent unknown-size conversion later in the paper. Source, §2.

The question concerns both efficient computation and execution on physical resources. A short schedule selected with knowledge of future jobs is not an online algorithm. Conversely, an online rule can make legitimate decisions yet produce schedules much longer than the best off-line schedule. The target retains both requirements and asks how much a scheduler's makespan guarantee deteriorates when advance knowledge of arrivals is removed.

Setting

A job is a finite processing task using some of a fixed finite collection of exclusive resources. An instance has a finite set $J$ of job identities, job data $d_j$ , and real release dates $r_j\ge0$ . Jobs cannot receive processing before release. Each execution specifies resource allocations over time and must satisfy the job's processing rules. Two jobs cannot occupy the same resource simultaneously. The completion time $C_j$ is the time at which job $j$ has finished, and the makespan is

C_{\max}=\max(\{0\}\cup\{C_j:j\in J\}).

The empty instance therefore has makespan zero. The off-line optimum $C_{\max}^{*}(I)$ is the least makespan among feasible schedules for the original released instance $I$ , with the same resources and processing rules as the online scheduler.

For parallel machines, a job of size $p_j>0$ runs at speed $s_{ij}>0$ on machine $i$ . Identical machines have all speeds equal to one; uniformly related machines have $s_{ij}=s_i$ ; unrelated machines allow job-dependent speeds. Nonpreemptive execution completes a job in one uninterrupted interval. A restart discards a canceled attempt's work. Preemption permits work to accumulate across interruptions and migration. These are separate choices of the environment, not extra powers granted to the transformed scheduler. Source, §1, p. 1314.

The base scheduler $\mathcal A$ receives a finite admitted batch whose local release dates are all zero. Its admission information may include job sizes, or sizes may remain hidden until processing reveals completion. The transformation preserves that choice. In particular, removing advance knowledge of release dates does not remove information about sizes that the base scheduler needs.

Formalization targets

The goal is the complete algorithm transformation of Theorem 2.1. Let $\rho\ge1$ and suppose one uniform polynomial-time scheduler $\mathcal A$ always produces feasible zero-release schedules satisfying

C_{\max}^{\mathcal A}(B)\le\rho C_{\max}^{*}(B).

Then the paper's batching controller supplies one causal polynomial-time scheduler $\mathcal A'$ for unknown release dates such that, for every finite valid released instance,

C_{\max}^{\mathcal A'}(I)\le2\rho C_{\max}^{*}(I).

Both guarantees are pure multiplicative inequalities, without an unspecified additive constant. The constant $2\rho$ is fixed by the published theorem. Source, Theorem 2.1, pp. 1316–1317.

The formal target also requires an execution of the controller, feasibility of its assembled schedule, disjoint batches covering every job, and at most $n=|J|$ nonempty base invocations. If the base computation costs at most $c(N+1)^d$ , the stated metered bound for the wrapper is

n c\bigl(N(I)+1\bigr)^d+10\bigl(N(I)+1\bigr)^2.

Here $N(I)$ counts encoded data cells, including release-date cells for analysis, and the constants $c,d$ and the base code are independent of both the instance and the resource configuration. The Lean statement indexes configurations explicitly, allowing their approximation factor $\rho(v)$ while retaining exactly $2\rho(v)$ in each configuration. The coefficient 10 is a bookkeeping allowance, not an approximation factor. There are no numbered milestone dependencies in the source proof; the only theorem target is Theorem 2.1.

Significance

The result allows zero-release scheduling guarantees to be used in settings with unpredictable arrivals while controlling their loss. Its scope is broader than one machine class. The source explicitly includes shop scheduling and allocation of submeshes, reflecting the independence of the conversion from particular speed formulas or off-line optimization methods. Source, p. 1317.

The mathematical theorem was proved in 1995. The task here is to formalize its statement and eventually its known guarantee. The local draft provides resource-use schedules, processing observations, an interpreted base program, and a controller whose execution is tied to the resulting schedule. Its theorem remains an open Lean proof obligation. These definitions also expose precisely what future formal proofs may assume about release information, processing information, and computation.

Difficulty

A makespan inequality alone cannot certify the result. A construction may accidentally inspect future job data, use an unattainable off-line value as an action, or count physical processing time as computational effort. Each produces a different statement from a polynomial-time online transformation.

Boundary events require equally careful treatment. Arrivals at a completion time must belong to the next batch; an arrival after an idle interval must not be lost; simultaneous arrivals must be collected together in a fixed job-ID order. Empty input and a single batch must remain meaningful without referring to nonexistent preceding batches. The source's compact stage notation is not itself an implementation specification for these cases. None of these concerns permits dropping the computational or information clauses from the target.

Formalization scope

The development uses nonnegative real time, natural-number job identities, finite profiles of half-open resource-use intervals, and a static job-local legality relation. It includes concrete parallel, shop, and mesh environment definitions. Whole-job restriction, time translation, release relaxation, and joining disjoint schedules preserve the intended static semantics. Time-dependent resource availability, external deadlines, and cross-job setup requirements are outside that interface.

The optimum inequality is written against every feasible off-line comparator. Valid jobs each have a finite legal execution; sequential execution after all releases supplies a finite feasible comparator, and makespans are bounded below by zero. Universal comparison is consequently equivalent to comparison with their infimum, including empty input.

The computational convention is an explicit finite-cell real RAM, with a finite program, a costed interpreter, and a processing port restricted to requested allocations and completion feedback. It permits arbitrary real times and does not assert a bit-complexity bound for arbitrary reals. Known and hidden sizes have separate admission encodings. Waiting for completion is a physical operation, whereas arithmetic, queue copying and message handling contribute to computation. No future job count or terminal-arrival notification is given to the controller.

The goal expressly rules out scalar-trace and vacuous-algorithm formalizations: it requires actual feasible resource schedules generated by the fixed controller using the interpreted base program, together with the computational bound. No variant theorem is posed. The recorded two-unit-job example demonstrates sharpness of this batching construction's factor 2, without claiming optimality among all possible online schedulers.

Selected references

David B. Shmoys, Joel Wein and David P. Williamson, Scheduling Parallel Machines On-Line, SIAM Journal on Computing 24(6), 1313–1331, 1995. Published article. Theorem 2.1 and its proof occupy printed pp. 1316–1317, PDF pp. 4–5.

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Operations Research Optimization Probability+1·Captain: Shuze Chen

Modeling, Measuring and Managing Risk XII: The Value-of-Information Process Is a SupermartingaleTextbook

Motivation

Multi-stage stochastic optimization models a decision maker who acts at times $0,1,\dots,T-1$ while a random scenario unfolds, and who may use at each time only the information revealed so far. This nonanticipativity constraint is what separates a genuine multi-stage problem from a collection of deterministic ones. A standard way of measuring its cost, the expected value of perfect information of decision analysis and stochastic programming (Birge & Louveaux 2011, Ch. 4), is to compare the decision maker's optimal value with that of a clairvoyant who sees the whole scenario before deciding.

Pflug and Römisch (Modeling, Measuring and Managing Risk, World Scientific 2007, §5.2) carry out this comparison not only at the root of a scenario tree but at every node, with the expectation replaced by a general acceptability functional $\mathcal A$ . This produces three stochastic processes on the tree: the optimality process, the clairvoyant process, and their difference, the value-of-information process. The book shows that, for functionals that behave linearly under mixing of distributions, the value of information is a supermartingale: the expected remaining value of perfect information decreases as the scenario is revealed, and vanishes at the horizon.

Setting

Let $(\Omega,\mathcal F,P)$ be a probability space and $T\ge0$ . A tree process is a process $(\nu_t)_{t=0,\dots,T}$ , $\nu_t$ taking values in a finite set $N_t$ of nodes, with $\sigma(\nu_t)=\sigma(\nu_s:\,s\le t)$ for every $t$ : the current node determines the whole path that led to it. The root stage $N_0$ consists of one node. The standard filtration is $\boldsymbol{\mathcal F}=(\mathcal F_0,\dots,\mathcal F_T)$ with $\mathcal F_t=\sigma(\nu_t)$ , and the clairvoyant's filtration is $\boldsymbol{\mathcal F}^T=(\mathcal F_T,\dots,\mathcal F_T)$ .

Scenario variables are $\xi_t=f_t(\nu_t)$ , $t=1,\dots,T$ . A profit function $H(x_0,\xi_1,x_1,\dots,x_{T-1},\xi_T)$ , written $H(x,\xi)$ , depends on a decision sequence $x=(x_0,\dots,x_{T-1})$ lying in a decision space $\mathcal X$ . A decision rule $x$ is a random sequence; it is non-anticipative, $x\lhd\boldsymbol{\mathcal F}$ , if each $x_t$ is $\mathcal F_t$ -measurable, and clairvoyant, $x\lhd\boldsymbol{\mathcal F}^T$ , if each $x_t$ is $\mathcal F_T$ -measurable.

A version-independent probability functional $\mathcal R$ assigns a real number to each distribution in its domain $\mathcal D$ of distributions on $\mathbb R$ . For a Markov kernel $K(\cdot\mid v)$ on $\mathbb R$ and a distribution $G$ on $\mathbb R$ , the compound distribution is $K\circ G=\int K(\cdot\mid v)\,dG(v)$ . $\mathcal R$ is compound linear, compound convex or compound concave if

\mathcal R\{K\circ G\}\ \ =,\ \le,\ \ge\ \ \int\mathcal R\{K(\cdot\mid v)\}\,dG(v)

respectively, for all $K$ , $G$ with $K(\cdot\mid v)\in\mathcal D$ for $G$ -almost every $v$ and $K\circ G\in\mathcal D$ , for which the right-hand integral exists ("compound linear on $\mathcal D$ "; the book's Example 2.13 uses exactly this domain-relative form for expected utility).

Fix such a functional $\mathcal A$ (an acceptability functional; larger is better), whose domain contains the law of every feasible profit $H(x,\xi)$ under the conditional probabilities below. For a node $v$ of $\nu_t$ with $P\{\nu_t=v\}>0$ let $P^{\{\nu_t=v\}}=P(\cdot\mid\nu_t=v)$ and let $\mathcal A_{P^{\{\nu_t=v\}}}[Y]$ be $\mathcal A$ applied to the law of $Y$ under it. The conditional problems are

A_t(v)=\max\{\mathcal A_{P^{\{\nu_t=v\}}}[H(x,\xi)] : x\lhd\boldsymbol{\mathcal F},\ x\in\mathcal X\},\qquad C_t(v)=\max\{\mathcal A_{P^{\{\nu_t=v\}}}[H(x,\xi)] : x\lhd\boldsymbol{\mathcal F}^T,\ x\in\mathcal X\}.

At the root these are the multi-stage problem $A_0$ and the clairvoyant's problem $C_0$ . The optimality process is $A_t=A_t(\nu_t)$ , the clairvoyant process $C_t=C_t(\nu_t)$ , and the value-of-information process $D_t=C_t-A_t$ .

Formalization targets

Goal: Corollary 5.3

If $\mathcal A$ is compound linear on its domain, then

D_t=C_t(\nu_t)-A_t(\nu_t),\qquad t=0,\dots,T,

is a supermartingale under $P$ with respect to $(\mathcal F_t)$ .

Milestones

Proposition 2.12. For a sub- $\sigma$ -algebra $\mathcal F_1$ and $\mathcal R(Y\mid\mathcal F_1)$ the functional applied to the conditional distribution of $Y$ given $\mathcal F_1$ : compound linearity gives $\mathcal R(Y)=\mathbb E[\mathcal R(Y\mid\mathcal F_1)]$ , compound convexity $\le$ , compound concavity $\ge$ .
Lemma 5.1. If $\mathcal A$ is compound convex, $(C_t)$ is a submartingale; if compound linear, a martingale.
Lemma 5.2. If $\mathcal A$ is compound convex, $(A_t)$ is a submartingale.

The goal is the weakest statement of the chain: it fixes no constants and asserts only the direction of drift of $D_t$ .

Significance

$D_0=C_0-A_0$ is the value of perfect information of the multi-stage problem, which the book interprets as a multi-period deviation-type risk functional (the multi-period average value-at-risk deviation of its Chapter 3 is an instance). The supermartingale property gives this risk a dynamic meaning: as the scenario is revealed, the remaining value of information decreases in conditional mean, and it is zero at the horizon. The two lemmas separate the effect of information into its two sources: the clairvoyant's value is a fair game for compound linear functionals, while the non-anticipative decision maker's value can only improve in conditional mean, because a decision fixed at time $t$ remains feasible after more is revealed.

The results are proved in the book. No formal version is known: the platform has no item on value-of-information processes, clairvoyant problems or compounding properties of functionals; the closest related items are a complete-information bound in partially observed Markov decision processes and a submartingale property of stable multi-period risk processes, both in other settings. A formalization adds a precise setting (finite scenario trees, conditioning on nodes, attained maxima) in which the chain from Definition 2.11 to Corollary 5.3 is checked.

Difficulty

The supermartingale inequality is a statement about conditional expectations on a tree, while compound convexity is a statement about functionals of mixtures of distributions; the work is to identify the law of the profit under $P^{\{\nu_t=v\}}$ as a compound distribution of the laws under the children's conditional probabilities, with the children encoded as a real mixing parameter. The book's proof of Lemma 5.1 interchanges the maximum and the functional, a step it justifies elsewhere only for monotone functionals, which Lemma 5.1 does not assume; its proof of Lemma 5.2 applies an inequality stated for nonrandom decisions to random ones. A solver should not rely on either step as printed. The martingale half of Lemma 5.1 needs the reverse inequality, for which the optimizers of the child problems must be combined into one feasible clairvoyant decision; the same combination is not available for non-anticipative decisions, which is why the optimality process is only a submartingale. Proposition 2.12 additionally needs regular conditional distributions.

Formalization scope

Lean 4 with Mathlib. Time is Fin (T + 1), stage $0$ being the single root node ([Unique (N 0)]), so $\mathcal F_0$ is trivial. Node types are finite with measurable singletons; this is what makes $P(\cdot\mid\nu_t=v)$ meaningful at every node of positive probability (Mathlib's ProbabilityTheory.cond) and every process integrable. The filtration is a Mathlib Filtration assumed equal to $\sigma(\nu_t)$ at each stage, and sub-, super- and martingales are Mathlib's Submartingale, Supermartingale, Martingale. Decisions take values in a measurable space with measurable singletons, $x\in\mathcal X$ holds for every $\omega$ , and $H$ is jointly measurable. The functional is 𝒜 : Measure ℝ → ℝ, version-independent by construction, evaluated only at probability measures; it is real-valued (the book allows $\overline{\mathbb R}$ ) with a domain 𝒟 : Set (Measure ℝ), and each compounding (in)equality is required only when the mixed and the mixing distributions lie in $\mathcal D$ and $v\mapsto\mathcal R\{K(\cdot\mid v)\}$ is integrable. The tree theorems assume that $\mathcal D$ contains the law of every feasible profit under every conditional probability of a positive-probability node; Proposition 2.12 assumes $Y\in\operatorname{dom}\mathcal R$ and that the conditional distributions are in $\mathcal D$ almost surely. Mixing is over Markov kernels from $\mathbb R$ to $\mathbb R$ (K ∘ₘ G). Proposition 2.12 uses Mathlib's condExpKernel on a standard Borel space and assumes $\mathbb E[\mathcal R(Y\mid\mathcal F_1)]$ exists.

The maxima $A_t(v)$ , $C_t(v)$ are not defined through sSup: the theorems take them as given numbers and assume each is the greatest element of its value set at every node of positive probability. This rules out the trivializing formalization in which a junk supremum of an empty or unbounded set makes the processes constant and the claim empty. The book's remark "in particular if $\mathcal A$ is the expectation" is covered as an instance and not stated separately: the expectation is compound linear on the distributions with a finite mean, and on a finite tree every feasible profit has finitely many values. Taking $\mathcal D$ to be all distributions instead would exclude the expectation, because Lean's Bochner integral of a non-integrable function is $0$ .

Needed infrastructure: finite partitions and conditional expectation on atoms, mixtures of conditional laws, measurable selection of child optimizers on a finite tree, and the disintegration property of condExpKernel. The definitions CompoundLinear/CompoundConvex/CompoundConcave, IsTreeProcess, and the decision-problem sets are reusable for other results of the book on conditional acceptability and multi-stage problems. Proofs of any milestone, and sorry-free instances showing the hypotheses are satisfiable (e.g. a bounded expected utility on a two-stage tree), are welcome.

Selected references

G. Ch. Pflug, W. Römisch, Modeling, Measuring and Managing Risk, World Scientific, 2007, §2.1.2 and §5.2. https://doi.org/10.1142/6478
J. R. Birge, F. Louveaux, Introduction to Stochastic Programming, 2nd ed., Springer, 2011 (value of perfect information and of the stochastic solution). https://doi.org/10.1007/978-1-4614-0237-4

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Modeling, Measuring and Managing Risk XI: The Mean-Variance Efficient Portfolio in Closed FormTextbook

Why mean-variance portfolios

An investor who splits a budget across $M$ risky assets faces a trade-off between expected return and risk. Markowitz proposed measuring risk by the variance of the portfolio return and selecting, for each required expected return, the portfolio of least variance (Markowitz 1952). Tobin added a risk-free asset (Tobin 1958), and Sharpe, Lintner and Mossin turned the resulting one-fund structure into the capital asset pricing model (CAPM), whose beta equation is still the standard first model of expected returns in finance (Sharpe 1964).

This mission formalizes the closed-form part of that theory as presented in §4.2.3 of Pflug and Römisch, Modeling, Measuring and Managing Risk (World Scientific 2007, doi:10.1142/6478), pp. 189–195: the explicit efficient portfolio without a risk-free asset (Proposition 4.8), the solution with a risk-free asset (Proposition 4.9), and the CAPM equation with its variance bound (Proposition 4.10).

Setting

Let $(\Omega,\mathcal F,\mathbb P)$ be a probability space and $\xi=(\xi^{(1)},\dots,\xi^{(M)})$ a random vector of returns, each coordinate with a finite second moment. Returns are profits: larger is better. The vector of expected returns is $r=\mathbb E\xi$ and the covariance matrix is the $M\times M$ matrix $C$ with $C_{ij}=\mathbb{C}\mathrm{ov}(\xi^{(i)},\xi^{(j)})$ ; it is symmetric and positive semidefinite. A portfolio is a weight vector $x\in\mathbb R^M$ (negative weights, i.e. short positions, are allowed) with return $Y_x=\sum_m x_m\xi^{(m)}$ , so that $\mathbb E(Y_x)=r^\top x$ and $\mathbb{V}\mathrm{ar}(Y_x)=x^\top Cx$ . Write $\mathbb 1$ for the all-ones vector.

The CAPM problem without a risk-free asset fixes a required return $\mu$ and asks for

\text{minimize } x^\top Cx\quad\text{subject to}\quad r^\top x=\mu,\ \ \mathbb 1^\top x=1. \tag{4.20}

With a risk-free asset of constant return $r_0$ , a portfolio is a pair $(x_0,\tilde x)$ , where $x_0$ is the holding of the risk-free asset and $\tilde x$ the holdings of the risky assets $\tilde\xi$ , with mean $\tilde r$ and covariance $\tilde C$ . Its return is $Y_x=x_0r_0+\tilde x^\top\tilde\xi$ , and the problem is

\text{minimize } \tfrac12\tilde x^\top\tilde C\tilde x\quad\text{subject to}\quad \tilde r^\top\tilde x+r_0x_0=\mu,\ \ \mathbb 1_M^\top\tilde x+x_0=1. \tag{4.23}

A portfolio is efficient if it solves the respective problem for some $\mu$ . The market portfolio $x_+$ holds nothing in the risk-free asset and $\tilde C^{-1}(\tilde r-r_0\mathbb 1)/\mathbb 1^\top\tilde C^{-1}(\tilde r-r_0\mathbb 1)$ in the risky ones; $Y_+$ is its return, $\mu_+=\mathbb E(Y_+)$ and $\sigma_+^2=\mathbb{V}\mathrm{ar}(Y_+)$ .

Formalization targets

Goal: Proposition 4.8, the efficient portfolio in closed form

Assume $C$ is invertible and $r$ is not a multiple of $\mathbb 1$ . With

a=r^\top C^{-1}r,\quad b=r^\top C^{-1}\mathbb 1,\quad c=\mathbb 1^\top C^{-1}\mathbb 1,\quad d=ac-b^2, \tag{4.22}

the unique solution of (4.20) is

x^*(\mu)=\mu\Big[\frac cd C^{-1}r-\frac bd C^{-1}\mathbb 1\Big]+\Big[\frac ad C^{-1}\mathbb 1-\frac bd C^{-1}r\Big], \tag{4.21}

any two efficient portfolios for different returns span all efficient portfolios by affine combinations (the Two Fund Theorem in this model), and

\mathbb E(Y_\mu)=\mu,\qquad \mathbb{V}\mathrm{ar}(Y_\mu)=\frac{\mu^2c-2\mu b+a}{d}.

The efficient frontier is therefore a parabola in the (variance, mean) plane.

Milestones and further results

Propositions 4.10 (i) and (ii) are milestones; Proposition 4.9 is a further item of the mission on which they stand.

Proposition 4.9. If $\tilde C$ is invertible and $\tilde r\ne r_0\mathbb 1$ , then with $q=(\tilde r-r_0\mathbb 1)^\top\tilde C^{-1}(\tilde r-r_0\mathbb 1)$ the unique solution of (4.23) is

x^*(\mu)=\frac1q\begin{pmatrix}(\tilde r-\mu\mathbb 1)^\top\tilde C^{-1}(\tilde r-r_0\mathbb 1)\\(\mu-r_0)\tilde C^{-1}(\tilde r-r_0\mathbb 1)\end{pmatrix},

and, when $\mathbb 1^\top\tilde C^{-1}(\tilde r-r_0\mathbb 1)\ne0$ , every efficient portfolio is an affine combination of $(1,0,\dots,0)$ and $x_+$ . 2. Proposition 4.10 (i), the CAPM equation (4.25). For every portfolio with $x_0+\mathbb 1^\top\tilde x=1$ ,

\mathbb E(Y_x)=r_0+\beta(x)(\mu_+-r_0),\qquad \beta(x)=\frac{\mathbb{C}\mathrm{ov}(Y_x,Y_+)}{\mathbb{V}\mathrm{ar}(Y_+)}.

Proposition 4.10 (ii). For every such portfolio

\mathbb{V}\mathrm{ar}(Y_x)\ge\Big(\frac{\mathbb E(Y_x)-r_0}{\mu_+-r_0}\Big)^2\sigma_+^2,

with equality if and only if the portfolio is efficient, and, as printed, if and only if $\mathbb{C}\mathrm{orr}(Y_x,Y_+)=1$ .

Significance

Proposition 4.8 is the explicit description of the mean-variance efficient set: the efficient portfolios form a line in weight space, and the minimal variance is a quadratic function of the required return. Proposition 4.9 shows that a risk-free asset turns this line into one through the risk-free portfolio (the one-fund, or separation, property), and Proposition 4.10 derives from it the CAPM relation between a portfolio's expected excess return and its beta, together with the capital-market-line bound. These are the statements on which the later sections of the chapter, and most textbook treatments of portfolio theory, build: risk-return diagrams, the market portfolio, beta as a risk measure.

The results are classical and their proofs are elementary, but they are not in Mathlib, and no Prove2Me item states them; the closest platform item is the multi-period mean-variance mutual fund theorem of Bäuerle and Rieder (MDPFinance.MeanVariance.mutual_fund_theorem), a different, dynamic statement. A formalization makes the hypotheses explicit that the printed proofs leave implicit (invertibility and nondegeneracy conditions, the budget constraint behind "any portfolio") and checks the printed equality conditions, one of which is false as stated.

Difficulty

The Lagrange conditions of the printed proof are necessary conditions only. Turning them into the statement requires showing that the stationary point is the global minimizer and the only one, which uses positive definiteness of $C$ ; the hypothesis gives only invertibility, and positive definiteness has to come from $C$ being a covariance matrix. Dividing by $d$ needs $d>0$ , which is not among the hypotheses and must be derived from " $r$ is not a multiple of $\mathbb 1$ " together with the structure of $C$ . The variance identities require computing variances and covariances of linear combinations of square-integrable random variables. In Proposition 4.10 (ii) the equality case is where the printed statement goes wrong: a portfolio short in the market portfolio attains equality with correlation $-1$ .

Formalization scope

All statements live in the namespace RiskFunctionals.Frontier. Assets are indexed by Fin M; vectors are Fin M → ℝ, matrices Matrix (Fin M) (Fin M) ℝ, and $C^{-1}$ is Mathlib's matrix inverse. The mean vector and covariance matrix are computed from a random vector with MemLp 2 coordinates on a probability space using Mathlib's covariance and variance, so " $C$ invertible" is the printed hypothesis and positive definiteness is a consequence, not an assumption. "The solution" is stated as existence and uniqueness of a global minimizer on the feasible set (IsSolution420, IsSolution423). The divisions by $q$ , by $\mathbb 1^\top\tilde C^{-1}(\tilde r-r_0\mathbb 1)$ and by $d$ are guarded by the printed or implicit nondegeneracy hypotheses; a statement in which $d=0$ or $q=0$ were allowed would make the closed forms the zero vector, so these guards are part of the statement rather than conveniences. The correlation is used only where the portfolio variance is positive, since it is undefined otherwise.

Beyond the paper. One variant item is posed beside the faithful statements: capm_variance_bound_corrected (corrected, finding F4) replaces the equality condition $\mathbb{C}\mathrm{orr}(Y_x,Y_+)=1$ of Proposition 4.10 (ii) by $|\mathbb{C}\mathrm{orr}(Y_x,Y_+)|=1$ ; with one risky asset of mean 2 and variance 1, $r_0=1$ and $x=(2,-1)$ , equality holds while the correlation is $-1$ . The faithful Proposition 4.10 (ii) is therefore expected to be disproved.

A complete development needs the variance and covariance of finite linear combinations of $L^2$ random variables, positive semidefiniteness of covariance matrices, and the minimization of a positive definite quadratic form under linear equality constraints. These are reusable well beyond this mission; proofs of them as separate lemmas are welcome.

Selected references

G. Ch. Pflug and W. Römisch, Modeling, Measuring and Managing Risk, World Scientific, 2007, §4.2.3, pp. 189–195. https://doi.org/10.1142/6478
H. Markowitz, Portfolio Selection, Journal of Finance 7(1), 77–91, 1952. https://doi.org/10.2307/2975974
J. Tobin, Liquidity Preference as Behavior Towards Risk, Review of Economic Studies 25(2), 65–86, 1958. https://doi.org/10.2307/2296205
W. F. Sharpe, Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk, Journal of Finance 19(3), 425–442, 1964. https://doi.org/10.1111/j.1540-6261.1964.tb02865.x
R. C. Merton, An Analytic Derivation of the Efficient Portfolio Frontier, Journal of Financial and Quantitative Analysis 7(4), 1851–1872, 1972. https://doi.org/10.2307/2329621

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Modeling, Measuring and Managing Risk VIII: Dual Representation of Multi-Period Acceptability FunctionalsTextbook

Why multi-period acceptability needs a dual description

Risk management in finance, insurance and energy rarely concerns one payoff at one date. A pension fund, a hydro-power producer or a bank holds an income process $Y=(Y_1,\dots,Y_T)$ , observed together with an increasing flow of information, a filtration $\boldsymbol{\mathcal F}=(\mathcal F_0,\dots,\mathcal F_T)$ . Chapter 3 of G. Ch. Pflug and W. Römisch, Modeling, Measuring and Managing Risk (World Scientific, 2007, doi:10.1142/6478), argues that risk can only be measured for the pair (process, filtration). It axiomatizes multi-period acceptability functionals, the profit-side counterpart of the dynamic risk measures of Artzner, Delbaen, Eber, Heath and Ku (Ann. Oper. Res. 152 (2007)), Riedel (Stoch. Proc. Appl. 112 (2004)) and Ruszczyński and Shapiro (Math. Oper. Res. 31 (2006)).

In one period, dual representations of acceptability functionals as infima over probability densities are what make them usable in stochastic optimization: they turn a risk-averse objective into a min–max problem with an explicit dual set. Section 3.2.1 of the book proves the multi-period analogue. Each version of translation-equivariance corresponds to an explicit set of dual processes.

Setting

Let $(\Omega,\mathcal F,P)$ be a probability space and $\boldsymbol{\mathcal F}=(\mathcal F_t)_{t=0}^T$ a filtration in $\mathcal F$ with $\mathcal F_0=\{\emptyset,\Omega\}$ . Fix $p\in[1,\infty)$ and $q\in(1,\infty]$ with $1/p+1/q=1$ . The process space is $\mathcal Y=\times_{t=1}^TL_p(\mathcal F_t)$ , normed by $\|Y\|_p=(\sum_t\mathbb E|Y_t|^p)^{1/p}$ . Its dual is $\mathcal Z=\times_{t=1}^TL_q(\mathcal F_t)$ , with the pairing $\langle Z,Y\rangle=\sum_{t=1}^T\mathbb E(Y_tZ_t)$ . Functionals take values in $\overline{\mathbb R}=[-\infty,+\infty]$ . The conjugate of $\mathcal A:\mathcal Y\to\overline{\mathbb R}$ is $\mathcal A^+(Z)=\inf\{\langle Z,Y\rangle-\mathcal A(Y):Y\in\mathcal Y\}$ (3.19).

A family $\boldsymbol{\mathcal F}\mapsto\mathcal A(\cdot\,;\boldsymbol{\mathcal F})$ is a multi-period acceptability functional (Definition 3.14) if every member is proper ( $<+\infty$ everywhere, $>-\infty$ somewhere) and:

(MA0) information monotonicity: $\mathcal F_t\subseteq\mathcal F'_t$ for all $t$ implies $\mathcal A(Y;\boldsymbol{\mathcal F})\le\mathcal A(Y;\boldsymbol{\mathcal F}')$ ;
(MA1) predictable translation-equivariance: $\mathcal A(Y_1,\dots,Y_t+C_t,\dots,Y_T)=\mathbb E(C_t)+\mathcal A(Y)$ for every $\mathcal F_{t-1}$ -measurable $C_t\in L_p$ ;
(MA2) concavity, and (MA3) monotonicity: $Y_t\le\tilde Y_t$ a.s. for all $t$ implies $\mathcal A(Y)\le\mathcal A(\tilde Y)$ .

The weaker forms are (MA1′), the same identity for constants $c_t$ only, and (MA1″), the identity for constants in the first period only. A multi-period probability functional is any functional $\mathcal Y\to\overline{\mathbb R}$ that does not take both values $\pm\infty$ (p. 27).

Section 3.1.3.2 adds information to a single-period functional $\mathcal A(Y)=\inf\{\mathbb E(YZ)-A(Z):\mathbb E Z=1,\ Z\in\mathcal Z_{\mathcal A}\}$ (3.7) through its dual extension $\mathcal A(Y;\mathcal F_1)=\inf\{\mathbb E(YZ)-A(Z):\mathbb E(Z\mid\mathcal F_1)=1,\ Z\in\mathcal Z_{\mathcal A}\}$ (3.8).

Formalization targets

Goal: Theorem 3.20 (p. 136)

If $\mathcal A$ is an upper semicontinuous multi-period acceptability functional, then for every $Y\in\mathcal Y$

\mathcal A(Y)=\inf_{Z\in\mathcal Z}\Big\{\sum_{t=1}^T\mathbb E(Y_tZ_t)-\mathcal A^+(Z)\ \Big|\ Z_t\ge0,\ \mathbb E(Z_t\mid\mathcal F_{t-1})=1,\ t=1,\dots,T\Big\}.\tag{3.21}

Conversely, if $\mathcal A$ can be represented in the form (3.21) for some proper $\mathcal A^+:\mathcal Z\to\overline{\mathbb R}$ , then $\mathcal A$ is proper, upper semicontinuous and satisfies (MA0)–(MA3).

Milestones

Proposition 3.10 (pp. 129–130): the dual extension (3.8) is information monotone, reduces to (3.7) for the trivial σ-algebra, equals $\mathbb E(Y)-A(\mathbb 1)$ for $\mathcal F_1$ -measurable $Y$ when $A$ is isotonic w.r.t. concave dominance, and is predictable translation-equivariant.
Eq. (3.20) (p. 136): a proper, concave, upper semicontinuous $\mathcal A$ on $\mathcal Y$ satisfies $\mathcal A(Y)=\inf\{\langle Z,Y\rangle-\mathcal A^+(Z):Z\in\mathcal Z\}$ .
Theorem 3.21 (p. 138): under (MA1′), (MA2), (MA3) and u.s.c., the representation holds over $\{Z_t\ge0,\ \mathbb E Z_t=1\}$ (3.22), together with a converse.
Theorem 3.22 (p. 139): under (MA1″), (MA2), (MA3) and u.s.c., the representation holds over $\{\mathbb E Z_1=1,\ Z_t\ge0\}$ (3.23), together with a converse.

Significance

Theorem 3.20 identifies the dual variables of a multi-period acceptability functional. They are nonnegative processes with $\mathbb E(Z_t\mid\mathcal F_{t-1})=1$ , so their running products $Z_1Z_2\cdots Z_t$ are, when integrable, density processes: changes of measure built period by period along the filtration. The three theorems 3.20–3.22 together show how the strength of translation-equivariance, (MA1), (MA1′) or (MA1″), shrinks or enlarges the dual set. This is the basis on which the book later represents the nested and the multi-period average value-at-risk (§3.3) and polyhedral functionals as linear programs. It also underlies the min–max reformulations of multi-stage risk-averse stochastic programs.

The results are proved in the book. None of them, and none of the axioms (MA0)–(MA3), has a machine-checked statement on the platform or in Mathlib. The related platform items, MultiperiodRisk.Bellman.stable_tfae (stability and the Bellman principle for sets of probabilities) and ConvexRiskFn.Dual.theorem_2_2 (single-period, loss-side, measures as duals), are different statements. Formalizing the chapter fixes the encoding of a filtration-dependent functional, which later chapters on nested functionals need.

Difficulty

The forward directions rest on Fenchel–Moreau duality on a product of $L_p(\mathcal F_t)$ spaces, which Mathlib does not provide in this form: the dual of the subspace of $\mathcal F_t$ -measurable classes must be identified with $L_q(\mathcal F_t)$ for every $t$ simultaneously, including $p=1$ , $q=\infty$ . The functionals are extended-real-valued, so every step must handle $\pm\infty$ , and the constraint $\mathbb E(Z_t\mid\mathcal F_{t-1})=1$ involves conditional expectations of $L_q$ elements.

Read literally, the statements are not all true:

The converse of Theorem 3.20 claims (MA0), which compares two filtrations, from a representation at each filtration separately. A family can be of the form (3.21) at two nested filtrations and still violate (MA0).
The converses' conclusion " $\mathcal A$ is proper" fails when $\mathcal A^+$ is proper but $-\infty$ on the whole constraint set.
Theorem 3.22 depends on the convention that a probability functional does not take both $\pm\infty$ .

The faithful statements are kept as printed; a solver should expect to disprove those parts.

Formalization scope

The filtration is a Mathlib Filtration indexed by Fin (T+1). A process is a function on Fin T whose component $t$ lies in lpMeas ℝ ℝ (𝔽 t.succ) p P, the subspace of $L_p(\Omega,\mathcal F,P)$ of classes with an $\mathcal F_{t+1}$ -measurable version. Lean's index $t$ is therefore the book's $t+1$ . The product topology on this type is the norm topology of $\|\cdot\|_p$ . In Proposition 3.10, $\mathcal Z_{\mathcal A}$ is the supergradient set $\{Z: A(Z)>-\infty\}$ of $A$ , as the book defines it on p. 43, not an independent set. Values are in EReal, and every infimum is taken in EReal over an explicit constraint set, so an empty set gives $+\infty$ , the book's value. Constraints hold almost surely, and conditional expectations are Mathlib's condExp. Concavity is the EReal inequality $\lambda\mathcal A(Y)+(1-\lambda)\mathcal A(\tilde Y)\le\mathcal A(\lambda Y+(1-\lambda)\tilde Y)$ .

Because (MA0) compares filtrations, $\mathcal A$ in Theorem 3.20 is a family indexed by all filtrations with trivial $\mathcal F_0$ . Part 1 assumes Definition 3.14 for the family, and part 2 assumes a representation at every filtration. Theorems 3.21 and 3.22 concern one filtration and keep the probability-functional convention as a hypothesis.

A trivializing formalization is ruled out: the infima in (3.21)–(3.23) are over the printed constraint sets, never all of $\mathcal Z$ (which would collapse them into (3.20)), and $\mathcal A^+$ in part 1 is the conjugate (3.19), not an arbitrary witness.

Not stated: Propositions 3.8 and 3.9 (expected conditional functionals, which need the conditional acceptability mappings of §2.3) and Theorem 3.23 (local Lipschitz continuity and Hadamard directional differentiability).

Beyond the paper. One corrected variant is posed beside the faithful items. multiperiod_representation_corrected drops (MA0) and properness from the converse of Theorem 3.20. A two-filtration family built from $\mathbb E Y_1+\mathbb E Y_2$ and the expected conditional $\mathrm{AV@R}_{1/2}$ violates (MA0), and $\mathcal A^+=-\infty$ off $Z=0$ gives $\mathcal A\equiv+\infty$ .

Reusable beyond this mission: Fenchel–Moreau duality on products of lpMeas spaces, and the identification of $\mathcal F_t$ -measurable duals. Contributions of either, or of proofs and disproofs of the printed converses, are welcome.

Selected references

G. Ch. Pflug, W. Römisch, Modeling, Measuring and Managing Risk, World Scientific, 2007. https://doi.org/10.1142/6478
P. Artzner, F. Delbaen, J.-M. Eber, D. Heath, H. Ku, Coherent multiperiod risk adjusted values and Bellman's principle, Annals of Operations Research 152, 2007. https://doi.org/10.1007/s10479-006-0132-6
F. Riedel, Dynamic coherent risk measures, Stochastic Processes and their Applications 112, 2004. https://doi.org/10.1016/j.spa.2004.03.004
A. Ruszczyński, A. Shapiro, Conditional risk mappings, Mathematics of Operations Research 31, 2006. https://doi.org/10.1287/moor.1060.0204
A. Ruszczyński, A. Shapiro, Optimization of convex risk functions, Mathematics of Operations Research 31, 2006. https://doi.org/10.1287/moor.1050.0186

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Modeling, Measuring and Managing Risk VI: Choquet Representation of Distortion FunctionalsTextbook

Motivation

A distortion functional values an uncertain profit $Y$ by integrating its quantiles against a reweighting of the probability scale: small (bad) quantiles receive more weight, large ones less. The construction appears in actuarial pricing (the distortion premium principles of Denneberg and Wang), in decision theory (Yaari's dual theory of choice), and in finance as spectral risk measures (Acerbi) or weighted values-at-risk (Cherny). Its central structural fact is that every concave distortion is a mixture of average values-at-risk, the lower-tail averages $\mathbb{AV@R}_\alpha$ : this Choquet representation reduces properties of the whole class (concavity, monotonicity with respect to second-order dominance, comonotone additivity) to those of one family. The representation was proved by Acerbi (2002), Jones and Zitikis (2003) and Pflug (2006). This mission formalizes it as Proposition 2.64 of Pflug and Römisch, Modeling, Measuring and Managing Risk (World Scientific 2007, §2.4.2), together with the lemmas of that section.

Setting

All random variables are profit variables (larger is better) on a probability space $(\Omega,\mathcal F,\mathbb P)$ . For a random variable $Y$ with distribution function $G(u)=\mathbb P\{Y\le u\}$ and left limit $G(u-)=\mathbb P\{Y<u\}$ , the quantile function is the left-continuous inverse

G^{-1}(p)=\inf\{u: G(u)\ge p\},\qquad 0<p\le1,

and the average value-at-risk at level $0<\alpha\le1$ is

\mathbb{AV@R}_\alpha\{G\}=\frac1\alpha\int_0^\alpha G^{-1}(u)\,du .

A distortion function $H$ is a bounded, right continuous, increasing function on $[0,1]$ , the distribution function of a non-negative measure $dH$ on $[0,1]$ (not necessarily a probability measure). Writing $\mathcal G_H$ for the distributions $G$ for which $G^{-1}$ is $dH$ integrable, the distortion acceptability functional is

\mathcal A_H\{G\}=\int_0^1 G^{-1}(p)\,dH(p),\qquad G\in\mathcal G_H,

also written $\mathcal A_H(Y)$ . $H(p)=p$ gives the expectation, $H=\mathbb 1_{[\alpha,1]}$ the value-at-risk $G^{-1}(\alpha)$ , and $H(p)=\min(p/\alpha,1)$ the average value-at-risk. $H$ is concave when $H(p)=\int_0^p h(\alpha)\,d\alpha$ with a nonincreasing density $h$ , which may be unbounded near $0$ .

Two orders on random variables enter: $Z\prec_{CXD}Z^*$ (convex order) when $\mathbb E\,U(Z)\le\mathbb E\,U(Z^*)$ for every convex $U$ for which both expectations exist; and $Y_1,Y_2$ are comonotone when their joint distribution function is $\min(G_1(u),G_2(v))$ . A functional is comonotone additive when it is additive on comonotone pairs.

Formalization targets

Goal: the Choquet representation (Proposition 2.64)

If $H$ is concave, there is an increasing function $M$ on $[0,1]$ with $M(0)=0$ and $M(1)=H(1)$ such that, for every $G\in\mathcal G_H$ ,

\mathcal A_H\{G\}=\int_0^1\mathbb{AV@R}_\alpha\{G\}\,dM(\alpha).

Conversely, every comonotone additive functional having such a Choquet representation is a distortion functional. The goal leaves $M$ otherwise unspecified; one $M$ serves every $G$ .

Milestones

Examples 2.63: the three distortions $H(p)=p$ , $\mathbb 1_{[\alpha,1]}(p)$ and $\min(p/\alpha,1)$ give $\mathbb E(Y)=-\int_{-\infty}^0G+\int_0^\infty(1-G)$ , $G^{-1}(\alpha)$ and $\mathbb{AV@R}_\alpha$ .
Lemma 2.62: for $G\in\mathcal G_H$ , with $\bar H(u)=H(1)-H(1-u)$ ,

\int_0^1 G^{-1}(p)\,dH(p)=-\int_{-\infty}^0H(G(u))\,du+\int_0^\infty\bar H(1-G(u))\,du .

Eq. (2.68): for $U$ uniform on $[0,1]$ and independent of $Y$ , the generalized quantile transform $(1-U)G(Y-)+UG(Y)$ is uniform on $[0,1]$ .
Eq. (2.69): for differentiable $H$ with $H(p)=\int_0^ph$ , $\mathcal A_H(Y)=\mathbb E\big[Y\,h\big((1-U)G(Y-)+UG(Y)\big)\big]$ .
Proposition 2.65, as printed: for differentiable $H$ , $\mathcal A_H(Y)=\inf\{\mathbb E(YZ): Z=h(U),\ U\text{ uniform}\}=\inf\{\mathbb E(YZ):Z\prec_{CXD}h(U)\}$ , attained at $Z=h\big((1-U)G(Y-)+UG(Y)\big)$ .

Significance

The Choquet representation identifies the concave distortion functionals with the closed convex cone generated by the average values-at-risk. Downstream, this gives the coherence properties of spectral risk measures, their consistency with second-order stochastic dominance, and, through Proposition 2.65, a dual description by rearrangements of the distortion density that is used to optimize portfolios under such criteria. It also separates the class from the larger family of law-invariant coherent functionals, which are infima of such mixtures over a family of mixing measures (Kusuoka) rather than single mixtures.

The results are classical and proved in the literature; none is machine-checked in this form. A related loss-side layer exists on the platform (MultistageStochastic.distortion_avar_mixture, proved, for a spectrum $\sigma$ on $L_\infty$ with the upper-tail AV@R), but its sign, tail and integrability conventions differ, so it does not state this book's theorem. The formalization adds a quantile calculus with atoms (the left-continuous $G^{-1}$ , $G(u-)$ , the generalized quantile transform) and Lebesgue–Stieltjes integration against distortion measures, both reusable for the rest of the series.

Difficulty

The forward direction is not a finite-dimensional extreme-point argument: $h$ may be unbounded at $0$ , so the mixing measure $dM$ must be obtained as a limit and the exchange of integrals justified for quantile functions that are only $dH$ -integrable, with $G^{-1}$ unbounded at both ends. Distributions with atoms make $G^{-1}(G(u))<u$ on flat stretches, which is exactly where Lemma 2.62 and (2.69) need care.

Proposition 2.65 is false as printed (finding F1): for the differentiable convex $H(p)=p^2$ and $Y$ uniform on $[0,1]$ , $\mathcal A_H(Y)=2/3$ , while $Z=2(1-Y)=h(1-Y)$ gives $\mathbb E(YZ)=1/3$ . Its printed proof uses that $h$ is nonincreasing. The faithful milestone is therefore expected to be disproved; a corrected variant is posed beside it.

Formalization scope

Distributions are probability measures $\mu$ on $\mathbb R$ ; for a random variable, $\mu$ is the image measure of $\mathbb P$ under $Y$ . $G^{-1}$ is computed by sInf; its junk value $0$ (at $p=1$ when $G<1$ everywhere, and at $p\le0$ ) is never charged. $H$ and $M$ are Mathlib StieltjesFunctions; $\mathcal A_H$ integrates over $(0,1]$ against $dH$ . Membership in $\mathcal G_H$ contains $dH$ -integrability of $G^{-1}$ , $H(0)=0$ (the mass $dH$ puts on $p=0$ , where $G^{-1}=-\infty$ ), and finiteness of $G^{-1}(1)$ when $dH$ charges $p=1$ . $M$ vanishes on $(-\infty,0]$ , so $dM$ has no atom at $0$ , and the representation integrates over $[0,1]$ ; $dM$ -integrability of $\alpha\mapsto\mathbb{AV@R}_\alpha$ is part of the conclusion, and the converse is stated on integrable $Y$ for which that integrability holds. Infima in Proposition 2.65 are greatest lower bounds of value sets restricted to integrable products, with attainment stated separately. Uniform variables are measurable with law Lebesgue measure on $[0,1]$ , and $U$ is independent of $Y$ in (2.68)–(2.69), as Proposition 1.3 requires. The book's $V_Y=(1-U)[G(Y-)-G(Y)]$ has a sign typo and is used as $(1-U)[G(Y)-G(Y-)]$ .

A trivializing formalization would integrate a quantile function that is not $dH$ - or $dM$ -integrable, where Lean's integral returns $0$ , or take an infimum of an empty set; every such integral is guarded by an integrability hypothesis or an asserted integrability, and every infimum is an IsGLB of a set shown nonempty by the attainment clause.

Beyond the paper. One variant: [Corrected] Proposition 2.65 adds the hypothesis " $H$ concave" and changes nothing else (finding F1, the counterexample above). A further finding (F2) records that the converse half of Proposition 2.64 does not use comonotone additivity; no variant is posed for it.

Contributions welcome: proofs of the milestones, a disproof of the printed Proposition 2.65, and reusable lemmas on quantile functions with atoms and on Stieltjes integration on $[0,1]$ .

Selected references

G. Ch. Pflug, W. Römisch, Modeling, Measuring and Managing Risk, World Scientific, 2007. https://doi.org/10.1142/6478
C. Acerbi, Spectral measures of risk: a coherent representation of subjective risk aversion, Journal of Banking & Finance 26 (2002). https://doi.org/10.1016/S0378-4266(02)00281-9
B. L. Jones, R. Zitikis, Empirical estimation of risk measures and related quantities, North American Actuarial Journal 7 (2003). https://doi.org/10.1080/10920277.2003.10596118
G. Ch. Pflug, On distortion functionals, Statistics & Decisions 24 (2006). https://doi.org/10.1524/stnd.2006.24.1.45
M. E. Yaari, The dual theory of choice under risk, Econometrica 55 (1987). https://doi.org/10.2307/1911158
S. S. Wang, A class of distortion operators for pricing financial and insurance risks, Journal of Risk and Insurance 67 (2000). https://doi.org/10.2307/253675

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Convex Optimization Operations Research Optimization+1·Captain: Shuze Chen

Modeling, Measuring and Managing Risk III: Representations of the Average Value-at-RiskTextbook

Why the average value-at-risk

Banks, insurers and planners who optimize under uncertainty need a single number that measures how bad the bad outcomes of an uncertain profit are. The value-at-risk, a quantile of the profit distribution, is the regulatory standard but ignores how far the worst outcomes fall below it and is not concave. The average value-at-risk (also called conditional value-at-risk, tail value-at-risk or expected shortfall) averages the worst $\alpha$ -fraction of outcomes instead. It is concave, isotonic with respect to second-order stochastic dominance, and, through the minimization formula of Rockafellar and Uryasev (2000), it turns risk-averse stochastic programs into linear or convex programs. It is also the building block of most other version-independent risk functionals: distortion and Kusuoka representations write them as mixtures of average values-at-risk.

The axiomatic view goes back to Artzner, Delbaen, Eber and Heath (1999); the treatment of distributions with atoms, where several plausible definitions disagree, is due to Acerbi and Tasche (2002) and Rockafellar and Uryasev (2002). This mission follows Section 2.2.3 of Pflug and Römisch, Modeling, Measuring and Managing Risk (World Scientific 2007, doi:10.1142/6478), pp. 48–58, together with Proposition 2.74 on p. 109.

Setting

All outcomes are profit variables: larger values are better. Let $(\Omega,\mathcal F,\mathbb P)$ be a probability space and $Y$ an integrable real random variable with distribution function $G(u)=\mathbb P\{Y\le u\}$ and left-continuous quantile function

G^{-1}(p)=\inf\{u : G(u)\ge p\},\qquad 0<p\le1 .

For $0<\alpha\le1$ the average value-at-risk is the lower-tail average

\mathbb{AV@R}_\alpha(Y)=\frac1\alpha\int_0^\alpha G^{-1}(u)\,du ,

so that $\mathbb{AV@R}_1(Y)=\mathbb EY$ , and the average value-at-risk deviation is $\mathbb{AV@R}\mathbb D_\alpha(Y)=\mathbb EY-\mathbb{AV@R}_\alpha(Y)$ . When $G$ jumps at $G^{-1}(\alpha)$ , the two levels

\alpha^+_G=\inf\{G(u):G(u)\ge\alpha\},\qquad \alpha^-_G=\sup\{G(u):G(u)<\alpha\}

satisfy $\alpha^-_G\le\alpha\le\alpha^+_G$ . Write $[y]^-=-\min(y,0)$ and $[y]^+=\max(y,0)$ . Second-order stochastic dominance $Y^{(1)}\prec_{SSD}Y^{(2)}$ means $\mathbb E\,U(Y^{(1)})\le\mathbb E\,U(Y^{(2)})$ for every nondecreasing concave $U$ for which both expectations exist; $Y^{(1)}$ , $Y^{(2)}$ are comonotone if $\mathbb P\{Y^{(1)}\le u, Y^{(2)}\le v\}=\min(G_1(u),G_2(v))$ for all $u,v$ .

In the Lean development, in the namespace RiskFunctionals.AVaR, these objects are distributionFunction, quantileFunction (both of the law P.map Y), averageValueAtRisk P Y α, avarDeviation P Y α, alphaPlus, alphaMinus, SSD and Comonotone.

Formalization targets

Goal: Theorem 2.34

For integrable $Y$ and $0<\alpha\le1$ :

$\displaystyle\mathbb{AV@R}_\alpha(Y)=\max\Big\{x-\frac1\alpha\mathbb E([Y-x]^-) : x\in\mathbb R\Big\}$ , the maximum attained;
$\displaystyle\mathbb{AV@R}_\alpha(Y)=\min\Big\{\mathbb E(YZ) : \mathbb EZ=1,\ 0\le Z\le\frac1\alpha\Big\}$ , the minimum attained;

and, if $G^{-1}(\alpha)$ is finite,

$\displaystyle\mathbb{AV@R}_\alpha(Y)=G^{-1}(\alpha)-\frac1\alpha\mathbb E([Y-G^{-1}(\alpha)]^-)=G^{-1}(\alpha)-\frac1\alpha\int_{-\infty}^{G^{-1}(\alpha)}(G^{-1}(\alpha)-u)\,dG(u)$ ;
$\displaystyle\mathbb{AV@R}_\alpha(Y)=\frac{\alpha^-_G}{\alpha}\mathbb E[Y\mid Y<G^{-1}(\alpha)]+\frac{\alpha-\alpha^-_G}{\alpha}G^{-1}(\alpha)$ ;
$\displaystyle\mathbb{AV@R}_\alpha(Y)=\frac{\alpha^+_G}{\alpha}\mathbb E[Y\mid Y\le G^{-1}(\alpha)]-\frac{\alpha^+_G-\alpha}{\alpha}G^{-1}(\alpha)$ .

The goal is the conjunction of the five parts, stated as printed. $G^{-1}(\alpha)$ is finite for every $\alpha<1$ , and for $\alpha=1$ exactly when $Y$ is essentially bounded above; parts 3–5 presuppose it. At $\alpha=1$ with $Y$ not essentially bounded above, the attainment claim of part 1 is false (see Beyond the paper), so the goal as printed is expected to be disprovable there.

Milestones

p. 49 and (2.18): $\alpha^+_G=G(G^{-1}(\alpha))$ , $\alpha^-_G=G(G^{-1}(\alpha)-)$ , $\alpha^-_G\le\alpha\le\alpha^+_G$ , $\mathbb P\{Y\le G^{-1}(\alpha)\}=\alpha^+_G$ , $\mathbb P\{Y<G^{-1}(\alpha)\}=\alpha^-_G$ .
(2.24): $G^{-1}(\alpha)$ maximizes $x\mapsto x-\frac1\alpha\mathbb E([Y-x]^-)$ .
(2.17): $\mathbb EY=\alpha\,\mathbb{AV@R}_\alpha(Y)-(1-\alpha)\,\mathbb{AV@R}_{1-\alpha}(-Y)$ for $0<\alpha<1$ .
Proposition 2.35 (i)–(vii): translation equivariance, concavity, isotonicity with respect to $\prec_{SSD}$ and $\mathbb{AV@R}_\alpha(Y)\le\mathbb{AV@R}_\alpha(\mathbb E(Y\mid\mathcal F_1))$ , positive homogeneity, $\mathbb{AV@R}\mathbb D_\alpha\ge0$ , Lipschitz continuity on $L_1$ , comonotone additivity.
Proposition 2.36: $Y^{(1)}\prec_{SSD}Y^{(2)}$ iff $\mathbb{AV@R}_\alpha(Y^{(1)})\le\mathbb{AV@R}_\alpha(Y^{(2)})$ for all $0<\alpha\le1$ .
Proposition 2.37: for nonconstant integrable $Y$ , $\alpha\mapsto\mathbb{AV@R}_\alpha(Y)$ is continuous and nondecreasing on $(0,1]$ and tends to $\operatorname{ess\,inf}Y$ as $\alpha\downarrow0$ .
Example 2.40 (newsboy problem): with purchase price $b$ , selling price $s>b$ and refund $r<b$ , the maximal expected profit is $(s-b)\,\mathbb{AV@R}_\alpha(Y)$ with $\alpha=\frac{s-b}{s-r}$ , attained at the order size $G^{-1}(\alpha)$ .
Proposition 2.74: $\min_a\{\mathbb E([Y-a]^+)+\frac{1-\alpha}{\alpha}\mathbb E([Y-a]^-)\}=\mathbb{AV@R}\mathbb D_\alpha(Y)$ , the minimum attained, for $0<\alpha\le1$ as printed.

Beyond the paper

A strengthened variant of Proposition 2.37 drops the hypothesis that $Y$ is nonconstant: for $Y\equiv c$ every average value-at-risk equals $c=\operatorname{ess\,inf}Y$ , and the book's argument never uses the hypothesis.

Two corrected variants address the endpoint $\alpha=1$ . For $Y$ integrable but not essentially bounded above (for instance exponential), $x-\mathbb E([Y-x]^-)=\mathbb E\min(Y,x)<\mathbb EY=\mathbb{AV@R}_1(Y)$ for every $x$ , so the maximum in (2.19) is not attained; likewise $\mathbb E([Y-a]^+)>0=\mathbb{AV@R}\mathbb D_1(Y)$ for every $a$ , so the minimum in (2.112) is not attained. The corrected Theorem 2.34 asserts part 1 only when $G^{-1}(\alpha)$ is finite, and the corrected Proposition 2.74 adds the same hypothesis; both leave everything else unchanged.

Significance

Theorem 2.34 is the toolkit through which the average value-at-risk is used. Representation 1 makes it the optimal value of a one-dimensional concave maximization, which is how it enters stochastic programs and the newsboy-type models of Examples 2.39–2.40. Representation 2 exhibits it as an infimum of linear functionals over an explicit dual set, which yields concavity, $L_1$ -Lipschitz continuity and the link to the general dual representation of acceptability functionals. Representations 3–5 are the computational formulas for discrete distributions, where the naive "average of the outcomes below the quantile" is wrong. Proposition 2.36 shows that the family $(\mathbb{AV@R}_\alpha)_\alpha$ characterizes second-order dominance, and Proposition 2.74 identifies the deviation as a minimal-loss functional.

The results are classical and proved in the literature. What the mission adds is a machine-checked version in the book's conventions: profit variables, the lower tail, the left-continuous quantile, the dual set in $L_\infty$ with bound $1/\alpha$ , and general distributions with atoms. The platform holds related results in other conventions: the Rockafellar–Uryasev minimization formula for loss functions and upper tails, an open loss-side dual representation on $L_\infty$ , and expected-shortfall properties on finite state spaces; none states these representations for profit variables on a general probability space.

Difficulty

Everything happens where $G$ has an atom at $G^{-1}(\alpha)$ . A proof that pretends $G$ is continuous, so that $\mathbb P\{Y\le G^{-1}(\alpha)\}=\alpha$ , covers none of the cases the theorem was written for: the optimal dual variable must put a fractional weight on the event $\{Y=G^{-1}(\alpha)\}$ , and representations 4 and 5 differ from each other precisely by the jump. Passing between integrals over $\Omega$ , integrals against the law, and Lebesgue integrals of the quantile function over $(0,\alpha]$ needs the quantile transform for arbitrary distributions. The endpoint $\alpha=1$ is genuinely different: there $G^{-1}(1)$ may be infinite and the maximum in (2.19) and the minimum in (2.112) are then not attained: the book's statements fail there, and the corrected variants carry the finiteness condition.

Formalization scope

A random variable is a measurable, integrable $Y:\Omega\to\mathbb R$ on a probability space; $\mathbb{AV@R}_\alpha(Y)$ is computed from the law of $Y$ and so depends only on its distribution. The quantile function is real-valued; wherever it is evaluated at a point $\alpha$ , the statement assumes $G(u)\ge\alpha$ for some $u$ . Maxima and minima are stated as attained greatest and least elements of the value sets, never as bare suprema or infima. In the dual representation $Z$ ranges over measurable functions with $\int Z\,d\mathbb P=1$ and $0\le Z\le1/\alpha$ almost surely. Conditional expectations given an event are integrals against the conditional probability measure; when the event is null its coefficient $\alpha^\pm_G$ is $0$ as well. The limit at $\alpha\downarrow0$ is taken in the extended reals, since $\operatorname{ess\,inf}Y$ may be $-\infty$ .

A trivializing formalization is ruled out: writing (2.19) or (2.20) with sSup/sInf and no attainment, or using an upper-tail, loss-side conditional value-at-risk in place of the book's lower-tail functional, would change or empty the statement; neither is done.

Contributions welcome: the quantile transform for general laws, the change of variables between $\Omega$ , the law and $(0,1)$ , and the Rockafellar–Uryasev argument are reusable across the rest of this series (Kusuoka and distortion representations, nested average value-at-risk).

Selected references

G. Ch. Pflug, W. Römisch, Modeling, Measuring and Managing Risk, World Scientific, 2007. https://doi.org/10.1142/6478
R. T. Rockafellar, S. Uryasev, Optimization of conditional value-at-risk, Journal of Risk 2(3), 2000. https://doi.org/10.21314/JOR.2000.038
R. T. Rockafellar, S. Uryasev, Conditional value-at-risk for general loss distributions, Journal of Banking & Finance 26(7), 2002. https://doi.org/10.1016/S0378-4266(02)00271-6
C. Acerbi, D. Tasche, On the coherence of expected shortfall, Journal of Banking & Finance 26(7), 2002. https://doi.org/10.1016/S0378-4266(02)00283-2
P. Artzner, F. Delbaen, J.-M. Eber, D. Heath, Coherent measures of risk, Mathematical Finance 9(3), 1999. https://doi.org/10.1111/1467-9965.00068

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Functional Analysis Operations Research Optimization+1·Captain: Shuze Chen

Modeling, Measuring and Managing Risk IV: Kusuoka Representations of Version-Independent Acceptability FunctionalsTextbook

Motivation

Risk measures used in portfolio optimization, insurance pricing and regulation are almost always law invariant: they assign a number to a random profit $Y$ that depends only on its distribution, not on the scenario in which each outcome occurs. Kusuoka (Kusuoka 2001) showed that on a non-atomic space every law-invariant coherent risk measure is the worst case, over a family of weightings, of mixtures of one elementary functional, the average value-at-risk (also called conditional value-at-risk or expected shortfall). This reduces the design and the computation of law-invariant risk measures to the choice of a set of probability measures on the interval of confidence levels, and it underlies the use of the average value-at-risk as the building block of spectral and distortion risk measures.

This mission formalizes the version of Kusuoka's theorem proved in G. Ch. Pflug and W. Römisch, Modeling, Measuring and Managing Risk (World Scientific 2007, §2.2.4, Theorem 2.45), stated for acceptability functionals of profit variables on $L_p$ , $1\le p<\infty$ , together with the coupling and realization results its proof uses and a worked example from the same section.

Setting

Let $(\Omega,\mathcal F,\mathbb P)$ be a probability space. It is non-atomic if no measurable set $A$ with $\mathbb P(A)>0$ has only subsets of probability $0$ or $\mathbb P(A)$ . Fix $1\le p<\infty$ and the conjugate exponent $q$ , $1/p+1/q=1$ ; $L_p$ is the space of real random variables with finite $p$ -th moment (identified when equal almost surely) and $L_q$ its dual, paired by $\mathbb E(YZ)$ .

For $Y$ with distribution function $G(u)=\mathbb P\{Y\le u\}$ , the quantile function is $G^{-1}(p)=\inf\{u: G(u)\ge p\}$ , $0<p\le1$ , and the average value-at-risk at level $0<\alpha\le1$ is the lower-tail average

\mathbb{AV@R}_\alpha(Y)=\frac1\alpha\int_0^\alpha G^{-1}(u)\,du .

The book uses profit variables, so larger values are better and $\mathbb{AV@R}_\alpha$ averages the worst $\alpha$ -fraction of outcomes; $\mathbb{AV@R}_1(Y)=\mathbb EY$ .

A real valued $\mathcal A:L_p\to\mathbb R$ is an acceptability functional if it is translation equivariant, $\mathcal A(Y+c)=\mathcal A(Y)+c$ (A1), concave (A2), and monotone, $Y^{(1)}\le Y^{(2)}$ a.s. $\Rightarrow\mathcal A(Y^{(1)})\le\mathcal A(Y^{(2)})$ (A3). It is version-independent if $\mathcal A(Y^{(1)})=\mathcal A(Y^{(2)})$ whenever $Y^{(1)}$ and $Y^{(2)}$ have the same distribution. A dual representation is

\mathcal A(Y)=\inf\{\mathbb E(YZ)-A(Z): Z\in\mathcal Z_{\mathcal A}\},\qquad \mathcal Z_{\mathcal A}\subseteq L_q,

and $(A,\mathcal Z_{\mathcal A})$ is version-independent if equally distributed $Z,\tilde Z$ are both or neither in $\mathcal Z_{\mathcal A}$ , with $A(Z)=A(\tilde Z)$ . Theorem 2.45 adds the conditions (C1) $A=0$ , (C2) every $Z\in\mathcal Z_{\mathcal A}$ is non-negative, (C3) every $Z\in\mathcal Z_{\mathcal A}$ has $\mathbb E Z=1$ .

Formalization targets

Goal: Theorem 2.45(i)

Under (C1), (C2) and (C3) there is a family $\mathcal G$ of probability measures on $(0,1]$ , fixed independently of $Y$ , such that for every $Y\in L_p$

\mathcal A(Y)=\inf\Big\{\int_{(0,1]}\mathbb{AV@R}_\alpha(Y)\,dm(\alpha): m\in\mathcal G\Big\}. \tag{2.37}

Milestones

Theorem A.2 (p. 266): on a non-atomic probability space every probability law on a complete separable metric space is the law of some random variable.
Proposition 1.8 (p. 11): for given marginals, the covariance is maximal under the comonotone coupling and minimal under the antimonotone coupling.
Eq. (2.38) (p. 61): under (C1), $\mathcal A(Y)=\inf\{\int_0^1F^{-1}(u)G^{-1}(1-u)\,du: G\in\mathcal G\}$ , with $F$ the law of $Y$ and $\mathcal G$ the laws of the $Z\in\mathcal Z_{\mathcal A}$ .
Theorem 2.45(ii)–(iv) (p. 60): the representations with non-negative measures under (C1)+(C2), with two families of measures under (C1),

\mathcal A(Y)=\inf\Big\{\int\mathbb{AV@R}_\alpha(Y)\,dm^{(1)}_G(\alpha)-\int\mathbb{AV@R}_{1-\alpha}(-Y)\,dm^{(2)}_G(\alpha)\Big\},

and in general with an additional penalty $-\tilde A\{G\}$ and $\mathbb{AV@R}_\alpha(-Y)$ in the second integral.

Example 2.46, Eq. (2.41) (p. 64): $\mathbb E(Y)-\tfrac12\mathbb E|Y-\mathbb EY|=\inf\{\int_{(0,1]}\mathbb{AV@R}_\alpha(Y)\,dm(\alpha): m\in\mathcal P(0,1],\ \int_{(0,1)}v^{-1}\,dm(v)\le1\}$ .

Significance

The result. The Kusuoka representation says that the average values-at-risk are the extreme points of law-invariant acceptability: any such functional is determined by a set of mixing measures on the confidence levels. Consequences used in the rest of the book and in the literature include the identification of comonotone additive functionals with single mixtures (Choquet representations), explicit representations of mean–deviation functionals such as (2.41), and the reduction of optimization problems with law-invariant risk constraints to families of average value-at-risk constraints. The statement for $L_p$ , rather than for bounded variables, is what makes it applicable to models with unbounded returns.

Formalizing it. The theorem is proved in the book and in the literature; it has not been machine-checked. Formalizing it requires infrastructure that is reusable beyond this mission: the quantile function and the average value-at-risk on $L_p$ , the realization of a prescribed joint law on an arbitrary non-atomic probability space, and the Hardy–Littlewood-type rearrangement inequality behind the antimonotone coupling. Related platform results are in other conventions: a distortion functional of losses on $L_\infty$ as a single mixture of average values-at-risk (MultistageStochastic.distortion_avar_mixture, proved) and Schmeidler's comonotonic characterization on finite uniform spaces (RiskUncSets.Distortion.theorem_4_1, open). Neither covers the infimum over a family, profit variables, or $L_p$ .

Difficulty

The obvious argument is to write each $Z\in\mathcal Z_{\mathcal A}$ as a mixing measure directly. This fails because $\mathbb E(YZ)$ depends on the joint law of $(Y,Z)$ , not on the two marginals; the infimum must first be reduced to antimonotone couplings, which requires (a) that every coupling of the two marginals can be realized on the given space without leaving $\mathcal Z_{\mathcal A}$ , using version-independence of both $\mathcal A$ and $(A,\mathcal Z_{\mathcal A})$ and non-atomicity, and (b) the extremal property of the antimonotone coupling for products of an $L_p$ and an $L_q$ variable, where quantile functions may be unbounded at both ends. The passage from $G^{-1}(1-u)$ to a mixing measure is a Stieltjes integration by parts on $(0,1)$ that must handle a point mass at $1$ , and the integrability of $\alpha\mapsto\mathbb{AV@R}_\alpha(Y)$ , which tends to the essential infimum of $Y$ (possibly $-\infty$ ) as $\alpha\to0$ . Non-atomicity cannot be dropped: the book's Counterexample on p. 63 is a version-independent functional on a two-point space without representation (2.37).

Formalization scope

Probability spaces are (Ω, ℱ, P) with IsProbabilityMeasure P; non-atomicity is the book's notion (no atoms), which is stronger than Mathlib's NoAtoms. Random variables are elements of Lp ℝ p P with 1 ≤ p, p ≠ ⊤, and dual variables elements of Lp ℝ q P with p.HolderConjugate q. The quantile function is the left-continuous inverse of the book's (1.3), evaluated only on $(0,1)$ or inside integrals over $(0,\alpha]$ ; $\mathbb{AV@R}_\alpha$ is the lower-tail average (2.15), not the upper-tail loss CVaR. Version-independence is "equal laws give equal values" (IdentDistrib). Every infimum is stated as a greatest lower bound in $\mathbb R$ (IsGLB), which forces the indexing set to be nonempty and bounded below; measures on $(0,1]$ are measures on $\mathbb R$ with no mass outside $(0,1]$ . The family $\mathcal G$ is quantified before $Y$ : a family allowed to depend on $Y$ would reduce (2.37) to matching the single number $\mathcal A(Y)$ by some mixture, a much weaker statement, and that reading is excluded. Each representation also asserts that every integrand is integrable against every measure of the family, so that no integral can take Lean's default value $0$ .

Beyond the paper. Two variant items accompany the goal. kusuoka_representation_strong ([Strengthened]) drops (C2) and (C3): for a real valued acceptability functional they follow from (A1), (A3) and the dual representation, since $\mathbb E(YZ)\ge\mathcal A(Y)$ for every $Y$ forces $\mathbb EZ=1$ (constants) and $Z\ge0$ (scaled indicators). kusuoka_representation_probe ([Sharpness probe]) drops non-atomicity; it is expected to be false by the p. 63 Counterexample, and a disproof is the sought outcome. The printed proof of Theorem 2.45(iii) has a sign gap (it derives $+\int\mathbb{AV@R}_{1-\alpha}(-Y)\,dm^{(2)}$ ); the statements (iii) and (iv) are kept as printed.

Contributions welcome: the quantile transform and realization of joint laws on non-atomic spaces, the rearrangement inequality on $L_p\times L_q$ , and the integration-by-parts construction of the mixing measure, each of which can be submitted as a separate lemma.

Selected references

G. Ch. Pflug, W. Römisch, Modeling, Measuring and Managing Risk, World Scientific, 2007, §2.2.4 and Appendix A. https://doi.org/10.1142/6478
S. Kusuoka, On law invariant coherent risk measures, Advances in Mathematical Economics 3, 83–95, 2001. https://doi.org/10.1007/978-4-431-67891-5_4
W. Hoeffding, Masstabinvariante Korrelationstheorie, Schriften des Mathematischen Instituts der Universität Berlin 5, 179–233, 1940.
S. T. Rachev, Probability Metrics and the Stability of Stochastic Models, Wiley, 1991 (Lemma 2.5.1, cited for Theorem A.2).

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Control Theory Operations Research Probability+1·Captain: Shuze Chen

Stochastic Networks XV: Positive Recurrence under MaxWeight SchedulingTextbook

Scheduling shared service capacity

A switch or wireless network can have several queues ready for service while its hardware permits only certain combinations to be served together. A scheduler must choose a feasible combination repeatedly as new packets arrive. The question is whether this local sequence of decisions keeps the queue process stable whenever the arrival rates fit within the network's service capacity. Kelly and Yudovina formulate this question for finite service schedules and independent Poisson arrivals in §7.8 of Stochastic Networks.

MaxWeight scheduling gives larger queues greater influence over the choice of schedule. An exponent controls how strongly queue length affects priority. The chapter's target is a stability guarantee for every positive exponent, rather than a guarantee confined to the familiar linear weighting. The supplied Lecture Notes PDF states this as Theorem 7.13, printed page 180, and develops supporting analytic questions in Appendix D.

The queueing model

Let $R$ be a finite nonempty set of queues. A schedule is a vector $\sigma\in\mathbb N^R$ describing how many packets each queue can serve in one time slot. The feasible schedule family $S$ is finite and nonempty. It is downward closed: reducing any schedule coordinate leaves a feasible schedule. This includes the possibility of unused service when a selected queue has fewer packets than the schedule permits.

The admissible region is

\Lambda=\left\{v\in\mathbb R^R:v\ge0,\ \exists(c_\sigma)_{\sigma\in S},\ c_\sigma\ge0,\ \sum_{\sigma\in S}c_\sigma=1,\ v\le\sum_{\sigma\in S}c_\sigma\sigma\right\}.

It describes arrival rates dominated by an average of feasible schedules. The interior used here is the ordinary interior in $\mathbb R^R$ .

Fix $\alpha>0$ . At queue vector $q$ , a schedule is drawn from a fixed probability distribution supported on the maximizers of $\sum_r\sigma_rq_r^\alpha$ . New arrivals $A_r$ in the slot are independent Poisson variables of means $\lambda_r$ , independent of the scheduling randomization and the past. The next queue vector is

q'_r=(q_r-\sigma_r)^++A_r.

These laws specify a homogeneous Markov chain on the countable state space $\mathbb N^R$ . They follow the general model in §7.8 and the explicit empty-queue convention in Example D.4, printed page 213.

Formalization targets

The goal is Theorem 7.13: for $\lambda\in\operatorname{int}\Lambda$ , the MaxWeight- $\alpha$ queue chain is positive recurrent. In return-time notation,

\forall q\in\mathbb N^R,\qquad \mathbb E_q[\tau_q^+]<\infty,\qquad \tau_q^+=\inf\{n\ge1:q(n)=q\}.

This retains every positive value of $\alpha$ and the full queue process.

Three supporting targets are selected from the same supplied source. Proposition D.1 states the Foster–Lyapunov criterion for an irreducible countable chain and an arbitrary finite exception set. A nonnegative function with drift at most $-\epsilon$ outside that set and at most $b-\epsilon$ inside it yields finite expected return to the set from every starting state and positive recurrence.

Exercise 7.18 concerns the weighted drift

D(q)=\sum_rq_r^\alpha\mathbb E[q'_r-q_r\mid q].

It asks for constants $\epsilon,K>0$ and finite $b$ such that

\|q\|_2>K\implies D(q)<-\epsilon\|q\|_2^\alpha,\qquad \|q\|_2\le K\implies D(q)<b.

Exercise D.1(3) asks for finite moments of $Y=\max(A,B)$ , where $B$ is a finite service bound and $A$ is Poisson, and for the tail term

\mathbb E\!\left[\left(\left|(1+Y/K)^{\alpha+1}\right|+1\right)\mathbf1_{\{Y>\delta K\}}\right]

to tend to zero as $K\to\infty$ for fixed $\delta>0$ . The source locations are printed pages 210–211, 182, and 214–215 respectively.

What the result establishes

The recurrence conclusion concerns repeated operation over an unlimited number of slots. It asserts a finite mean return time to each queue state under the specified policy. It does not require an explicit formula for the stationary distribution, nor does it prescribe a numerical delay guarantee. The result therefore identifies a capacity regime in which this queue-based scheduling rule is stable, while leaving performance within that regime as a separate question.

The formalization contribution is the connection between a concrete random scheduling transition law and recurrence on an infinite state space. Its reusable components include the capacity region of finite schedules, capped-service queue transitions, extended return expectations, and the Poisson envelope estimates. The textbook presents the theorem as an established result with a proof sketch. The theorem items here are statements awaiting proofs; compilation verifies their Lean syntax and types, not their mathematical proofs.

Where faithfulness requires care

Negative drift by itself is insufficient for the recurrence conclusion. The distinctions between negative drift, drift bounded away from zero, and control of upward increments matter in the chapter's discussion, printed pages 180–181. Arbitrary positive real exponents also require attention at empty queues and to unbounded Poisson arrivals.

The sketch's equality between actual departures and a chosen schedule fails when a positive queue is shorter than the proposed service. This is a proof gap, not a counterexample to the stability theorem. Appendix D additionally switches between conditioning on the full vector and conditioning on one coordinate in (D.5). Those conditional expectations need not agree for coupled schedules. The selected statements retain actual capped departures and avoid relying on that ambiguous display. Completing the analytic link to the goal remains part of the proof task.

Formalization scope

Queue and schedule coordinates are natural numbers, rates are nonnegative real numbers, and the exponent is an arbitrary positive real. The schedule family is a finite set of vectors. Random tie rules are stationary probability kernels supported on the maximizing schedules; the source specifies no particular distribution among ties. The norm in Exercise 7.18 is made Euclidean explicitly. Constants there are chosen before the queue vector is quantified, so they cannot depend on the current state or time.

Return times are represented by extended nonnegative sums of survival probabilities. Consequently, an event of never returning has infinite expected cost. Real drift sums have explicit absolute-integrability conditions. In the queue drift and Poisson moment items, integrability is a conclusion, not an assumed substitute for analysis. The Poisson tail term is an unconditional tail integral, including when its conditioning event would have probability zero.

No selected statement contains an unspecified $O(\cdot)$ term or a high-probability claim; there are therefore no numerical asymptotic constants to instantiate. Exercise D.1 is included only through its explicitly numbered part (3). The remaining parts, (D.5), congestion-control optimization results, and the switch-specific matching example are outside this mission's theorem list. No variants are posed.

Stability is not defined as negative drift, and the goal is not restricted to $\alpha=1$ or a finite truncation of the queue space. Mathlib supplies probability mass functions, product measures, Poisson laws, real powers, and topology. The local definitions supply the scheduling and recurrence interfaces needed to state the targets.

Selected references

F. P. Kelly and E. Yudovina, Stochastic Networks, Cambridge University Press, 2014. DOI: 10.1017/CBO9781139565363. Exact indices and quotations refer to the supplied Lecture Notes on Stochastic Networks PDF, SHA-256 ec271d555059aee58613e5e9a98b8346214b16185c527314f5d94c1ac8b17b6a; correspondence with a particular published printing has not been verified.

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Convex Optimization Operations Research Probability·Captain: Shuze Chen

Stochastic Networks XIII: The Capacity Region of Distributed Random AccessTextbook

Motivation

Stations sharing a communication channel must decide when to transmit. If nearby stations transmit together, their signals may interfere; if they all wait, capacity is wasted. A central scheduler can coordinate activity, but distributed access assigns decisions to individual stations. The mathematical question is whether simple random local decisions can supply all the service combinations that the interference constraints permit. This mission concerns the idealized continuous-time model in Section 5.4 of Kelly and Yudovina, Stochastic Networks, using the supplied Lecture Notes PDF.

The model isolates a capacity question from implementation issues. Stations sense active neighbors immediately, so collisions do not occur. Transmission durations and attempt clocks are exponential. The question concerns the equilibrium activity of this process for fixed parameters. Delayed sensing, packet queues, and parameter adaptation require additional results and are outside the theorem's claim.

Setting

An interference graph is a finite simple undirected graph whose vertices are stations. An edge means that its two endpoints cannot transmit together. Write $R$ for the station set and $S$ for the set of binary activity vectors $n=(n_r)_{r\in R}$ satisfying this restriction. The value $n_r=1$ means that station $r$ is transmitting. The all-zero vector is always feasible.

Each station has a finite real parameter $\theta_r$ . An active station stops transmitting at rate $1$ . An inactive station starts at rate $\exp(\theta_r)$ when all its neighbors are inactive. These transitions define a continuous-time Markov chain on $S$ . Its generator is $Q$ , with negative total outgoing rates on the diagonal, and its transition matrices are $P(t)=\exp(tQ)$ for $t\ge0$ .

A distribution $p$ on feasible schedules offers mean activity $\sum_n p(n)n_r$ at station $r$ . The capacity region includes every nonnegative arrival vector that some such distribution can dominate:

\Lambda=\left\{\lambda\ge0:\exists p\ge0,\quad\sum_{n\in S}p(n)=1,\quad \sum_{n\in S}p(n)n_r\ge\lambda_r\quad(r\in R)\right\}.

This definition permits arbitrary distributions on feasible schedules. The random-access process produces a much more specific family, indexed by one real number per station.

Formalization targets

Equation (5.8) identifies the stationary distribution of the specified chain:

\pi_\theta(n)=\frac{\exp(\sum_r n_r\theta_r)}{\sum_{m\in S}\exp(\sum_r m_r\theta_r)}, \qquad \pi_\theta P(t)=\pi_\theta\quad(t\ge0).

The goal is Theorem 5.17, printed page 126 of the supplied PDF. Every arrival vector in the ordinary interior of $\Lambda$ has strictly smaller coordinates than the equilibrium service vector for some finite parameters:

\lambda\in\operatorname{int}\Lambda \quad\Longrightarrow\quad \exists\theta\in\mathbb R^R,\quad \lambda_r<\sum_{n\in S}n_r\pi_\theta(n)\quad(r\in R).

Three Appendix C milestones supply the accompanying optimization statements. For a convex objective $f$ and componentwise convex constraint map $h$ on a convex set $X$ , Theorem C.7 asserts convexity of the extended value function $\phi(b)=\inf\{f(x):x\in X,h(x)\le b\}$ . Theorem C.3 supplies a nonvertical support at an interior point of its finite-value domain. Theorem C.4 states the equivalence between such support and the strong Lagrangian principle for equality constraints, with

L_b(x;y)=f(x)+y^T(b-h(x)),\qquad \phi(b)=\inf_{x\in X}L_b(x;y).

The faithful C.4 statement is retained without an unprinted finiteness hypothesis. Its infinite-value defect and the separately labelled correction are described below. The goal itself retains the source's strict service inequalities.

Significance

The capacity theorem says that the exponential random-access family does not leave a gap inside the region attainable by arbitrary mixtures of independent schedules. It provides a service margin in every coordinate at once. Its quantifiers allow the parameters to depend on the graph and arrival vector, but require every parameter to remain a finite real number. The theorem does not provide a uniform parameter bound as the arrival vector approaches the boundary.

The known mathematical result is attributed to Jiang and Walrand in the supplied source. The formalization target is a checked statement of that result and its supporting mathematical interfaces. The proposal's theorem declarations remain proof tasks. The finite exclusion process, its invariant law, and the distinction between feasible schedule mixtures and exponential-family marginals are reusable in later work on scheduling. The extended-value optimization layer can also support problems unrelated to communication networks.

Difficulty

Normalizing positive exponential weights only constructs a probability distribution. It does not establish that the distribution is invariant under the given dynamics, nor that its marginal vectors cover every strictly feasible demand. Changing one station's parameter changes the other stations' equilibrium opportunities through interference. Coordinatewise scalar tuning therefore does not settle simultaneous coverage.

The optimization statements require care at infeasible and unbounded problems. A real-valued infimum with a default value would conceal these cases. In particular, the literal C.4 equivalence fails for an empty domain: both primal and Lagrangian infima are $+\infty$ , but no finite-height hyperplane touches that value. This is an issue with the unrestricted statement, not a new hypothesis for the capacity theorem.

Formalization scope

Stations are indexed by Fin d, and activity coordinates by Fin 2. Simple graphs enforce symmetry and exclude self-interference. A decidability instance is computational bookkeeping for a finite adjacency relation. The empty station set is permitted: it has one empty schedule, and the service conclusion has no coordinates. For one station the mean activity is $\exp(\theta)/(1+\exp(\theta))$ . All normalizing sums are finite and strictly positive because the empty schedule contributes weight one.

Interior means ambient Euclidean interior. Arrival vectors in the closed region alone are insufficient: at the one-station boundary demand $\lambda=1$ , no finite parameter supplies service greater than one. Finding F1 records this sharpness example; no extra sharpness item is posed.

The CTMC is represented by its finite generator and matrix-exponential transition semigroup. Its stationary-law theorem asserts invariance for every nonnegative time. Optimization infima use EReal, so infeasibility gives $+\infty$ and unboundedness below gives $-\infty$ . Convexity is expressed through the real epigraph; supporting affine functions have real heights and slopes. Slack elimination in (C.2) preserves the inequality $h(x)\le b$ . No asymptotic constants or failure probabilities require instantiation in these targets.

Beyond the paper. The single corrected variant of C.4 adds exactly that $\phi(b)$ is finite. Finding F2 gives the empty-domain counterexample to the unrestricted equivalence and the supporting argument for the correction. The original remains the milestone anchor. Findings F3 and F4 separately record the slack-multiplier sign error on printed page 209 and the word “maximized” in the minimization argument on printed page 127; neither changes the goal or creates a variant.

The definitions have concrete mathematical bodies without proof placeholders. The theorem types expose their defining formulas for independent auditing, and use Mathlib's finite sums, graph types, topology, convexity, extended reals, and matrix exponential. A formulation that assumes the desired service vector or proves only normalization would not capture this mission.

Selected references

F. P. Kelly and E. Yudovina, Stochastic Networks, Cambridge University Press, 2014. DOI. All numbered claims and page citations refer to the supplied Lecture Notes on Stochastic Networks PDF: Section 5.4, printed pages 125–128 (PDF pages 133–136), and Appendix C, printed pages 207–209 (PDF pages 215–217). Equivalence of this supplied file to a particular published printing is unverified.

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Algorithmic Game Theory Economics Operations Research·Captain: Shuze Chen

Impact of Traceability Technology Adoption in Food Supply Chain Networks: Under the Break Point Criterion, Traceability Lowers the Retailer's Equilibrium Profit Exactly on an Interval (r₁, r₂)Research Paper

Motivation

Food-borne outbreaks are traced slowly: when contaminated produce reaches shelves, the source is often unknown for weeks, and retailers recall everything. Traceability technology (for example blockchain ledgers) identifies the contaminated source at once, so only its units are discarded. Regulators and large retailers have pushed for it, and the intuitive view is that every member of the chain gains. Dong, Jiang & Xu (Management Science, 2022) show in a three-tier pricing game that this is not so: the chain's most powerful member, the retailer, can earn less in equilibrium with traceability than without it. The reversal happens for an explicit interval of retail prices, under an explicit condition on the primitives.

Setting

A retailer sells $n \ge 2$ units at an exogenous retail price $r$ . She buys them at a wholesale price $w$ from one tier-1 supplier, who buys one unit at a procurement price $p$ from each of $n$ tier-2 suppliers (farms). Each farm produces at unit cost $c_p > 0$ and is contaminated independently with probability $\rho$ , or $\rho_e$ if it exerts a risk-reduction effort at cost $e > 0$ , where $0 \le \rho_e < \rho < 1$ and $\rho_\tau = \rho - \rho_e$ . Assumption 1 of the paper is $c_p < \bar c_p = (1-\rho)e/\rho_\tau$ . A farm's survival probability is $s = 1-\rho_e$ with effort and $1-\rho$ without.

Case N (no traceability). Any contamination voids every payment. Farm $i$ 's payoff under an effort profile $a$ is $\big(\prod_j s(a_j)\big)p - (e\mathbb 1\{a_i\} + c_p)$ .
Case T (traceability). Only contaminated units are lost. Farm $i$ 's payoff is $s(a_i)p - (e\mathbb 1\{a_i\} + c_p)$ .

Tier-1 earns (expected units sold) $\cdot(w-p)$ and the retailer (expected units sold) $\cdot(r-w)$ ; expected units sold are $n\prod_j s(a_j)$ in Case N and $\sum_j s(a_j)$ in Case T. The retailer chooses $w$ , tier-1 then $p$ , and the farms then choose effort simultaneously. The solution concept is subgame-perfect Nash equilibrium (SPNE) with the paper's selection rules: the Pareto-dominant tier-2 equilibrium is played; an indifferent farm exerts effort; an indifferent tier-1 or retailer induces effort; an agent with zero profit still trades.

The paper solves both games in closed form. With $\ell = (1-\rho_e)^n - (1-\rho)^n$ , the effort thresholds of the retail price are

r^e_N = \max\Big\{\tfrac{(1-\rho_e)^{n+1}e-[2(1-\rho_e)^n-(1-\rho)^n]\rho_\tau c_p}{\rho_\tau\ell^2},\ \tfrac{e}{\rho_\tau(1-\rho_e)^{n-1}}\Big\},\qquad r^e_T = \tfrac{(1-\rho_e)^2e-(1+\rho-2\rho_e)\rho_\tau c_p}{\rho_\tau^3},

and the retailer's profit on the relevant branches (Tables EC.1–EC.2) is $\pi^e_{0N}(r) = n(1-\rho_e)^n(r - w^e_N)$ , $\pi^o_{0T}(r) = n(1-\rho)r - nc_p$ and $\pi^e_{0T}(r) = n(1-\rho_e)(r - w^e_T)$ , with $w^e_N = \max\{\frac{(1-\rho_e)e-\rho_\tau c_p}{\rho_\tau\ell}, \frac{e}{\rho_\tau(1-\rho_e)^{n-1}}\}$ and $w^e_T = \frac{(1-\rho_e)e-\rho_\tau c_p}{\rho_\tau^2}$ . The paper's Definition 1:

DEFINITION 1 (BREAK POINT CRITERION). The Break Point Criterion (BPC) is satisfied if traceability adoption decreases the retailer's equilibrium profit when the retail price is at the effort threshold with traceability, i.e., $\pi^e_{0T}(r^e_T) < \pi^e_{0N}(r^e_T)$ .

Formalization targets

Goal: Proposition 3 (p. 21)

Under the standing assumptions and BPC there exist $r_1, r_2$ with

r^e_N < r_1 < r^e_T < r_2,\qquad \pi^o_{0T}(r_1) = \pi^e_{0N}(r_1),\qquad \pi^e_{0T}(r_2) = \pi^e_{0N}(r_2),

such that both games have an SPNE at every retail price, and for all SPNE $s_N$ , $s_T$ at $r$ ,

r_1 < r < r_2 \implies \Pi_T(r; s_T) < \Pi_N(r; s_N),\qquad r \notin (r_1, r_2) \implies \Pi_N(r; s_N) \le \Pi_T(r; s_T),

where $\Pi_N$ , $\Pi_T$ are the retailer's equilibrium profits.

Milestones

In proof order: Lemma 1 (tier-2 equilibrium, Case N, p. 9); Lemma 2 (tier-1's optimal price, Case N, p. 10); Proposition 1 (SPNE, Case N, p. 11); Lemma 3, Lemma 4 (Case T, p. 13); Proposition 2 (SPNE, Case T, p. 14); Lemma 5 (BPC implies $r^e_N < r^e_T$ and $(1-\rho_e)^n > 1-\rho$ , p. 21).

Stronger: the exact sign pattern

The variant proposition_3_strong adds strict increase with traceability on $(r^o_T, r_1) \cup (r_2, \infty)$ , where $r^o_T = c_p/(1-\rho)$ , and equality on $(-\infty, r^o_T] \cup \{r_1, r_2\}$ .

Significance

Proposition 3 is the paper's headline result (Abstract, p. 1): removing the cascade threat does not make every member better off, and the losses concentrate on retail prices around the traceability effort threshold $r^e_T$ . It underlies the paper's later welfare discussion (Observation 1, Proposition 4), where every member is worse off on part of the interval, and the policy point that adoption may need subsidies or contract adjustments.

The result is proved in the paper, partly in sketch form; none of it is formalized. A formal development produces the first fully specified multi-tier pricing-and-effort game on the platform: an $n$ -player effort game with a selection rule, two nested best-response problems with tie rules, and subgame-perfect equilibrium as a predicate, together with a welfare comparison stated on equilibria rather than on formulas.

Difficulty

Each step is elementary, but the equilibrium has to be established, not assumed. The tier-2 stage is an $n$ -player game in Case N whose equilibria above $p^e_N$ can be multiple (a stag hunt), so the selection rule matters. Tier-1 and the retailer optimize a margin whose effect on effort is a step function of the price; their maxima are attained only because the thresholds are closed and the tie rules settle the boundary, and they must be identified among all real prices. The final comparison is between two piecewise-linear functions with different breakpoints in the two cost branches. The paper prints no proof of Lemma 5, Lemma 4 or Proposition 2, and Proposition 3's printed proof is a short sketch that treats the high-cost branch as "similar".

Formalization scope

Everything lives in the namespace FoodTrace.Adoption (definition module Def_FoodTrace_Adoption_Model). Conventions:

Parameters Params = (n, ρ, ρe, e, cp); Standing collects $n \ge 2$ , $0 \le \rho_e$ , $\rho_e < \rho$ , $\rho < 1$ , $e > 0$ , $c_p > 0$ and Assumption 1. The implicit hypotheses $0 \le \rho_e$ , $\rho < 1$ , $e > 0$ , $c_p > 0$ are made explicit.
Effort is a Bool per farm; profiles are Fin n → Bool, with no symmetry built in.
A viable tier-2 equilibrium follows (EC.9) of p. ec22: nonnegative payoffs; effort weakly preferred by farms exerting it; no effort strictly preferred by the others. The selected outcome weakly Pareto-dominates every viable equilibrium; "no sale" is selected exactly when none exists.
"No trade" is none with profit 0; prices range over all of $\mathbb R$ .
Best responses and retailer optimality are maximizers over all prices and no trade, with the tie rules as explicit clauses (induce effort, then trade, whenever some maximizer does).
Case T revenue for asymmetric profiles is $\sum_j s(a_j)$ times the margin; the paper prints only the symmetric case.
$w^e_N$ and $r^e_N$ carry the max, so $\pi^e_{0N}$ is one formula in both cost branches. $\underline c_p$ is cpLow, $\bar c_p$ is cpBar.
Proposition 3's "Otherwise, traceability adoption increases the retailer's profit" is stated as $\Pi_N \le \Pi_T$ : the profits are equal at $r_1$ , $r_2$ and for $r \le r^o_T$ .

The retailer's profit in the goal is her payoff at an SPNE of the game, not the closed forms of Tables EC.1–EC.2; the goal asserts the existence of SPNE in both games at every retail price, so the comparison over all equilibria is not vacuous; and BPC is satisfiable (the Figure 4 instance $n = 2$ , $\rho = 0.4$ , $\rho_e = 0.1$ , $c_p = e = 0.4$ gives $\pi^e_{0T}(r^e_T) = 36/5 < 198/25 = \pi^e_{0N}(r^e_T)$ ).

Beyond the paper. One variant is posed beside the faithful goal: proposition_3_strong (strengthened, finding F2), the exact sign pattern above. Evidence: a four-region comparison of the equilibrium profits, confirmed at 1.87 million sampled retail prices on 18,820 random BPC instances in both cost branches.

Needed infrastructure: finite products and sums over Fin n, real inequalities with powers, and case analysis on Option. The equilibrium layer (selection, best responses with tie rules, SPNE as a predicate) is reusable for other sequential supply-chain games. Proofs of the stage results (Lemmas 1–4, Propositions 1–2) and of Lemma 5 are welcome as independent contributions.

Selected references

Y. Dong, B. Jiang, M. Xu, Impact of Traceability Technology Adoption in Food Supply Chain Networks, Management Science, 2022. https://doi.org/10.1287/mnsc.2022.4440 (accepted manuscript: SSRN 3484664, https://ssrn.com/abstract=3484664)

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Convex Optimization Operations Research Optimization·Captain: Shuze Chen

Robust Optimization XIII: Exact Quadratic Decision Rules on EllipsoidsTextbook

Decisions made with partial information

An adjustable decision is a rule that responds to information revealed after an optimization problem is specified. In production planning, for example, a later quantity may depend on observed demand, while an earlier commitment must remain constant. A robust counterpart requires the resulting decisions to satisfy the constraints for every realization in a prescribed uncertainty set. The information available to each decision is part of the model: observing one linear combination of uncertain data does not permit a policy to use all the data coordinates.

Quadratic rules provide a finite family of responses with more flexibility than affine rules. The resulting robust constraints still quantify over infinitely many uncertain realizations. This mission concerns the case in which that infinite family has an exact finite semidefinite representation: fixed recourse and ellipsoidal uncertainty, as stated in Corollary 14.3.8 of Ben-Tal, El Ghaoui and Nemirovski. The result is about optimization over the selected quadratic rules; it makes no claim that these rules attain the best value among all possible responses.

The model and the information available to a policy

Let $z\in\mathbb R^L$ denote the uncertain data and let

Z=\{z:\|Qz\|_2\le1\},\qquad Q\in\mathbb R^{r\times L},\quad\ker Q=\{0\}.

The uncertain linear problem has $n$ decision variables and $m$ constraint rows:

\min_x\{c(z)^Tx+d(z):A(z)x\ge b(z)\}.

All four data functions are affine in $z$ . For variable $j$ , a specified matrix $P_j\in\mathbb R^{k_j\times L}$ determines its observations. A quadratic decision rule has the form

X_j(z)=p_j+q_j^TP_jz+(P_jz)^TR_j(P_jz).

The optimization variables are the coefficients $p_j,q_j,R_j$ and an objective threshold $t$ . The matrices $P_j$ are fixed data. The coefficient matrices $R_j$ are unrestricted: using full arrays retains every quadratic polynomial, with harmless redundancy in their skew-symmetric parts. If $P_j=0$ , the policy is constant.

Fixed recourse means that whenever $P_j\ne0$ , both $c_j(z)$ and column $A_j(z)$ are independent of $z$ . Coefficients of nonadjustable variables may still be uncertain. This is Definition 14.3.1, printed p. 371, including the condition on the objective. Robust epigraph feasibility means that $A(z)X(z)\ge b(z)$ and $c(z)^TX(z)+d(z)\le t$ hold for every $z\in Z$ .

Formalization targets

The supporting geometric target is Lemma 14.3.7, printed p. 384. The quadratic lift of $z$ is the symmetric matrix with top-left entry zero, first column $[0;z]$ , and lower-right block $zz^T$ . Its ordinary convex hull is compact and satisfies

\operatorname{conv}\left\{\begin{bmatrix}0&z^T\\z&zz^T\end{bmatrix}:z\in Z\right\} =\left\{\begin{bmatrix}0&v^T\\v&W\end{bmatrix}:\begin{bmatrix}1&v^T\\v&W\end{bmatrix}\succeq0,\ \operatorname{Tr}(QWQ^T)\le1\right\}.

The goal is Corollary 14.3.8. For every policy coefficient vector and threshold, robust epigraph feasibility is equivalent to feasibility of the explicit SDP after adding one nonnegative multiplier for each constraint row and one for the objective row. If the signed slack in row $i$ is written $\alpha_i+\beta_i^Tz+z^T\Gamma_i z$ , its matrix constraint is

\begin{bmatrix}\alpha_i-\lambda_i&\beta_i^T/2\\\beta_i/2&\Gamma_i+\lambda_iQ^TQ\end{bmatrix}\succeq0,\qquad\lambda_i\ge0.

The model defines these coefficients directly from the original data and the observation matrices. The theorem also asserts that the matrices depend affinely on the SDP decision vector. There are $m+1$ matrix constraints of order $L+1$ , and

N=1+\sum_j(1+k_j+k_j^2)+(m+1)

scalar decision variables. With $K=\sum_jk_j$ , the bound $N\le1+n(1+K+K^2)+(m+1)$ and the corresponding dense matrix coefficient bound make the polynomial-size conclusion explicit.

What the exact representation provides

The equivalence retains every original policy coefficient and objective threshold. It therefore preserves the entire robust feasible epigraph, including instances with no feasible policy and optimization problems whose values are unbounded. The goal also states equality of attainable thresholds and preservation of minimizers for the same linear objective $t$ in the finite SDP. It does not assume that either problem has a minimizer. Equality of a pair of real-valued infima would express less information and could conceal degenerate cases.

The mathematical results are established in the textbook. This mission supplies open Lean theorem statements for their formal verification and concrete definitions for their objects. The homogeneous S-lemma and the inhomogeneous S-procedure are already Proved on Prove2Me and are retained as references. The moment-hull identity and the complete quadratic-policy representation are the new targets. Their definition layer can also support work on polynomial policies and other lifted uncertainty sets.

Difficulty

Checking a quadratic constraint on an ellipsoid is a global requirement. Evaluating it at a finite collection of sample points does not establish the required universal inequality. Likewise, the image of a convex uncertainty set under the quadratic lift is generally not convex. The target must identify its whole convex hull, including compactness and boundary points.

The formalization must also preserve information restrictions through every matrix coefficient. Full observation cannot replace $P_jz$ , even if the resulting semidefinite program is easier to describe. The objective has uncertain affine data and participates in the same robust requirement as the constraints. These issues distinguish the complete statement from a single quadratic inequality or a result that assumes all coefficient columns are certain.

Formalization scope

All spaces and matrices are finite dimensional over the reals. The code uses finite functions for coordinates, but defines the uncertainty set through a sum of squares, so the norm is Euclidean. Matrix indices have a distinguished first coordinate. The unshifted lift has zero there; the positive-semidefinite moment block has one. Positive semidefiniteness also enforces symmetry of the moment matrix.

The model, uncertainty set, policy evaluation, original robust epigraph, and proposed SDP constraints are separate concrete definitions. Neither feasibility equivalence nor the moment-hull identity is assumed in the definitions. The theorem quantifies over every finite dimension, including zero; an empty observation list produces a constant policy. No division by a dimension or a policy-dependent denominator occurs.

The polynomial-size statement concerns the explicit real SDP representation, as in the book. It does not add a machine model or a bit-complexity theorem for an SDP solver. The formalization uses full quadratic coefficient arrays, so its stated counts need not be minimal. The bound includes all these redundant coefficients. There are no optional variants in this mission. The recorded sharpness observation about $Q$ explains why its injectivity is retained in the compactness theorem.

Selected references

A. Ben-Tal, L. El Ghaoui and A. Nemirovski, Robust Optimization, Princeton University Press, 2009. DOI: 10.1515/9781400831050. Definition 14.3.1, printed p. 371; equation (14.3.6), printed p. 375; Lemma 14.3.7 and Corollary 14.3.8, printed pp. 384–385. The pinned PDF pages are respectively 397, 401, and 410–411.

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Robust Optimization XIV: Extreme-Scenario Reduction in Multistage InventoryTextbook

Inventory decisions under uncertain demand

An inventory manager chooses replenishment orders before knowing all future demand. Orders made today affect both the stock available tomorrow and the cost of later decisions. A flexible commitment contract adds another consideration: deviations from previously announced orders can incur penalties. The uncertainty therefore concerns a sequence of observations, and the object being optimized is a sequence of decision rules. Chapter 15 of Ben-Tal, El Ghaoui and Nemirovski, Robust Optimization uses this setting to compare the cost of unrestricted adaptive policies with the cost of affine policies.

For such a comparison, an exact benchmark matters. Evaluating a policy against every trajectory in a continuous uncertainty set can require infinitely many constraints. The chapter's Lemma 15.2.1 identifies a class of problems whose optimal value is preserved by restricting uncertainty to finitely many extreme scenarios. This mission formalizes that general lemma, which the book applies to inventory management, rather than the numerical experiments or a particular contract data set.

A multistage convex model

A history at stage $t$ is the tuple $d^{t-1}=(d_0,\ldots,d_{t-1})$ , where each observation $d_j$ is a real vector of dimension $n_j$ . A policy specifies a real decision vector $S_t(d^{t-1})$ of dimension $m_t$ for every possible history. Dependence only on observations already available is called nonanticipativity. Two full trajectories with identical observed prefixes necessarily produce the same decision at that stage.

There are $T+1$ decision stages and uncertainty sets $F_0,\ldots,F_T$ . All data except the observations are fixed. Linear maps $A_t,B_{t+1},C_{t+1}$ and vectors $b_t$ impose the initial and linking inequalities

A_1S_1(d^0)\ge b_1,

A_{t+1}S_{t+1}(d^t)\ge B_{t+1}d^t+C_{t+1}S_t(d^{t-1})+b_{t+1},\qquad 1\le t\le T.

Vector inequalities hold coordinatewise. The term $B_{t+1}d^t$ uses the whole observed prefix. A fixed real bound $R$ restricts the infinity norm of each decision at stages $1,\ldots,T$ . The terminal decision $S_{T+1}$ still appears in the last linking constraint, but the printed norm restriction does not cover it.

Each decision incurs a stage cost $f_t$ . The costs are lower semicontinuous convex functions with polyhedral effective domains. A cost may be $+\infty$ outside its domain; within the domain it is finite. Polyhedrality of the domain does not require the cost itself to be piecewise linear. The total cost includes all $T+1$ stages.

Formalization target

Write $\operatorname{Opt}(F_0,\ldots,F_T)$ for the infimum of all real bounds $E$ for which some policy satisfies the preceding constraints and

\sum_{t=1}^{T+1}f_t(S_t(d^{t-1}))\le E \qquad\text{for every }d^T\in F_0\times\cdots\times F_T.

For nonempty polytopes $D_t\subset\mathbb R^{n_t}$ , the target is the complete value equality in Lemma 15.2.1, printed p. 430:

\operatorname{Opt}(D_0,\ldots,D_T) =\operatorname{Opt}(\operatorname{Ext}D_0,\ldots,\operatorname{Ext}D_T).

Here $\operatorname{Ext}D$ denotes the set of extreme points of $D$ . Both sides optimize over unrestricted nonanticipative policies. The target does not restrict those policies to affine rules, assert that every fixed policy has its worst cost at an extreme trajectory, or assert attainment of the optimum. The source states this lemma without a local numbered proof sequence, so the mission has one goal and its concrete model definitions, with no additional milestone chain.

What the result makes possible

For interval uncertainty, the extreme observations are interval endpoints. The book uses this consequence to replace a box of demand trajectories by a finite scenario set. It then describes a finite optimization problem whose decisions must agree whenever their available information agrees. These are exact comparisons of optimal values for the specified models, not estimates of sampling error. The number of scenarios can still grow exponentially with the horizon, as the chapter discusses on printed pp. 430–432 of the source.

The formalization makes the policy information pattern, terminal-stage treatment, and extended-value convention explicit. These details are useful beyond the inventory application because the lemma permits arbitrary compatible linear constraints and convex stage costs. The current artifacts state the known textbook result in Lean; a successful statement build is not a proof of the lemma. The reusable definitions are concrete and contain no unproved mathematical assertions.

Why the statement requires care

The uncertainty set is a product, but decisions at different stages are connected by constraints. A decision chosen for one trajectory cannot be chosen independently of another trajectory sharing the same history. Consequently, a statement about a single convex function on a polytope does not by itself express the multistage result. The target must retain the order of policy choice and universal quantification over trajectories, the complete cost sum, and the information available at each decision.

The final stage introduces a separate boundary issue: its cost and feasibility constraints remain present even though the printed bound stops one stage earlier. Omitting the final cost or extending the bound would change the problem. Degenerate and infeasible instances also require a value convention that does not assign them an arbitrary real number.

Formalization scope

Lean represents observations and decisions by finite real coordinate spaces, with dimensions allowed to vary by stage. Histories are finite dependent tuples; a policy's argument type contains exactly the observed prefix. Linear maps represent the book's matrices, including maps from stacked histories. Nonempty polytopes are convex hulls of nonempty finite sets, allowing singletons and lower-dimensional polytopes. The cost-domain predicate uses finitely many nonstrict matrix inequalities.

The extended optimal value is an EReal infimum of achievable real objective bounds. An empty set of bounds gives $+\infty$ , while bounds unbounded below give $-\infty$ . No subtraction of infinities occurs. Regular costs exclude $-\infty$ , so the finite sum in the objective has no mixed-infinity ambiguity. The definition does not assume feasibility, boundedness below, or attainment. Zero horizon and zero-dimensional coordinate spaces follow the same displayed formulas.

Formalization Note No singleton assumption is placed on $D_0$ , and no measurability, continuity, or affine restriction is imposed on policies. The theorem expands its concrete model for independent auditing. A Lean type check verifies agreement with the separately named reusable definitions. The required mathematical infrastructure includes convex hulls, extreme points, lower semicontinuity, finite linear algebra, and extended-real order. Completion requires a proof of the stated value equality with precisely these hypotheses.

Selected references

A. Ben-Tal, L. El Ghaoui and A. Nemirovski, Robust Optimization, Princeton University Press, 2009, Chapter 15, §15.2.4.1, Lemma 15.2.1, printed p. 430 (PDF p. 456). DOI: 10.1515/9781400831050.

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Robust Optimization XI: Finite Reformulations of Worst-Case Convex LossTextbook

Robust fitting with uncertain measurements

Regression and classification fit a parameter vector to observed data. When measurements can vary within a prescribed set, a fit judged only at the nominal observations need not control its loss on the possible data. A worst-case loss constraint requires the loss bound to hold for every admissible matrix of measurements. Such a requirement can contain infinitely many inequalities even when the parameter vector has finitely many coordinates. Chapter 12 of Ben-Tal, El Ghaoui and Nemirovski, Robust Optimization (2009) identifies structural assumptions under which this requirement has an exact finite convex representation.

This mission concerns the representation in Theorem 12.2.5. Its application is to robust fitting problems whose uncertainty support function separates into a feature-dependent factor and the infinity norm of a residual-space vector. The result applies to both symmetric losses, such as norms of regression residuals, and asymmetric losses formed from positive residual parts.

Data, residuals, and loss

Let $n,\ell,m$ be positive integers. The nominal data matrix $Z^n\in\mathbb R^{n\times m}$ has one column per measurement and one row per feature. For a parameter vector $\theta\in\mathbb R^n$ , its residual vector is $r=(Z^n)^T\theta\in\mathbb R^m$ . A fixed matrix $U\in\mathbb R^{n\times\ell}$ describes the structure of permitted perturbations, and a nonnegative scalar $\rho$ controls their size. The uncertain data matrices are

\mathcal Z=\{Z^n+\rho U\Delta:\Delta\in D\},

where $D\subset\mathbb R^{\ell\times m}$ is convex, compact, and contains zero. No independence of its columns is imposed.

The loss function $L:\mathbb R^m\to\mathbb R$ is finite and convex. Assumption L requires one of two forms: $L(r)=\pi(|r|)$ for every $r$ , or $L(r)=\pi(r_+)$ for every $r$ . Absolute value and positive part act coordinatewise. The function $\pi$ is convex and nondecreasing in every coordinate on the nonnegative orthant. The choice of form is global, rather than chosen separately at each residual.

Assumption A specifies the support of $D$ in every rank-one direction:

\max_{\Delta\in D}u^T\Delta v=\phi(u)\psi(v).

Here $\phi$ is finite, nonnegative, convex, and positively homogeneous, and $\psi$ is a norm. The goal uses $\psi(v)=\|v\|_\infty$ . These are the standing objects and assumptions of Section 12.2, printed pages 311–319 of the source.

Formalization targets

For any real threshold $\tau$ , the goal is the exact equivalence

\forall Z\in\mathcal Z:\ L(Z^T\theta)\le\tau

if and only if there exists a real $\kappa$ satisfying

\rho\phi(U^T\theta)\le\kappa,\qquad L((Z^n)^T\theta+\kappa e_i)\le\tau,\qquad L((Z^n)^T\theta-\kappa e_i)\le\tau\quad(1\le i\le m).

The goal also states convexity in $(\theta,\kappa)$ of the scalar functions on the left of these constraints. There are $2m$ shifted-loss inequalities and one amplitude inequality.

The numbered supporting targets preserve the source's identities. Equation 12.2.11 gives the Fenchel biconjugate representation of a finite convex loss. Equation 12.2.12 equates the worst-case realized loss with its conjugate expression involving the support of $D$ . Equation 12.2.15 replaces that support by its factorization under Assumption A. Equation 12.2.18 equates the resulting worst-case loss with a maximum over the dual unit ball:

L_{\rm wc}(r,\kappa)=\max_{\psi^*(\xi)\le1}L(r+\kappa\xi),\qquad \kappa\ge0.

The maxima in these identities are required to be attained. The Fenchel representation itself uses the source's supremum notation.

What the representation provides

Theorem 12.2.5 replaces a constraint over a continuum of data matrices with finitely many evaluations of the original loss and the uncertainty gauge. It retains the dependence of uncertainty on the fitted parameter through $\phi(U^T\theta)$ . Consequently, the representation describes exactly the robust feasible parameters, including zero uncertainty, rather than merely giving a sufficient condition. Evaluation efficiency remains relative to the supplied loss and gauge, as in the book's tractability discussion.

The mathematical result is established in the textbook. The contribution sought here is a machine-checked formalization of its statement and supporting identities. The general finite-dimensional Fenchel biconjugacy theorem already has a proved platform item and is reused as the first milestone; the other three identities and the capstone are open theorem statements in this draft.

Mathematical difficulty

The central issue is relating uncertainty in a matrix to uncertainty in the residual vector while preserving exactness. A uniform bound on individual matrix entries does not by itself express Assumption A. Conjugate functions also need extended values: replacing an infinite conjugate by a real default would change the optimization problem. Establishing equality of optimal values alone is insufficient for identities written with maxima; their attainment is part of the targets. The zero-amplitude case must agree with the nominal loss throughout.

Formalization scope

Vectors use finite-dimensional real Euclidean spaces and matrices have explicitly ordered feature and measurement indices. Coordinates use Fin m, so the first coordinate is indexed by zero. The infinity norm in the goal is taken on the plain coordinate function; it is not the Euclidean norm of the residual vector. The statements include both signs for every coordinate.

The published fenchelConjugate and fenchelBiconjugate definitions are reused. Conjugate objectives take values in EReal. Every claimed maximum is expressed using IsGreatest at a common finite real value, so existence, boundedness, equality, and attainment remain visible. The inner support supremum in Equation 12.2.12 is real-valued and ranges over a nonempty compact image. The dual unit ball in Equation 12.2.18 is represented directly by $\xi^Tv\le\psi(v)$ for all $v$ .

All hypotheses about the loss, gauge, norm, and uncertainty set are explicit in each new theorem. None assumes the desired finite reformulation. Assumption L is retained despite the recorded observation that a weaker loss hypothesis suffices for the mathematical equivalence. There are no variant targets. A computational cost model and an optimization algorithm are outside this formalization; the explicit finite formulas and their convexity are the representation being formalized. Reusable work includes finite-dimensional conjugacy, compact support maximization, and dual-norm geometry.

Selected references

A. Ben-Tal, L. El Ghaoui and A. Nemirovski, Robust Optimization, Princeton University Press, 2009. DOI: 10.1515/9781400831050. Section 12.2, printed pp. 311–319; Equations 12.2.11, 12.2.12, 12.2.15, 12.2.18 and Theorem 12.2.5.

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Robust Optimization II: Safe Robust Counterparts for Bounded Random PerturbationsTextbook

Motivation: reliability under random data

A linear constraint can fail when its coefficients vary. In an engineering or planning model, requiring it to hold for every imaginable realization may exclude useful decisions, while checking it only at an average realization may give little control over failure. A chance constraint specifies an acceptable probability of violation. Chapter 2 of Ben-Tal, El Ghaoui and Nemirovski, Robust Optimization studies deterministic convex conditions that guarantee this probabilistic requirement for a single uncertain linear inequality.

The chapter distinguishes a guarantee from an exact description of chance feasibility. Its robust constraints are safe approximations: satisfying them guarantees reliability, but a reliable decision need not satisfy them. The goal here concerns how known bounds on the perturbations can be incorporated into such an approximation without losing an explicit convex representation.

Setting: an uncertain scalar inequality

A decision is a vector $x\in\mathbb R^n$ . The deterministic data are vectors $a^0,a^1,\ldots,a^L$ and scalars $b^0,b^1,\ldots,b^L$ . Define the residual coordinates by $z_i(x)=(a^i)^Tx-b^i$ , including the nominal coordinate $i=0$ . A realized perturbation $\zeta\in\mathbb R^L$ satisfies the constraint exactly when

z_0(x)+\sum_{i=1}^L\zeta_i z_i(x)\le0.

For a probability law $P$ and a tolerance $0<\varepsilon<1$ , chance feasibility means that the strict reverse inequality has probability at most $\varepsilon$ . Robust feasibility over a deterministic set $Z$ means that the displayed inequality holds for every $\eta\in Z$ .

The perturbation coordinates are independent measurable real random variables. Their moment-generating functions obey the two-sided bound

\mathbb E_P e^{t\zeta_i}\le\exp\left(\max\{\mu_i^-t,\mu_i^+t\}+\frac12\sigma_i^2t^2\right)\qquad(t\in\mathbb R),

where $\mu_i^-\le\mu_i^+$ and $\sigma_i\ge0$ . These are the book's assumptions P.1–2. The goal additionally assumes almost-sure support bounds $a_i^-\le\zeta_i\le a_i^+$ , with finite endpoints satisfying $a_i^-\le\mu_i^-\le\mu_i^+\le a_i^+$ . No positive lower bound on $\sigma_i$ is imposed.

Formalization targets

Write $B(y)=\sum_i\max\{\mu_i^-y_i,\mu_i^+y_i\}$ , $D(y)=\sqrt{\sum_i\sigma_i^2y_i^2}$ , and $r_\varepsilon=\sqrt{2\log(1/\varepsilon)}$ . The general safety statement, Proposition 2.4.1, asserts that

z_0+B(z)+r_\varepsilon D(z)\le0\quad\Longrightarrow\quad P\{z_0+\sum_i z_i\zeta_i>0\}\le\varepsilon.

Its feasible set is convex. Proposition 2.4.2 states the tail bound at every nonnegative safety level $r$ :

P\{\sum_i z_i\zeta_i>B(z)+rD(z)\}\le e^{-r^2/2}.

Theorem 2.4.3 identifies the first deterministic inequality with robust feasibility over the moment uncertainty set

Z_{\rm mom}=\{\eta:\exists w,\ \mu_i^-\le\eta_i-\sigma_iw_i\le\mu_i^+\ (1\le i\le L),\ \sqrt{\sum_iw_i^2}\le r_\varepsilon\}.

The capstone is Theorem 2.4.4. Let $Z$ be the intersection of $Z_{\rm mom}$ with the support box $[a^-,a^+]$ . Then $x$ is robust feasible over $Z$ if and only if there exist extended residual vectors $u,v\in\mathbb R^{L+1}$ satisfying

z_i(x)=u_i+v_i\quad(0\le i\le L),

u_0+\sum_i\max\{a_i^-u_i,a_i^+u_i\}\le0,\qquad v_0+B(v)+r_\varepsilon D(v)\le0.

This system is convex in its decision and auxiliary variables. Every feasible triple has violation probability at most $\varepsilon$ . The theorem includes both the deterministic equivalence and the probabilistic guarantee, as on printed page 48.

Two supporting targets retain the chapter's bounded mean-zero specialization. Proposition 2.3.1 gives the tail estimate $P\{\sum_i z_i\zeta_i>r\sqrt{\sum_i z_i^2}\}\le e^{-r^2/2}$ for independent mean-zero perturbations in $[-1,1]$ . Proposition 2.3.3 identifies robustness over the intersection of the unit box and a Euclidean ball with its lifted conic system and establishes the corresponding chance guarantee. These are supporting statements from the source, not replacements for the general capstone.

Significance: explicit conditions with a probability guarantee

The capstone connects a deterministic set of allowed perturbations with a stochastic model specified through moment bounds and support information. It gives a finite convex system whose feasible decisions satisfy the requested reliability level for every law meeting those assumptions. The statement does not require choosing a particular parametric distribution or estimating a probability separately for every decision.

These are known textbook results. The formalization targets are their Lean statements and eventual machine-checked proofs, including zero scales and boundary feasibility. The reusable content comprises finite-dimensional uncertainty sets, explicit exponential-moment hypotheses, and the relationship between lifted convex constraints and strict-event probabilities. The present authoring pass leaves the six theorem proofs open; compilation checks their types, not their truth.

Difficulty: boundary cases carry mathematical content

The exact robust representation includes equality cases in the endpoint order and permits degenerate ellipsoids. A formulation restricted to positive scales or strict interior intersections would omit cases of the stated theorem. The chance guarantee must also include the boundary of the explicit safe set; it cannot be replaced by a condition requiring an additional strict margin.

The printed proofs have local discrepancies recorded in the mission findings: the proof of Proposition 2.4.2 omits a sum and invokes an open-interval risk bound at its excluded endpoint when the safety level is zero, the first line of the goal's proof uses a different radius, and the limiting argument on page 50 asserts a zero limit that fails for strictly feasible residuals. These findings concern the proofs, not counterexamples to the theorem statements. The goal retains the printed constants and hypotheses.

Formalization scope

The namespace is RobustOpt2009.ScalarChance. Vectors use real-valued functions on Fin L; the nominal coordinate is a separate real component. All Euclidean lengths are explicit square roots of sums of squares, avoiding the default function-space sup norm. Probability is evaluated under a probability measure, and all random coordinates are measurable. The exponential-moment definition includes integrability, preventing nonintegrable Bochner integrals from supplying default zero values.

Scaled coordinates $\sigma_iw_i$ implement the book's zero-denominator convention: a zero scale forces zero displacement. Robust feasibility is a universal inequality, so no empty or unbounded real supremum is used. The exponential-certificate set, its topological closure, and the perturbation uncertainty set remain separate concrete objects. No definition assumes the desired equivalence or tail bound. Finite sums include the zero-dimensional case, where the statements reduce to their deterministic or empty-event versions.

Further work consists of proving the stated concentration, convexity, and exact representation claims using these definitions and Mathlib's probability and finite-dimensional analysis. No variants are posed in this chapter.

Selected references

Aharon Ben-Tal, Laurent El Ghaoui and Arkadi Nemirovski, Robust Optimization, Princeton University Press, 2009, Chapter 2, especially printed pp. 29–33 and 45–50. DOI: 10.1515/9781400831050. Published citation basis; PDF pages are printed pages plus 26.

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Robust Optimization IV: The Bernstein Uncertainty SetTextbook

Chance constraints and robust feasibility

An optimization model can require a linear inequality to hold with high probability when its coefficients are random. Such a chance constraint controls the probability of violation, but its feasible set need not be convex or easy to describe. A conservative convex approximation can make the requirement usable within a larger optimization model. Chapter 4 of Ben-Tal, El Ghaoui and Nemirovski, Robust Optimization studies when such approximations also have a robust interpretation: the inequality must hold for every perturbation in a deterministic uncertainty set.

The target here is the chapter's Bernstein construction. It starts from information about the joint exponential moments of a random vector, rather than a fixed list of scenarios. Its conclusion identifies exactly the uncertainty set associated with the conservative approximation. Exactness concerns the relationship between that approximation and robust feasibility; the approximation is still only a sufficient condition for the original chance constraint.

Probability model and conjugate data

Let $\zeta$ have probability distribution $P$ on $\mathbb R^L$ . A coefficient vector is a pair $z=(z_0,w)$ , where $z_0\in\mathbb R$ and $w\in\mathbb R^L$ . The requirement is

P\{z_0+\langle w,\zeta\rangle>0\}\le\epsilon,\qquad 0<\epsilon<1.

The violation event is strict. Equality in the underlying linear inequality therefore counts as feasible. The parameter $\epsilon$ controls the permitted probability of failure.

A logarithmic moment-generating majorant is a finite convex function $\Phi:\mathbb R^L\to\mathbb R$ satisfying $\Phi(0)=0$ and

\log\mathbb E_P\exp\langle w,\zeta\rangle\le\Phi(w)\quad\text{for all }w.

These are Assumption Q.1. They concern the joint distribution; independence of the coordinates is not required. Assumption Q.2 provides an affine conjugate representation

\Phi(w)=\sup_{u\in\mathbb R^M}\{\langle w,Au+a\rangle-\phi(u)\},

where $A$ is linear, $a\in\mathbb R^L$ , and $\phi:\mathbb R^M\to\mathbb R\cup\{+\infty\}$ is convex and lower semicontinuous. Every real sublevel set of $\phi$ is bounded. Allowing $+\infty$ permits restrictions on the effective domain without changing the ambient space.

Formalization targets

The goal is Theorem 4.2.5, printed pp. 85–86 of the book. Define the coefficient sets independently by

Z^o_\epsilon=\{(z_0,w):\exists\alpha>0,\ \alpha z_0+\Phi(\alpha w)\le\log\epsilon\},\qquad Z_\epsilon=\operatorname{cl}Z^o_\epsilon.

Define the auxiliary sublevel set and its affine image by

U_\epsilon=\{u:\phi(u)\le\log(1/\epsilon)\},\qquad \mathcal Z_\epsilon=AU_\epsilon+a.

The theorem says that $U_\epsilon$ is nonempty, convex, and compact and that

(z_0,w)\in Z_\epsilon\iff z_0+\langle\zeta,w\rangle\le0\quad\text{for every }\zeta\in\mathcal Z_\epsilon.

It also retains the displayed perspective characterization and its probabilistic consequence:

(z_0,w)\in Z_\epsilon\iff\inf_{\beta>0}\left[z_0+\beta\Phi(\beta^{-1}w)+\beta\log(1/\epsilon)\right]\le0,

and this condition implies the original chance constraint.

Four source-numbered milestones accompany the goal. Lemma B.1.1 identifies the closure of a homogenized convex sublevel set with the zero sublevel set of its perspective infimum, including finiteness and convexity of that function. Proposition 4.2.2 characterizes the Bernstein approximation and asserts its normal safety. Proposition 4.2.3 gives bounded sublevel sets for an affine conjugate representation whose linear map has trivial kernel. Theorem B.1.2 gives the general robust representation under its directional-limit assumption. Each statement retains its source hypotheses and all conclusion clauses.

Mathematical and formal significance

The chapter's result gives a deterministic meaning to a probabilistic sufficient condition. An uncertainty set chosen through the conjugate data describes precisely the same coefficient constraints as the Bernstein approximation. This connects the probability model to robust linear feasibility while preserving the distinction between a guarantee and an exact description of the original chance-feasible set.

The formalization supplies separate, concrete definitions for the probability condition, the coefficient cone, and the perturbation set. That separation makes the equality in the main result a substantive theorem. The appendix statements also expose reusable finite-dimensional convex analysis, including perspective values, closed cones, and affine images of extended-real sublevel sets. These are known mathematical results from the book; the current draft consists of type-checked statements awaiting Lean proofs. A successful statement build is not a proof of any of these assertions.

Difficulty

Closure and optimization at a boundary are central issues. The perspective infimum ranges over strictly positive scales, but its value may be approached only as the scale tends to zero. Replacing the infimum condition by the existence of one feasible scale would change the target. Similarly, an extended-real function can impose a proper effective domain, so a formalization restricted to everywhere-finite penalties would omit cases covered by the theorem.

The general appendix theorem includes a directional limit below a specified level. That limit can be negative infinity, including in the application to the chapter's coefficient variable. Requiring a finite real limit would exclude the intended application. Probability expressions also require attention to the strict violation event and to the meaning of an expectation when exponential moments diverge.

Formalization scope

Finite-dimensional real spaces are represented by functions on Fin n, with the standard topology. Coefficients are pairs $(z_0,w)$ , so coefficient feasibility and perturbation membership have different types. The linear map is explicit and its affine translation is applied before taking the pairing. No independence, Gaussian distribution, bounded support, or injectivity assumption is added to the goal. Injectivity belongs only to Proposition 4.2.3 and its variant.

Formalization Note Infima and suprema are computed in the complete extended-real lattice. The penalty's negative-infinity value is explicitly excluded where the source uses $\mathbb R\cup\{+\infty\}$ . Convexity is expressed by the real epigraph and lower semicontinuity by closed real sublevels. Q.1 uses the equivalent bound on the nonnegative integral of the exponential, avoiding a default real-integral value on nonintegrable inputs. The uncertainty set is the actual affine image of the specified sublevel set, not an assumed certificate for robust feasibility.

Beyond the paper. One additional target strengthens Proposition 4.2.3 by removing only lower semicontinuity. Finding F1 records that finite bounds in the positive and negative coordinate directions, together with injectivity, already bound each sublevel set. The faithful proposition retains lower semicontinuity and remains the milestone. This strengthening does not change the goal or remove the topological assumptions needed for compactness elsewhere.

Selected references

A. Ben-Tal, L. El Ghaoui and A. Nemirovski, Robust Optimization, Princeton University Press, 2009. DOI: 10.1515/9781400831050. Chapter 4, §§4.1–4.2.3, printed pp. 81–86 (PDF pp. 107–112); Appendix B, §§B.1.1–B.1.3, printed pp. 471–475 (PDF pp. 497–501). Citation basis: published.

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Convex Optimization Operations Research Optimization·Captain: Shuze Chen

Robust Optimization III: Global Sensitivity for Linear ConstraintsTextbook

Motivation: controlling violations beyond normal uncertainty

A resource constraint can remain useful even when an exceptional disturbance makes exact satisfaction impractical. In such cases the model needs to distinguish the disturbances against which feasibility is mandatory from those for which a controlled violation is acceptable. Ordinary robust optimization specifies a set of perturbations and requires the constraint to hold everywhere in that set. It does not itself describe what happens outside that set. Chapter 3 of Ben-Tal, El Ghaoui and Nemirovski, Robust Optimization introduces a quantitative way to make this distinction for uncertain linear inequalities.

The object of this mission is the globalized robust counterpart of a single scalar inequality. Its tolerance is proportional to distance from a normal uncertainty range, with a prescribed restriction on admissible displacement directions. The goal is Proposition 3.2.1, which characterizes this global requirement by two ordinary robust inequalities. The result is a known theorem of the book; the target here is its faithful Lean formalization.

Setting: a perturbation structure and a scalar constraint

Let $x\in\mathbb R^n$ be a decision vector. The data consist of vectors $a^0,a^1,\ldots,a^L\in\mathbb R^n$ and scalars $b^0,b^1,\ldots,b^L\in\mathbb R$ . A perturbation vector $\zeta\in\mathbb R^L$ determines the inequality

\left[a^0+\sum_{\ell=1}^L\zeta_\ell a^\ell\right]^Tx \le b^0+\sum_{\ell=1}^L\zeta_\ell b^\ell.

The normal range $Z$ is a nonempty closed convex subset of $\mathbb R^L$ . A closed convex cone $K$ containing zero describes the possible directions of departure. The physically possible range is the sum

Z+K=\{z+d:z\in Z,\ d\in K\}.

Here $K$ denotes the book's $\mathcal L$ ; $L$ remains the number of perturbation coordinates. A fixed norm $N$ measures displacement. These are the perturbation-structure assumptions (G.a–c) on printed page 68 of the source.

For a physically possible perturbation, its constrained distance from the normal range is

d(\zeta,Z\mid K)=\inf\{N(\zeta-z):z\in Z,\ \zeta-z\in K\}.

The displacement restriction is part of the definition. The admissible set of displacement lengths is nonempty when $\zeta\in Z+K$ and is bounded below by zero. Thus the distance has a finite real value on precisely the domain where the counterpart uses it.

A nonnegative real parameter $\alpha$ is the global sensitivity. Global robustness requires the residual of the uncertain inequality to be at most $\alpha d(\zeta,Z\mid K)$ for every physically possible perturbation. This is Definition 3.1.1 and equation (3.1.4) of the book.

Formalization targets

Write the residual and its directional part as

r_x(\zeta)=\left[a^0+\sum_\ell\zeta_\ell a^\ell\right]^Tx-b^0-\sum_\ell\zeta_\ell b^\ell, \qquad q_x(d)=\left[\sum_\ell d_\ell a^\ell\right]^Tx-\sum_\ell d_\ell b^\ell.

The goal is the complete equivalence of Proposition 3.2.1, printed page 69:

\bigl[\forall\zeta\in Z+K,\ r_x(\zeta)\le\alpha d(\zeta,Z\mid K)\bigr] \quad\Longleftrightarrow\quad \begin{cases} r_x(\zeta)\le0&\forall\zeta\in Z,\\ q_x(d)\le\alpha&\forall d\in K\text{ with }N(d)\le1. \end{cases}

Both right-hand requirements are universal inequalities. The nominal coefficients occur in the first and do not occur in the second. The directional condition includes the zero direction and every direction on the boundary of the unit ball.

There are no separate supporting theorem milestones in this mission. The numbered Chapter 1 representation results discussed after the proposition describe applications of its conclusion. They are not intermediate targets needed to state this capstone.

Significance: separating two kinds of robustness

The equivalence distinguishes feasibility throughout the normal range from control of how the residual varies along admissible directions. As explained in Remark 3.2.2 of the book, it brings the globalized counterpart into contact with representation results for ordinary robust constraints. The proposition itself remains valid without a boundedness assumption on the normal range and without choosing a special shape such as a box or ellipsoid.

The formalization fixes the exact relationship between the domain of the distance and the quantifiers in the counterpart. It also makes the treatment of the cone and the norm explicit. These conventions are useful when subsequent developments specialize the model. The mission asks for a proof of the displayed equivalence; finite reformulations for particular uncertainty geometries and the chapter's antenna-array application are outside this target.

Difficulty: distance domains and boundary cases

The constrained distance cannot generally be replaced by ordinary distance to $Z$ . That substitution changes which normal points are eligible when evaluating a disturbance. The normal range can also be unbounded, so compactness of $Z$ is not available as a standing assumption. A formal proof must handle the domain of the infimum and the full directional quantification with these features intact.

The parameter $\alpha$ may be zero. In that case the globalized condition requires exact feasibility throughout $Z+K$ , as noted after Definition 3.1.1 in the source. The cone may consist only of zero, and the relevant displacement can have zero norm. None of these cases is excluded by a positivity assumption added for convenience.

Formalization scope

All coefficients and sensitivities are real. Decision and perturbation vectors use the finite coordinate types Fin n and Fin L; their zero-based indexing is a relabeling of the source sums. The nominal coefficients are separate parameters. The goal writes the coordinate sums and constrained infimum explicitly, using only Mathlib objects, so its meaning does not depend on an unpublished definition from another chapter.

An arbitrary norm is represented by Mathlib's real seminorm type together with definiteness: $N(v)=0$ implies $v=0$ . Closedness uses the usual topology on finite-dimensional real coordinate space. Convexity of both sets, nonemptiness of $Z$ , membership of zero in $K$ , and closure of $K$ under nonnegative scaling are explicit hypotheses. The global robustness condition is the distance inequality itself, not a definition chosen to assume its desired characterization.

The real infimum is used only over a nonempty set of nonnegative lengths. Its value on an empty set is not used by the theorem. Zero coordinate dimensions are permitted with empty sums interpreted as zero. The faithful goal retains convexity of $Z$ ; the separately recorded observation that it is unnecessary does not alter this statement. The remaining formal work concerns this exact equivalence, using finite sums, convex sets, seminorms, topology and real infima from Mathlib.

Selected references

Aharon Ben-Tal, Laurent El Ghaoui and Arkadi Nemirovski, Robust Optimization, Princeton University Press, 2009. DOI: 10.1515/9781400831050. Chapter 3, Definition 3.1.1 and equations (3.1.2)–(3.1.4), printed page 68 (PDF page 94); Proposition 3.2.1 and Remark 3.2.2, printed page 69 (PDF page 95); proof continuation, printed page 70 (PDF page 96).

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Quantum Information·Captain: Goku

The Security of Practical Quantum Key DistributionResearch Paper

Motivation

Quantum key distribution (QKD) is the first quantum information task to reach the level of mature technology, already fit for commercialization. It aims at the creation of a secret key between authorized partners connected by a quantum channel and a classical authenticated channel. The security of the key can in principle be guaranteed without putting any restriction on the eavesdropper's power. The first two sections provide a concise up-to-date review of QKD, biased toward the practical side. The rest of the paper presents the essential theoretical tools that have been developed to assess the security of the main experimental platforms (discrete variables, continuous variables and distributed-phase-reference protocols).

This mission is the statement-level formalization of The Security of Practical Quantum Key Distribution (arXiv:0802.4155v3). Every numbered result of the paper was scanned. The results stated here are open sorry goals, ready for solvers to prove.

What is formalized

15 statements of the paper are in Lean 4 (Mathlib 0df444a). The goal theorem is I.B.5: intercept-resend information and disturbance (I.B.5: intercept-resend information and disturbance, PDF p. 5). The milestones are the other numbered results, in the paper's order, each quoting the paper.

The paper's own objects are the definition item of this mission. Objects shared across papers come from the prove2quantum library and from earlier papers' missions, listed as references.

Provenance: prove2quantum's pipeline drafted each statement (gpt-6.1-sol on Stanford's Sherlock cluster). A blind read-back, written without seeing the paper, is attached to every item. An independent moderator compared each statement against the paper and passed the whole paper without edits.

Deferred results

28 results of the paper are not stated yet, because the library lacks an interface they need:

Paper seed, p.1: practical BB84 without decoy states, IV.B.2, Eq.(31): Operational tagged/squashed no-decoy BB84 extraction against all compatible attacks (hard)
II.C.3: schematic finite-key bound, Eq.(7): Printed schematic claim: Prob[||rho_KE-tau_K tensor rho_E||_1>2epsilon] is bounded up to omitted constants by exp(ell-F(rho_KE,epsilon)). (source-ambiguity)
II.E.1: phase randomization yields Poisson photon statistics, Eqs.(11)–(13): Normalized full-Fock coherent states and trace-class phase-average equality (hard)
III.B.2.a: one-way individual-attack bound, Eqs.(21)–(22): Uniform-over-compatible individual-attack one-way EC–PA achievability (hard)
III.B.2.b: one-way collective-attack bound, Eqs.(21),(23),(24): Uniform ccq direct/reverse extraction with half trace-distance security (hard)
IV.B.1: BB84 photon components and optimized privacy cost, Eq.(28): Constructed photon-tagged attack states, Holevo extrema and attaining attacks (hard)
IV.B.3: exact-parameter decoy BB84 rate and summation, Eqs.(33)–(34): Operational separately processed class extraction including vacuum contribution (hard)
IV.B.5: safe-source EB conservative multipair bound, Eqs.(40)–(42): Safe-SPDC multipair tagging and evaluated extraction (hard)
IV.B.5: EB qubit rate from a basis-independent source, Eq.(43): Basis-independent uncharacterized source to secure evaluated protocol (hard)
IV.C.2: calibrated decoy upper estimate, Eq.(50): Restricted calibrated-attack upper estimate with operational comparison (hard)
IV.C.2: calibrated no-decoy upper estimate, Eqs.(51)–(52): Feasible inferred calibrated parameters and restricted-attack upper comparison (hard)
V.B.3: Gaussian conditional variance and mutual information, Eqs.(60)–(61): Physical Gaussian experiment and actual mutual information (hard)
V.B.4: Eve–Alice conditional-variance tradeoff, Eqs.(62)–(65): Purification-conditioned quadrature variances and minimizing ensemble (hard)
V.B.4: reverse-reconciled individual-attack rate, Eq.(66): Gaussian individual-attack rate and saturating entanglement-cloner construction (hard)
V.B.4: qualitative high-loss / large-modulation noise tolerance: Source reports positivity in a high-loss (t*eta -> 0), large-modulation (v -> infinity) regime when epsilon<1/2. (source-ambiguity)
V.B.5: computed collective Gaussian Holevo information, Eqs.(67)–(68): Actual Gaussian output and conditioned state entropies (hard)
V.B.5: Gaussian-modulated homodyne collective security, Eq.(69): Gaussian extremality plus operational reverse-reconciled extraction (hard)
V.B.6: conditioned binary-channel postselection, Eqs.(70)–(71): Announcement-conditioned Gaussian instrument and averaged EC–PA extraction (hard)
VI.B.1: coherent beam-splitting attack and output states: Vacuum beam-splitter isometry on full Fock spaces (hard)
VI.B.1: DPS beam-splitting Holevo information and rate, Eqs.(73)–(74): Full-Fock two-pulse conditional mixtures and their Holevo spectra (hard)
VI.B.1: COW beam-splitting Holevo information and rate, Eqs.(75)–(76): Two-mode coherent/vacuum key states and their Holevo information (hard)
VI.B.2: restricted two-pulse COW upper estimate, Eqs.(77)–(78): Reported restricted-family expression: xi=2sqrt[V(1-V)]. If exp(-mu)<=xi, I_E=1; otherwise I_E=epsilon+(1-epsilon)h((1+F_V(mu))/2), F_V=(2V-1)exp(-mu)-xisqrt(1-exp(-2mu)). K=R*(1-I_E-leak_EC(Q)), R=nu_tilde*(mutt_B*eta+2p_d). (source-ambiguity)
VII.A–B: numerical platform and calibration comparisons: Figures 4–6 compare optimized K/nu_S for specified Table II parameter sets. BB84 curves are unconditional bounds under IV’s assumptions; CV curves concern collective attacks; COW curves concern only VI.B.2 restricted attacks. The stated calibration improvements are numerical/qualitative. (source-ambiguity)
Appendix A: Bell-diagonal errors and Eve’s information, Eqs.(A1)–(A4): Measured Bell purification and computed conditional Eve entropies (hard)
Appendix A: general six-state privacy cost, Eq.(A5): Six-state operational reduction with boundary-safe computed Bell Holevo cost (hard)
Appendix A: depolarizing six-state key fraction, Eq.(A6): Depolarizing six-state secure fraction and implicit first zero (hard)
Appendix A: BB84 entropy maximization and 11% threshold, Eqs.(A7)–(A9): BB84 entropy maximum and operational symmetry/extraction reduction (hard)
Appendix B: qualitative four-link repeater claims: The source reports ideal R_4 proportional to t^(1/4), four memories released before T_M, three Bell measurements, an approximately 95% fidelity requirement, and effective memory transmission p_M_prime≈p_M*t^(1/4). (source-ambiguity)

Selected references

The Security of Practical Quantum Key Distribution, arXiv:0802.4155v3, https://arxiv.org/abs/0802.4155v3

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Quantum Information·Captain: Goku

Duality and Rational Modules in Hopf Algebras over Commutative RingsResearch Paper

Motivation

Let $A$ be an algebra over a commutative ring $R$ . If $R$ is noetherian and $A^\circ$ is pure in $R^A$ , then the categories of rational left $A$ -modules and right $A^\circ$ -comodules are isomorphic. In the Hopf algebra case, we can also strengthen the Blattner-Montgomery duality theorem. Finally, we give sufficient conditions to get the purity of $A^\circ$ in $R^A$ .

This mission is the statement-level formalization of Duality and Rational Modules in Hopf Algebras over Commutative Rings (arXiv:math/0011207v1). Every numbered result of the paper was scanned. The results stated here are open sorry goals, ready for solvers to prove.

What is formalized

18 statements of the paper are in Lean 4 (Mathlib 0df444a). The goal theorem is Proposition 2.1 (Proposition 2.1, PDF p. 4). The milestones are the other numbered results, in the paper's order, each quoting the paper.

The paper's own objects are the definition item of this mission. Objects shared across papers come from the prove2quantum library and from earlier papers' missions, listed as references.

Deferred results

11 results of the paper are not stated yet, because the library lacks an interface they need:

Abstract: rational left modules and finite-dual right comodules: Full arbitrary-dimensional rational-module and right-comodule categories with the canonical inverse functors (hard)
Theorem 2.11: Rational-part endofunctor, left-exact preradical, inverse comodule functors, and sigma[A C] subgeneration (hard)
Proposition 2.15: Canonical finite-dual bialgebra/Hopf upgrades, contravariant morphisms, and pure coalgebra subobjects (hard)
Theorem 2.17: Both canonical categorical isomorphisms, both sigma identifications, and the full restriction/Rat commutative diagram (hard)
Lemma 3.1: Constructed Hopf actions, both smash-product algebras, lambda and opposite-algebra rho (hard)
Theorem 3.2: Right-comodule algebra, induced H-action, iterated smash algebra, RL condition, and algebra duality isomorphism (hard)
Remark 3.4: group-algebra duality application: Rationality/local-finiteness of the arbitrary group action, needed for the printed application of Theorem 3.2 (source-ambiguity)
Corollary 4.3: Canonical finite-dual coalgebra with all conditional bialgebra/Hopf upgrades and rational-pairing compatibility (hard)
Example 4.7: strongly graded and crossed-product algebras: Internal strong finite-group gradings and general crossed, twisted, ordinary, and skew group algebras over noncommutative coefficients (hard)
§4.2: polynomial algebras and their finite duals: Two distinct canonical polynomial coalgebra structures and their finite-dual bialgebra/Hopf upgrades, plus coefficient-algebra closure (hard)
Theorem 4.14: Multivariate central Laurent algebra closure and the canonical multiplicative-group Hopf structure on its finite dual (hard)

Selected references

Duality and Rational Modules in Hopf Algebras over Commutative Rings, arXiv:math/0011207v1, https://arxiv.org/abs/math/0011207v1

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Quantum Information·Captain: Goku

Higher-Dimensional Algebra V: 2-GroupsResearch Paper

Motivation

A 2-group is a "categorified" version of a group, in which the underlying set G has been replaced by a category and the multiplication map has been replaced by a functor. Various versions of this notion have already been explored; our goal here is to provide a detailed introduction to two, which we call "weak" and "coherent" 2-groups. A weak 2-group is a weak monoidal category in which every morphism has an inverse and every object x has a "weak inverse": an object y such that x tensor y and y tensor x are isomorphic to 1. A coherent 2-group is a weak 2-group in which every object x is equipped with a specified weak inverse x* and isomorphisms i_x: 1 -> x tensor x* and e_x: x* tensor x -> 1 forming an adjunction. We describe 2-categories of weak and coherent 2-groups and an "improvement" 2-functor that turns weak 2-groups into coherent ones, and prove that this 2-functor is a 2-equivalence of 2-categories. We internalize the concept of coherent 2-group, which gives a quick way to define Lie 2-groups. We give a tour of examples, including the "fundamental 2-group" of a space and various Lie 2-groups. We also explain how coherent 2-groups can be classified in terms of 3rd cohomology classes in group cohomology. Finally, using this classification, we construct for any connected and simply-connected compact simple Lie group G a family of 2-groups G_hbar (for integral values of hbar) having G as its group of objects and U(1) as the group of automorphisms of its identity object. These 2-groups are built using Chern-Simons theory, and are closely related to the Lie 2-algebras g_hbar (for real hbar) described in a companion paper.

This mission is the statement-level formalization of Higher-Dimensional Algebra V: 2-Groups (arXiv:math/0307200v3). Every numbered result of the paper was scanned. The results stated here are open sorry goals, ready for solvers to prove.

What is formalized

8 statements of the paper are in Lean 4 (Mathlib 0df444a). The goal theorem is Strong monoidal functors preserve weak inverses (Proposition 3: strong monoidal functors preserve weak inverses, PDF p. 9). The milestones are the other numbered results, in the paper's order, each quoting the paper.

The paper's own objects are the definition item of this mission. Objects shared across papers come from the prove2quantum library and from earlier papers' missions, listed as references.

Deferred results

37 results of the paper are not stated yet, because the library lacks an interface they need:

Abstract; Theorem 17: improvement and forgetting are a 2-equivalence: Universe-correct W2G and C2G, strict improvement/forgetting 2-functors, and a 2-natural isomorphism (hard)
Proposition 11: free coherent 2-groups: The strict-preservation category c2g and its object-set forgetful functor (hard)
Theorem 12: coherence of free coherent 2-groups: A syntactic free coherent 2-group and substitution functor on c2g (hard)
Corollary 16: improvement extends to a 2-functor: The strict 2-category structures and actual Imp 2-functor (hard)
Proposition 20: internal coherent 2-groups in Cat: Internal coherent 2-group in Cat with natural covariant inverse functor (hard)
Proposition 32: internal strict 2-groups and crossed modules: Internal groups, kernels, semidirect products and internal category objects for arbitrary K (hard)
Proposition 39: every coherent 2-group equivalent to a special one: A source-corrected, explicitly scoped replacement for the printed assertion (source-ambiguity)
Lemma 40: transport coherent structure along tensor comparison: Dependent transport of full monoidal/coherent data with specified identity comparisons (hard)
Proposition 42: normalize homomorphisms of special 2-groups: Special coherent objects, normalized strong monoidal functors and monoidal natural isomorphisms (hard)
Theorem 43: cohomological description of objects, homomorphisms and 2-homomorphisms: A source-corrected, explicitly scoped replacement for the printed assertion (source-ambiguity)
Corollary 44: classification up to equivalence by H3 data: Intrinsic associator cohomology invariant and quotient of quadruples by compatible group isomorphisms (hard)
Proposition 45: strictification without preserving skeletalness: Fully strict 2-group structure and strong monoidal equivalence (hard)
Theorem 55: integer Chern–Simons family: A source-corrected, explicitly scoped replacement for the printed assertion (source-ambiguity)
Proposition 58: classification of special internal 2-groups: A source-corrected, explicitly scoped replacement for the printed assertion (source-ambiguity)
Theorem 59: special topological 2-groups with connected Lie data have vanishing associator class: Special topological 2-groups, global continuous H3 and continuous special equivalence (hard)
Corollary 60: no ordinary topology on nonzero Chern–Simons examples: Chern–Simons integral class injection and isomorphism to the topological family presentation (hard)
Example 33: Strict automorphism 2-group: Strict automorphism monoidal groupoid of an arbitrary strict 2-category (hard)
Example 34: Weak automorphism 2-group: Weak automorphism monoidal groupoid of an arbitrary bicategory (hard)
Example 35: Autoequivalence 2-group: Autoequivalence groupoid with nonadjoint equivalence data and compatible pairs of 2-isomorphisms (hard)
Example 36: Adjoint autoequivalence 2-group: Adjoint autoequivalence groupoid with paired compatible 2-isomorphisms (hard)
Example 47: Skeletal strict Lie 2-groups from smooth actions: Smooth crossed-module/internal Lie-category construction and converse for every skeletal strict Lie 2-group (hard)
Example 48: Tangent 2-group: Tangent Lie group, adjoint semidirect product identification and tangent groupoid (hard)
Example 49: Cotangent 2-group: Coadjoint semidirect product Lie groupoid and fixed cotangent trivialization (hard)
Example 50: Representations and Poincare 2-groups: Representation Lie 2-group and its defining Lorentz/Poincare specialization (hard)
Example 51: Strict Lie automorphism 2-group: A source-corrected, explicitly scoped replacement for the printed assertion (source-ambiguity)
Example 52: SU(2) and quaternionic automorphism examples: A source-corrected, explicitly scoped replacement for the printed assertion (source-ambiguity)
Example 53: Surjective crossed modules are precisely central extensions: Unique smooth descended conjugation action for a surjective smooth homomorphism (hard)
Example 54: Heisenberg 2-group: Heisenberg Lie group with coefficient-one multiplication and resulting Lie 2-group (hard)
Section 7: internal categories form a strict 2-category with products: KCat with internal functors/transformations, strict 2-category laws and finite products (hard)
Section 8.1: bitorsors are equivalent to the strict automorphism 2-group: Balanced-product monoidal groupoid of bitorsors and monoidal equivalence to Auts(G) (hard)
Section 8.2: fundamental coherent 2-group of a pointed space: Actual loops, relative-boundary bigon quotient and reverse-path adjunction (hard)
Section 8.2: realize an autoequivalence 2-group topologically: A source-corrected comparison between an adjoint-equivalence core and its fundamental 2-group, explicitly distinguishing Auteq from Autad (source-ambiguity)
Section 8.2: proposed biequivalence with pointed connected homotopy 2-types: A source-corrected, explicitly scoped replacement for the printed assertion (source-ambiguity)
Section 8.3: vanishing H3 class characterizes skeletal strictification: Intrinsic H3 class and simultaneous skeletal/strict representative equivalence (hard)
Section 8.4: strict Lie 2-groups and Lie crossed modules: Two categories of smooth internal groupoids/crossed modules with derivation 2-morphisms (hard)
Section 8.4: tangent groupoid is a product-preserving functor: Product-preserving Diff to DiffCat tangent-groupoid functor (hard)
Section 8.5 proof: connected compact Lie continuous cohomology vanishes in degree at least three: Global normalized continuous cochain complex for arbitrary connected abelian Lie coefficients (hard)

Selected references

Higher-Dimensional Algebra V: 2-Groups, arXiv:math/0307200v3, https://arxiv.org/abs/math/0307200v3

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Quantum Information·Captain: Goku

Composing Quantum Protocols in a Classical EnvironmentResearch Paper

Motivation

We propose a general security definition for cryptographic quantum protocols that implement classical non-reactive two-party tasks. The definition is expressed in terms of simple quantum-information-theoretic conditions which must be satisfied by the protocol to be secure. The conditions are uniquely determined by the ideal functionality F defining the cryptographic task to be implemented. We then show the following composition result. If quantum protocols pi_1,...,pi_k securely implement ideal functionalities F_1,...,F_k according to our security definition, then any purely classical two-party protocol, which makes sequential calls to F_1,...,F_k, is equally secure as the protocol obtained by replacing the calls to F_1,...,F_k with the respective quantum protocols pi_1,...,pi_k. Hence, our approach yields the minimal security requirements which are strong enough for the typical use of quantum protocols as subroutines within larger classical schemes. Finally, we show that recently proposed quantum protocols for oblivious transfer and secure identification in the bounded-quantum-storage model satisfy our security definition, and thus compose in the above sense.

This mission is the statement-level formalization of Composing Quantum Protocols in a Classical Environment (arXiv:0804.1059v2). Every numbered result of the paper was scanned. The results stated here are open sorry goals, ready for solvers to prove.

What is formalized

3 statements of the paper are in Lean 4 (Mathlib 0df444a). The goal theorem is Lemma 2.1: three approximate conditional-independence bounds (Lemma 2.1, PDF p. 4). The milestones are the other numbered results, in the paper's order, each quoting the paper.

The paper's own objects are the definition item of this mission. Objects shared across papers come from the prove2quantum library and from earlier papers' missions, listed as references.

Deferred results

13 results of the paper are not stated yet, because the library lacks an interface they need:

Paper headline / Theorem 5.1: composing quantum protocols in a classical environment: Operational two-party CPTP strategies, computed real/ideal execution, and the source-exact Definition 4.1 on the same protocol; computed sequential hybrid substitution (hard)
Proposition 4.2: security with classical side information and conditional quantum input: Operational two-party CPTP strategies, computed real/ideal execution, and the source-exact Definition 4.1 on the same protocol (hard)
Proposition 4.3: simulation for conditionally independent quantum input: Operational two-party CPTP strategies, computed real/ideal execution, and the source-exact Definition 4.1 on the same protocol (hard)
Corollary 5.2: security of a hybrid implementation after substitution: Admissibility of the hybrid simulators in the adversary classes required by hybrid security (source-ambiguity)
Proposition 6.2: identification security implies ideal functionality security: Operational two-party CPTP strategies, computed real/ideal execution, and the source-exact Definition 4.1 on the same protocol (hard)
Proposition 7.2: sender-randomized 1-2 OT security implies ideal functionality security: Operational two-party CPTP strategies, computed real/ideal execution, and the source-exact Definition 4.1 on the same protocol (hard)
Unnumbered §7: bootstrapping randomized-OT correctness from the older definition: A quantitative error conversion for the older randomized-OT correctness definition (source-ambiguity)
Proposition B.2: regular 1-2 OT security implies ideal functionality security: Operational two-party CPTP strategies, computed real/ideal execution, and the source-exact Definition 4.1 on the same protocol (hard)
Proposition B.4: fully randomized 1-2 OT security implies ideal functionality security: A source-justified sharp Markov estimate consistent with the printed 4ε headline (source-ambiguity)
Proposition B.6: randomized Rabin OT security implies ideal functionality security: Operational two-party CPTP strategies, computed real/ideal execution, and the source-exact Definition 4.1 on the same protocol (hard)
Proposition C.2: identification without unfairness: A source-justified ε Markov estimate or an independently justified correction of the headline constant (source-ambiguity)
Unnumbered introduction: composability of the cited bounded-storage protocols: Concrete cited protocol implementations, their memory bounds, and a source-exact security/correctness bridge (hard)
Unnumbered Appendix C: the cited identification protocol also satisfies the stronger definition: The concrete cited Q-ID execution and a complete match of its extractor/security experiment to Definition C.1 (hard)

Selected references

Composing Quantum Protocols in a Classical Environment, arXiv:0804.1059v2, https://arxiv.org/abs/0804.1059v2

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Quantum Information·Captain: Goku

Central Extensions of Smooth 2-Groups and a Finite-Dimensional String 2-GroupResearch Paper

Motivation

We provide a model of the String group as a central extension of finite-dimensional 2-groups in the bicategory of Lie groupoids, left-principal bibundles, and bibundle maps. This bicategory is a geometric incarnation of the bicategory of smooth stacks and generalizes the more naïve 2-category of Lie groupoids, smooth functors, and smooth natural transformations. In particular this notion of smooth 2-group subsumes the notion of Lie 2-group introduced by Baez-Lauda. More precisely we classify a large family of these central extensions in terms of the topological group cohomology introduced by G. Segal, and our String 2-group is a special case of such extensions. There is a nerve construction which can be applied to these 2-groups to obtain a simplicial manifold, allowing comparison with with the model of A. Henriques. The geometric realization is an $A_\infty$ -space, and in the case of our model, has the correct homotopy type of String(n). Unlike all previous models our construction takes place entirely within the framework of finite dimensional manifolds and Lie groupoids. Moreover within this context our model is characterized by a strong uniqueness result. It is a unique central extension of Spin(n).

This mission is the statement-level formalization of Central Extensions of Smooth 2-Groups and a Finite-Dimensional String 2-Group (arXiv:0911.2483v3). Every numbered result of the paper was scanned. The results stated here are open sorry goals, ready for solvers to prove.

What is formalized

3 statements of the paper are in Lean 4 (Mathlib 0df444a). The goal theorem is Lemma 51: double inverse is naturally identity (Lemma 51, PDF p. 20). The milestones are the other numbered results, in the paper's order, each quoting the paper.

The paper's own objects are the definition item of this mission. Objects shared across papers come from the prove2quantum library and from earlier papers' missions, listed as references.

Deferred results

36 results of the paper are not stated yet, because the library lacks an interface they need:

Theorem 1: classification of central extensions: Smooth central-extension bicategory, smooth Segal-Mitchison cohomology and symmetric monoidal biequivalence (hard)
Theorem 2: finite-dimensional String model and coherent uniqueness: Spin(n), fixed-class smooth extension bicategory, and String realization (hard)
Lemma 9: Cech groupoid equivalence over X: Smooth internal Cech groupoids, slice functors and natural isomorphisms (hard)
Proposition 23: bundlization criterion [Ler08]: Left-principal smooth bibundle, bundlization and bibundle isomorphism (hard)
Theorem 26: stacks, bibundles and localization (Pronk [Pro96]): Bibun, presentable smooth stacks and bicategorical localization at essential equivalences (hard)
Lemma 29: four action-base-change transversality statements: Transversality of target anchors and four smooth action-base-change domains (hard)
Lemma 30: transverse contracted product is a manifold: Reversed right-principal bibundle and smooth contracted coequalizer (hard)
Proposition 31: transverse weak pullbacks in Bibun: Bibun pullback cone and universal equivalence of cone categories (hard)
Theorem 57: three equivalent 2-group conditions: Weak pushout F of monoidal categories, map s and s-locality (hard)
Corollary 59: cocompleteness and monadic reflection: Bicategory of weak 2-groups, weak colimits, reflection and two-monadicity (hard)
Corollary 60: free 2-group adjunction is monadic: Forgetful pseudofunctor from weak 2-groups to groupoids and free weak adjunction (hard)
Theorem 61: 2-groups in localized groupoid presheaves: Reflective groupoid-presheaf bicategory, monoidal objects and evaluation adjoints (hard)
Lemma 76: kernel square of an extension: Principal smooth two-group extension and homotopy kernel cone (hard)
Proposition 79: initial kernel square: Kernel-square bicategory KS(f) with fixed-edge morphisms (hard)
Lemma 80: extension descent is a homotopy colimit: Augmented weak Cech simplicial two-group and bicolimit cone (hard)
Corollary 81: monadic bar resolution: Monadic weak adjunction, augmented comonadic bar and bicolimit (hard)
Lemma 82: induced conjugation action on an abelian kernel: Automorphism two-group and conjugation-compatible descent category (hard)
Lemma 84: skeletal discrete extension classification: Skeletal weak two-group Gamma(G,A,rho,alpha) and extension equivalence (hard)
Lemma 88: relative universal bundle: Universal groupoid bundle realization, contracted space and topological section space (hard)
Corollary 89: homotopy-functorial bibundle realization: Section-space family of realized bibundle maps and composition (hard)
Corollary 91: Morita invariance of realization: Morita bibundle equivalence and nerve realization homotopy equivalence (hard)
Theorem 94: natural A-infinity realization: Section-space nonsymmetric A-infinity operad and its algebra (hard)
Corollary 97: circle and integral coefficients: Smooth Segal-Mitchison cochain complex, coefficient connecting maps and BG comparison (hard)
Lemma 98: splitting a cochain bicategory: Strict symmetric monoidal cochain bicategory and its coherent splitting (hard)
Theorem 99: natural cocycle-level extension classification: Normalized smooth cocycle bicategory and cocycle-to-extension pseudofunctor (hard)
Theorem 100: generator extension models String(n): Smooth Spin(n), generator classification and realized String projection (hard)
Remark 58: groupoid characterization and F equivalent to Z: Isomorphism-class monoid of a monoidal groupoid and weak pushout F (hard)
Example 62: free 2-groups on all smooth stacks: All smooth stacks, stackification localization and free two-group adjunction (hard)
Unnumbered: abelian extension square is also a pushout: Abelian smooth two-group extension and bicategorical pushout cone (hard)
Unnumbered: symmetric monoidal bicategory of central extensions: Central-extension bicategory with pullback/pushforward and Baer sum (hard)
Unnumbered: functorial group-like H-space realization: Smooth two-group realization, group-like H-space and homotopy functor (hard)
Remark 95: homomorphisms realize to A-infinity morphisms: Compatible section-operad structures and A-infinity morphism (hard)
Unnumbered §4.3: cocycle edge class and realized extension: Topological Dold-Kan double delooping, edge map and realized pullback (hard)
Remark 101: opposite generators and coefficient inversion: Coefficient-inversion comparison of opposite-generator extensions (hard)
Unnumbered §4.4: extensions with nontrivial module action: Twisted smooth Segal-Mitchison cocycle extension with its induced action (hard)
Unnumbered §4.4: announced general cohomological classification: Bicategorical H2(G;A), connecting maps and degree-shift comparison (source-ambiguity)

Selected references

Central Extensions of Smooth 2-Groups and a Finite-Dimensional String 2-Group, arXiv:0911.2483v3, https://arxiv.org/abs/0911.2483v3

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