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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.

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.

≤ 1.9992Formalized record

2 provers on it1 of 1 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.

≤ 2.9983Formalized record≤ 2.99791Open frontier

3 provers on it2 of 3 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.

≤ 7.606309Formalized record

6 provers on it7 of 7 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.

≤ 84Formalized record≤ 80Open frontier

3 provers on it6 of 7 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.

≤ 41Formalized record≤ 5Open frontier

35 provers on it11 of 13 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?

≤ 2.37134Formalized record≤ 2.371177Open frontier

16 provers on it7 of 8 missions formalized

All missions

Get started

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

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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.

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.

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.

≤ 1.9992Formalized record

2 provers on it1 of 1 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.

≤ 2.9983Formalized record≤ 2.99791Open frontier

3 provers on it2 of 3 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.

≤ 7.606309Formalized record

6 provers on it7 of 7 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.

≤ 84Formalized record≤ 80Open frontier

3 provers on it6 of 7 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.

≤ 41Formalized record≤ 5Open frontier

35 provers on it11 of 13 missions formalized

Matrix multiplication exponent

≤ 2.37134Formalized record≤ 2.371177Open frontier

16 provers on it7 of 8 missions formalized

All missions

Completed

Operations Research Optimization Probability+1·Captain: mikedeng1

Single Machine Scheduling with Release Dates II: The Random α-Schedule with a Truncated Exponential DensityResearch Paper

Motivation

Minimizing the total weighted completion time $\sum_j w_j C_j$ of jobs with release dates on a single machine, written $1\,|\,r_j\,|\,\sum w_j C_j$ in scheduling notation, is strongly NP-hard. It is one of the basic models of machine scheduling, and it has served as a test case for a general technique in approximation algorithms: solve a linear programming relaxation, read off a preemptive schedule, and convert it into a nonpreemptive one using $\alpha$ -points, the times at which a given fraction of each job has been processed. Phillips, Stein and Wein (doi:10.1007/BF01585872) introduced ordering jobs by points of a preemptive schedule; Goemans (SODA 1997, reference [11] of the paper below) and Chekuri, Motwani, Natarajan and Stein (doi:10.1137/S0097539797327180) showed that choosing $\alpha$ at random improves the guarantee.

Goemans, Queyranne, Schulz, Skutella and Wang (doi:10.1137/S089548019936223X) combine two LP relaxations, shown to have equal value, with carefully chosen random $\alpha$ . This mission formalizes their Theorem 3.5: when a single $\alpha$ is drawn from a truncated exponential density, the resulting schedule costs in expectation at most $c < 1.7451$ times the LP lower bound.

Timeline:

1998: Phillips, Stein and Wein give the first constant-factor approximation (ratio $2$ ) for the unit-weight problem $1\,|\,r_j\,|\,\sum C_j$ by converting a preemptive schedule.
1997: Goemans (SODA) introduces randomly chosen $\alpha$ -points for the weighted problem, the conference precursor of the paper formalized here.
2001: Chekuri, Motwani, Natarajan and Stein give a randomized $e/(e-1) \approx 1.58$ -approximation for the unit-weight problem.
2002: Goemans, Queyranne, Schulz, Skutella and Wang give $1.7451$ for a single random $\alpha$ (Theorem 3.5) and $1.6853$ for job-dependent random $\alpha_j$ (Theorem 3.9), both against the same LP bound.
1999: Afrati et al. (doi:10.1109/SFFCS.1999.814574) give a polynomial-time approximation scheme for the problem. It is not LP-based, so the LP-relative factors of Theorems 3.5 and 3.9 remain of interest as bounds on the relaxations' integrality gaps.

Setting

There are $n$ jobs $N = \{1, \dots, n\}$ . Job $j$ has an integral processing time $p_j > 0$ , an integral release date $r_j \ge 0$ and a weight $w_j > 0$ . The jobs are indexed so that $w_1/p_1 \ge w_2/p_2 \ge \cdots \ge w_n/p_n$ .

A preemptive schedule gives each job a set $A_j \subseteq [r_j, \infty)$ of processing times of measure $p_j$ , the sets pairwise disjoint. The mean busy time of $j$ is $M_j = \frac{1}{p_j}\int_{A_j} t\,dt$ .

The LP schedule is the preemptive schedule that always processes the available job of smallest index, which under the indexing above is the available job of largest ratio $w_j/p_j$ . Its mean busy times are $M^{LP}_j$ .

The mean busy time relaxation (R) minimizes $\sum_j w_j (M_j + \tfrac12 p_j)$ subject to $\sum_{j \in S} p_j M_j \ge p(S)\big(r_{\min}(S) + \tfrac12 p(S)\big)$ for every nonempty $S \subseteq N$ , where $p(S) = \sum_{j\in S} p_j$ and $r_{\min}(S) = \min_{j\in S} r_j$ . Its optimal value is $Z_R$ , a lower bound on the optimum of $1\,|\,r_j\,|\,\sum w_j C_j$ .

For $0 < \alpha \le 1$ , the $\alpha$ -point $t_j(\alpha)$ is the first time at which $j$ has received $\alpha p_j$ units of processing in the LP schedule, and $t_j(0^+)$ is its start time. The $\alpha$ -schedule processes the jobs nonpreemptively, each as early as possible, in nondecreasing order of $t_j(\alpha)$ ; $C^\alpha_j$ is the completion time of $j$ in it. More generally, the $(\alpha_j)$ -schedule orders the jobs by $t_j(\alpha_j)$ for a vector $\boldsymbol\alpha = (\alpha_j)$ .

Let $0 < \gamma < 1$ solve $1 - \frac{\gamma^2}{1+\gamma} = \gamma + \ln(1+\gamma)$ ( $\gamma \approx 0.4675$ ), and set

c = \frac{1+\gamma}{1+\gamma-e^{-\gamma}}, \qquad \delta = 1 - \frac{\gamma^2}{1+\gamma}, \qquad f(\alpha) = \begin{cases}(c-1)e^{\alpha} & 0 < \alpha \le \delta,\\ 0 & \text{otherwise.}\end{cases}

Formalization targets

Goal: Theorem 3.5

If $\alpha$ is drawn with density $f$ , then $c < 1.7451$ , the expectation below is finite, and

\mathbb E_f\Big[\sum_{j} w_j C^\alpha_j\Big] \le c \cdot Z_R .

Milestones

Theorem 2.5: $M^{LP}$ is an optimal solution to (R), so $Z_R = \sum_j w_j (M^{LP}_j + \tfrac12 p_j)$ .
Eq. (3.1): $M^{LP}_j = \int_0^1 t_j(\alpha)\,d\alpha$ .
Corollary 3.2: $C^{\boldsymbol\alpha}_j \le t_j(\alpha_j) + \sum_{k:\,\alpha_k\le\eta_k(\alpha_j)} (1+\alpha_k-\eta_k(\alpha_j))\,p_k$ , where $\eta_k(\alpha_j)$ is the fraction of $k$ processed by $t_j(\alpha_j)$ in the LP schedule.
Eq. (3.10): $M^{LP}_j = t_j(0^+) + \sum_{k\in N_2}(1-\mu_k)p_k + \tfrac12 p_j$ , where $N_2$ is the set of jobs processed between the start and the completion of $j$ and $\mu_k$ the fraction of $j$ done before $k$ starts.
Eq. (3.11): the bound of Corollary 3.2 rewritten in terms of $N_1$ , $N_2$ and $\mu_k$ .
Lemma 3.6: $f$ is a density on $[0,1]$ with $\int_0^\eta f(\alpha)(1+\alpha-\eta)\,d\alpha \le (c-1)\eta$ and $\int_\mu^1 f(\alpha)(1+\alpha)\,d\alpha \le c(1-\mu)$ for $\eta, \mu \in [0,1]$ .

Significance

Theorem 3.5 is a randomized $1.7451$ -approximation for $1\,|\,r_j\,|\,\sum w_j C_j$ , and simultaneously a bound on the integrality gap of (R): every instance has $\mathrm{OPT} \le 1.7451\, Z_R$ . The paper also derandomizes it: there are at most $n$ distinct $\alpha$ -schedules (its Proposition 3.8), and the best of them can be found in $O(n^2)$ time. The paper remarks that the truncated exponential density is optimal for its analysis and that the factor is tight for it.

On the formal side, the mission produces a reusable model of single-machine preemptive schedules, the LP schedule, $\alpha$ -points and list scheduling, and the first machine-checked instance of the $\alpha$ -point rounding technique that recurs across scheduling approximation. The result itself is proved in the paper; no machine-checked proof of it or of the LP-schedule identities (3.1), (3.10) is known to exist.

Difficulty

The obvious approach, bounding each $C^\alpha_j$ separately by a multiple of $t_j(\alpha)$ , gives only the factor $\max\{1 + 1/\alpha, 1 + 2\alpha\}$ for fixed $\alpha$ (Theorem 3.3 of the paper), which is at least $1+\sqrt2$ . The improvement needs the precise structure of the LP schedule around each job: which jobs interrupt $j$ ( $N_2$ ) and which do not ( $N_1$ ), and how the fraction $\eta_k(\alpha)$ of each other job depends on $\alpha$ . Turning this structure into exact identities for piecewise-constant, piecewise-linear functions of $\alpha$ defined through a recursively built schedule is the bulk of the formal work. The analytic part (Lemma 3.6) is elementary but depends on the specific equation that defines $\gamma$ .

Formalization scope

Jobs are Fin n (0-based) with $p, r : \texttt{Fin } n \to \mathbb N$ and real weights. The sortedness $w_j/p_j \ge w_k/p_k$ for $j \le k$ , positivity $p_j > 0$ and $w_j > 0$ are hypotheses of every theorem about the LP schedule. Because the data are integral, the LP schedule is defined slot by slot on unit intervals $[\tau, \tau+1)$ ; a job's processing set is a finite union of such intervals. $\alpha$ -points are infima over reals, and every statement restricts $\alpha$ to $(0,1]$ . The $(\alpha_j)$ -schedule's completion times are given by the closed form of list scheduling, $C_j = \max_{k \preceq j}(r_k + \sum_{k \preceq i \preceq j} p_i)$ , with ties of $\alpha$ -points broken by index (they do not occur for $\alpha \in (0,1]$ ). $Z_R$ is the infimum of the objective of (R) over its feasible set. The random $\alpha$ has law $f(\alpha)\,d\alpha$ on $\mathbb R$ , and the expectation is a Lebesgue integral.

The goal asserts integrability of $\alpha \mapsto \sum_j w_j C^\alpha_j$ together with the bound, so the inequality cannot hold through the convention that a non-integrable function has integral $0$ . The bound is against $Z_R$ defined from (R), not against $\sum_j w_j (M^{LP}_j + \tfrac12 p_j)$ , which would build Theorem 2.5 into the goal; and the density's support $(0,\delta]$ is written out, so no mass is placed outside $(0,1]$ . The running-time claims are not formalized, and the uniqueness of $\gamma$ is not asserted: the statements hold for every solution in $(0,1)$ .

Useful infrastructure: finite unions of intervals and their measures, monotone piecewise-linear functions and their integrals, and list-scheduling identities. Contributions to any milestone, and general lemmas about the LP schedule (it is a preemptive schedule, has no idle time inside a job's span, processes interrupting jobs completely), are welcome.

Selected references

M. X. Goemans, M. Queyranne, A. S. Schulz, M. Skutella, Y. Wang, Single Machine Scheduling with Release Dates, SIAM J. Discrete Math. 15(2):165–192, 2002. doi:10.1137/S089548019936223X
C. Phillips, C. Stein, J. Wein, Minimizing average completion time in the presence of release dates, Math. Programming 82:199–223, 1998. doi:10.1007/BF01585872
M. X. Goemans, Improved approximation algorithms for scheduling with release dates, Proc. 8th ACM-SIAM SODA, 591–598, 1997 (no DOI; reference [11] of Goemans et al. 2002).
C. Chekuri, R. Motwani, B. Natarajan, C. Stein, Approximation techniques for average completion time scheduling, SIAM J. Comput. 31(1):146–166, 2001. doi:10.1137/S0097539797327180
F. Afrati et al., Approximation schemes for minimizing average weighted completion time with release dates, Proc. 40th IEEE FOCS, 32–43, 1999. doi:10.1109/SFFCS.1999.814574

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Linear Optimization Operations Research Optimization·Captain: mikedeng1

Revenue Management Under the Markov Chain Choice Model IV: Dimension Reduction Recovers a Choice-Based Solution with at Most n+1 Nested Offer SetsResearch Paper

Motivation

In network revenue management a firm sells $n$ products that consume $m$ capacitated resources over $T$ periods, and in each period it chooses which offer set of products to make available. When customers choose among the offered products according to a choice model, the standard planning tool is the choice-based linear program, which assigns a frequency $u_S\ge 0$ to every offer set $S\subseteq N$ (Liu and van Ryzin 2008). It has $2^n$ variables. Its optimal solution is what the firm actually implements: a randomization over offer sets.

For the Markov chain choice model of Blanchet, Gallego and Goyal (2016), Feldman and Topaloglu (2017) show that the choice-based program is equivalent to a reduced linear program with only $2n$ variables. The reduced program can be solved directly, but its solution is a pair of vectors $(\hat x,\hat z)$ , not a randomization over offer sets. This mission formalizes the paper's Section 7, which converts the reduced solution back into a randomization over offer sets: the Dimension Reduction algorithm, and Theorem 9, which says that the algorithm produces at most $n+1$ offer sets and that these sets are nested.

Setting

Products are indexed by $N=\{1,\dots,n\}$ . A customer arrives wanting product $j$ with probability $\lambda_j$ . If the product she visits is offered, she buys it. Otherwise she moves to product $i$ with probability $\rho_{j,i}$ , and leaves without buying with probability $1-\sum_{i\in N}\rho_{j,i}$ . Throughout, $\lambda_j>0$ , $\rho_{j,i}\ge0$ and $\sum_{i\in N}\rho_{j,i}<1$ for all $j,i$ .

For an offer set $S\subseteq N$ , the (Balance) equations

P_{j,S}+R_{j,S}=\lambda_j+\sum_{i\in N}\rho_{i,j}R_{i,S}\ \ \forall j\in N,\qquad P_{j,S}=0\ \ \forall j\notin S,\qquad R_{j,S}=0\ \ \forall j\in S

have a unique solution. $P_{j,S}$ is the probability that product $j$ is purchased, and $R_{j,S}$ is the expected number of visits to $j$ while it is not offered. Write $P_S=(P_{j,S})_j$ and $R_S=(R_{j,S})_j$ .

The set $\mathcal H$ consists of all pairs $(x,z)\in\mathbb R^n\times\mathbb R^n$ with $x,z\ge0$ and $x_j+z_j=\lambda_j+\sum_{i}\rho_{i,j}z_i$ for all $j$ . The (Reduced) linear program maximizes $\sum_j T r_j x_j$ over $(x,z)\in\mathcal H$ subject to the capacity rows $\sum_j T a_{q,j}x_j\le c_q$ for each resource $q$ .

The Dimension Reduction algorithm starts from the optimal solution $(\hat x,\hat z)$ of (Reduced) and sets $(x^1,z^1)=(\hat x,\hat z)$ . At iteration $k$ it forms $S^k=\{j: x^k_j>0\}$ . It stops with $\alpha^k=1$ if $S^k=\emptyset$ . Otherwise it sets $\alpha^k=\min\{x^k_j/P_{j,S^k}: j\in S^k\}$ , stops if $\alpha^k=1$ , and if not updates

x^{k+1}_j=\frac{x^k_j-\alpha^kP_{j,S^k}}{1-\alpha^k},\qquad z^{k+1}_j=\frac{z^k_j-\alpha^kR_{j,S^k}}{1-\alpha^k}.

The weights are $\gamma^k=(1-\alpha^1)\cdots(1-\alpha^{k-1})\alpha^k$ .

Formalization targets

Goal: Theorem 9

Let $(\hat x,\hat z)$ be optimal for (Reduced). The algorithm stops at some first iteration $K$ with $1\le K\le n+1$ . At that $K$ ,

\hat x=\sum_{k=1}^K\gamma^kP_{S^k},\qquad \hat z=\sum_{k=1}^K\gamma^kR_{S^k},\qquad \sum_{k=1}^K\gamma^k=1,\qquad S^1\supseteq S^2\supseteq\cdots\supseteq S^K .

Milestones

In the order the proof uses them:

(Balance) has a unique, nonnegative solution for every $S$ (p. 1325).
$P_{j,S}>0$ for $j\in S$ (p. 1325).
For $(\hat x,\hat z)\in\mathcal H$ : $\hat z_j\ge R_{j,S_{\hat x}}$ . This is Lemma 11 of the online appendix, as quoted on p. 1332.
For $(\hat x,\hat z)\in\mathcal H$ with nonempty support: $\alpha_{\hat x}\le1$ (p. 1332).
The two stopping configurations are (Balance) solutions: $S_{\hat x}=\emptyset$ gives $(P_\emptyset,R_\emptyset)$ , and $\alpha_{\hat x}=1$ gives $(P_{S_{\hat x}},R_{S_{\hat x}})$ (p. 1332).
Lemma 8: one step of the algorithm maps $\mathcal H$ into $\mathcal H$ and removes a minimizing index from the support (p. 1333).
The iterates stay in $\mathcal H$ , and $S^{k+1}\subseteq S^k\setminus\{j^k\}$ (pp. 1333–1334).

Significance

Theorem 9 makes the reduced program usable. Together with the equivalence of (Reduced) and the choice-based program, it produces an optimal choice-based solution that offers at most $n+1$ sets. By a basic-solution count the choice-based program has an optimal solution with at most $m+1$ nonzero offer sets. Theorem 9 bounds the number by the number of products instead, and adds a structural property: the offered sets form a chain. Each set is contained in the previous one, so the implemented policy offers a single nested sequence of assortments. The algorithm needs at most $n+1$ solutions of linear systems of size $n$ . It never enumerates offer sets.

The result is proved in the paper. As far as a search of the platform and of Mathlib shows, it has not been formalized. Mathlib's Carathéodory theorem (Analysis/Convex/Caratheodory) bounds the number of points in a convex combination. It does not construct this decomposition, and it does not give nestedness. Formalizing Theorem 9 therefore requires the paper's argument: a verified, terminating peeling procedure on the polyhedron $\mathcal H$ whose output is a convex combination of (Balance) solutions indexed by a chain of sets.

Difficulty

The algorithm divides twice. Step 2 divides by $P_{j,S^k}$ , which must be positive on $S^k$ . Step 3 divides by $1-\alpha^k$ , which must be nonzero whenever the algorithm continues. Both rest on nontrivial facts: the positivity of purchase probabilities, and the bound $\alpha\le1$ on $\mathcal H$ . That bound in turn requires the comparison $\hat z\ge R_{S_{\hat x}}$ . The paper proves this comparison only in its online appendix; it is a monotonicity property of the inverse of $I-\rho^\top$ restricted to the unoffered products. The update of Step 3 can make coordinates of $z$ negative unless this comparison holds, so the naive observation that " $(x^k,z^k)$ is a combination of points of $\mathcal H$ " does not by itself keep the iterates in $\mathcal H$ .

Termination is not a consequence of $\alpha^k<1$ alone. It needs the strict decrease of the support, which comes from choosing a minimizing index. The weights $\gamma^k$ are defined by products of the $(1-\alpha^l)$ . Their sum telescopes to $1$ only because the last step has $\alpha^K=1$ .

Formalization scope

Everything lives in the namespace MarkovChainChoice.DimReduction. Products are Fin n and offer sets are Finset (Fin n). The model is a structure carrying $\lambda_j>0$ , $\sum_i\rho_{j,i}<1$ and, as a disclosed implicit hypothesis, $\rho_{j,i}\ge0$ . The pair $(P_S,R_S)$ is chosen by Classical.epsilon among the solutions of (Balance); milestone 1 is what identifies it with the paper's unique solution. Revenues, capacities and consumptions carry no sign conditions. "Optimal for (Reduced)" means feasible and at least as good as every feasible point.

The iterates are indexed from $k=1$ as in the paper. The value at index $0$ is an unused copy of the input. "The algorithm stops at iteration $K$ " is the first $K\ge1$ with $S^K=\emptyset$ or $\alpha^K=1$ . Iterates after the first stop involve a division by zero, which Lean evaluates to $0$ , and no statement refers to them. The termination bound $K\le n+1$ is a conclusion of the goal, not a hypothesis. The goal is about the algorithm's actual iterates, not about an arbitrary sequence that satisfies the invariants. The paper's $\subset$ and $\supset$ denote non-strict inclusion and are rendered as $\subseteq$ and $\supseteq$ .

Two milestones are stated more generally than the page. The paper states the iteration invariants for the (Reduced) optimum; the Lean requires only $(\hat x,\hat z)\in\mathcal H$ , which every feasible (Reduced) solution satisfies. Lemma 8's $\arg\min$ may be a set; the statement holds for every minimizer. The page contains printed slips: on p. 1332, $\hat v_j$ has numerator $\hat x_j-\alpha R_{j,S}$ , and the paper writes $R_{\hat x}$ for $R_{S_{\hat x}}$ . The Lean uses the correct forms, as in Lemma 8 and Step 3. None of these slips appears in a milestone quotation.

A complete development needs a small theory of M-matrices, in the form of nonnegativity of $(I-Q)^{-1}$ for substochastic $Q$ , together with finite induction on supports. The comparison lemma (milestone 3) and the (Balance) existence and uniqueness result (milestone 1) are reusable across the other missions on this paper. Proofs of any milestone are welcome, as are alternative proofs of the goal.

Selected references

J. B. Feldman, H. Topaloglu, Revenue Management Under the Markov Chain Choice Model, Operations Research 65(5):1322–1342, 2017. https://doi.org/10.1287/opre.2017.1628
J. Blanchet, G. Gallego, V. Goyal, A Markov Chain Approximation to Choice Modeling, Operations Research 64(4):886–905, 2016. https://doi.org/10.1287/opre.2016.1505
Q. Liu, G. van Ryzin, On the Choice-Based Linear Programming Model for Network Revenue Management, Manufacturing & Service Operations Management 10(2):288–310, 2008. https://doi.org/10.1287/msom.1070.0169

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

Bargaining under Incomplete Information III: Trade Probability and Expected Profits in the Uniform Linear EquilibriumResearch Paper

Motivation

A buyer and a seller negotiate over a single indivisible good. Each knows what the good is worth to them but not what it is worth to the other side, and each shades their offer to exploit the other's uncertainty. Chatterjee and Samuelson (Bargaining under Incomplete Information, Operations Research 31(5), 1983) modelled this as a one-shot game in which both parties submit sealed offers simultaneously and a sale takes place at a weighted average of the two offers whenever the buyer's offer is at least the seller's.

The weight $k$ is a design parameter: $k = 1$ lets the buyer set the price, $k = 0$ the seller, and $k = 1/2$ splits the difference. For values uniform on a common interval the paper computes an explicit equilibrium for every $k$ (its Example 1) and then asks the questions a designer of the rule cares about: how often does trade happen, who gains when $k$ moves, and which $k$ maximises the expected gains of the two parties together. The answers are the subject of this mission.

The example became a benchmark for bilateral trade. Myerson and Satterthwaite (J. Econ. Theory 29, 1983) proved that no mechanism can guarantee efficient trade with two-sided private information, and that for uniform values the split-the-difference equilibrium of this game attains the largest expected gains from trade of any mechanism. The numbers $9/32$ and $\tfrac{9}{64}\bar v$ below are therefore the second-best benchmarks against which later work on the $k$ -double auction (Satterthwaite and Williams, J. Econ. Theory 48, 1989; Leininger, Linhart and Radner, J. Econ. Theory 48, 1989) measures inefficiency.

Setting

A seller with reservation price $v_s$ and a buyer with reservation price $v_b$ each know their own value. The two values are drawn independently and uniformly on $[0, \bar v]$ , with $\bar v > 0$ ; this law, written $\mathrm{unif}(\bar v)$ , is also each player's belief about the other's value ( $F_s(v) = F_b(v) = v/\bar v$ in the paper).

Under the Bargaining Rule with parameter $k \in [0,1]$ , the seller asks $s$ and the buyer offers $b$ . If $b \ge s$ the good is sold at price $P = kb + (1-k)s$ , the seller earns $P - v_s$ and the buyer earns $v_b - P$ ; otherwise both earn zero. Ties trade.

An offer strategy maps a value to an offer. The strategies of Example 1(a) are

S(v_s) = \frac{v_s}{2-k} + \frac{1-k}{2}\bar v \quad\text{for } 0 \le v_s \le \tfrac{2-k}{2}\bar v, \qquad S(v_s) \ge \text{the same expression for } \tfrac{2-k}{2}\bar v < v_s \le \bar v,

B(v_b) = \frac{v_b}{1+k} + \frac{k(1-k)}{2(1+k)}\bar v \quad\text{for } \tfrac{1-k}{2}\bar v \le v_b \le \bar v, \qquad B(v_b) \le \text{the same expression for } 0 \le v_b < \tfrac{1-k}{2}\bar v.

A pair $(S, B)$ with these four properties is said to have the shape of Example 1(a) (IsExample1Pair k v̄ S B). On the two inequality ranges a seller asks too much, or a buyer bids too little, for any trade to occur, so the strategy there is free apart from the bound.

For such a pair, with $(v_s, v_b) \sim \mathrm{unif}(\bar v) \otimes \mathrm{unif}(\bar v)$ , the trade probability is $\Pr[S(v_s) \le B(v_b)]$ (tradeProb), and the ex ante expected profits — taken before either value is drawn, as the paper specifies on p. 843 — are

\pi_s = \mathbb E\bigl[\mathbf 1\{S(v_s) \le B(v_b)\}\,(kB(v_b) + (1-k)S(v_s) - v_s)\bigr],\qquad \pi_b = \mathbb E\bigl[\mathbf 1\{S(v_s) \le B(v_b)\}\,(v_b - kB(v_b) - (1-k)S(v_s))\bigr]

(sellerExAnte, buyerExAnte).

Formalization targets

Goal: Example 1(c)(iii), total expected profit

For $0 \le k \le 1$ , $\bar v > 0$ and every pair of the shape of Example 1(a),

\pi_s + \pi_b = \frac{\bar v}{16}(1+k)(2-k),

and as a function of $k \in [0,1]$ this total attains its maximum $\tfrac{9}{64}\bar v$ at $k = 1/2$ . The goal is the paper's efficiency statement: among these equilibria, splitting the difference maximises expected group profit.

Milestones

Example 1(b). $\Pr[S(v_s) \le B(v_b)] = \dfrac{-k^2 + k + 2}{8}$ , with maximum $9/32$ at $k = 1/2$ .
Example 1(c)(i). $\pi_s(k) = \dfrac{\bar v}{48}(2-k)^2(1+k)$ , strictly decreasing in $k$ on $[0,1]$ .
Example 1(c)(ii). $\pi_b(k) = \dfrac{\bar v}{48}(1+k)^2(2-k)$ , strictly increasing in $k$ on $[0,1]$ .

The goal is the sum of milestones 2 and 3 together with a one-variable maximisation; milestone 1 describes the trade region over which both profits are integrated.

Significance

The formulas answer the design question for the rule. Moving $k$ toward the buyer's offer makes the price rule look more favourable to the seller, yet milestone 2 shows the seller's equilibrium profit falls and milestone 3 shows the buyer's rises: the paper (p. 844) uses this to show that an intuition which ignores the players' strategic response is mistaken. The comparison with truthful offers, which would trade with probability $1/2$ and earn expected group profit $\bar v/6$ , quantifies the cost of strategic misrepresentation: at best $9/32$ and $\tfrac{9}{64}\bar v$ .

The paper states these results as "straightforward computations" and prints no derivation. As far as is known, none of them has a machine-checked proof. A formalization settles the constants against the exact strategies of Example 1(a), including the non-linear no-trade branches the paper allows, and provides a worked example of computing trade probabilities and expected payoffs under a product of uniform laws, reusable for other double-auction and bilateral-trade examples.

Difficulty

The computation is elementary on paper, but the equilibrium strategies are only partly specified: on the seller's high range and the buyer's low range the offers are arbitrary functions subject to a bound, and need not be measurable. The obvious approach — substitute the linear formulas and integrate — is valid only after showing that these free branches never trade, so that the trade event and both integrands agree almost everywhere with their linear versions. The resulting integrals are over a product of two conditioned Lebesgue measures, not over Lebesgue measure on the plane, and the trade region depends on $k$ through both strategies. The monotonicity claims hold only on $[0,1]$ (the seller's cubic is not monotone on $\mathbb R$ ), and the derivative of each profit vanishes at an endpoint of the interval.

Formalization scope

Values are real numbers; the uniform law on $[0, \bar v]$ is Lebesgue measure conditioned on the interval (volume[|Icc 0 v̄]), and the joint law of $(v_s, v_b)$ is the product measure, with pairs ordered $(v_s, v_b)$ . The trade probability is the real number (P {p | S p.1 ≤ B p.2}).toReal; the profits are Bochner integrals over the product. Ties trade. The results are ex ante, not conditional on a player's own value. Every theorem assumes $0 \le k \le 1$ and $\bar v > 0$ . Maxima are stated with IsMaxOn on $[0,1]$ plus the value at $k = 1/2$ ; monotonicity with StrictAntiOn/StrictMonoOn on $[0,1]$ .

The statements do not assume that $(S, B)$ is an equilibrium; they are computations about any pair of the shape of Example 1(a). That this pair is an equilibrium is Example 1(a) itself, the goal of a companion mission. No measurability of $S$ or $B$ is assumed: the free branches never trade, so each integrand agrees almost everywhere with a bounded measurable function and the integrals are the paper's expectations. A formalization that assumes $S$ and $B$ linear everywhere, or that integrates over a single uniform variable, proves a different statement.

A complete development needs: the reduction of the trade event and the integrands to their linear versions on $[0,\bar v]^2$ ; Fubini for the product of conditioned measures; evaluation of polynomial integrals over a triangle; and elementary calculus on cubics. Lemmas on integrating over products of uniform laws are reusable. Contributions of intermediate lemmas, such as the explicit trade region, are welcome.

Selected references

K. Chatterjee and W. Samuelson, Bargaining under Incomplete Information, Operations Research 31(5):835–851, 1983. https://doi.org/10.1287/opre.31.5.835
R. B. Myerson and M. A. Satterthwaite, Efficient Mechanisms for Bilateral Trading, Journal of Economic Theory 29(2):265–281, 1983. https://doi.org/10.1016/0022-0531(83)90048-0
M. A. Satterthwaite and S. R. Williams, Bilateral Trade with the Sealed Bid k-Double Auction: Existence and Efficiency, Journal of Economic Theory 48(1):107–133, 1989. https://doi.org/10.1016/0022-0531(89)90120-8
W. Leininger, P. B. Linhart and R. Radner, Equilibria of the Sealed-Bid Mechanism for Bargaining with Incomplete Information, Journal of Economic Theory 48(1):63–106, 1989. https://doi.org/10.1016/0022-0531(89)90121-X

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Convex Optimization Graph Theory Operations Research+1·Captain: mikedeng1

The Traveling-Salesman Problem and Minimum Spanning Trees, Part II: Convergence of the Constant-Step Ascent to the 1-Tree BoundResearch Paper

Motivation

The traveling-salesman problem (TSP) asks for a cheapest cycle through all vertices of a weighted complete graph. Exact algorithms for it are branch-and-bound searches, and their size depends almost entirely on the quality of the lower bounds used to prune the search. In 1970 Held and Karp introduced the 1-tree bound: a Lagrangian relaxation of the degree-2 constraints of a tour, whose value can be evaluated by one minimum-spanning-tree computation (Held & Karp, Part I, 1970). Part II (Held & Karp, 1971) replaces the ascent procedure of Part I by an iterative method related to the relaxation method for linear inequalities of Agmon and of Motzkin and Schoenberg (1954), and with it solved to proven optimality every instance presented to it, up to 64 cities. The iteration is the prototype of what is now called the subgradient method with Polyak-type step sizes, and the 1-tree bound remains a standard lower bound in exact TSP codes.

Timeline:

1954 — Agmon; Motzkin and Schoenberg: the relaxation method for systems of linear inequalities, and convergence of Féjer-monotone sequences relative to full-dimensional sets.
1970 — Held and Karp (Part I): the 1-tree bound $\max_\pi w(\pi)$ and a column-generation / ascent method for it.
1971 — Held and Karp (Part II): the iteration $\pi^{m+1} = \pi^m + t_m v_{k(\pi^m)}$ , its relaxation-method analysis (Lemmas 1–3) and the constant-step guarantee (Theorem 1).
1974 — Held, Wolfe and Crowder validate the method as general subgradient optimization.

Setting

Let $n \ge 3$ and let $(c_{ij})$ be a symmetric real $n\times n$ matrix of weights on the edges of the complete graph $K_n$ with vertex set $\{1,\dots,n\}$ ; weights may be negative and need not satisfy the triangle inequality. A subgraph has weight equal to the sum of its edge weights. A tour is a cycle through every vertex exactly once; $C^*$ is the weight of a minimum tour.

A 1-tree is a tree on the vertex set $\{2,\dots,n\}$ together with two distinct edges at vertex $1$ . Index the 1-trees by $k$ ; let $c_k$ be the weight of the $k$ -th 1-tree, $d_{ik}$ the degree of vertex $i$ in it, and $v_k \in \mathbb R^n$ the degree-excess vector with components $d_{ik}-2$ . For $\pi \in \mathbb R^n$ define

w(\pi) = \min_k\,[c_k + \pi\cdot v_k].

A tour is a 1-tree with $v_k = 0$ , so $C^* \ge w(\pi)$ for every $\pi$ (Eq. (2)); the best bound is $\max_\pi w(\pi)$ . For a point $\pi$ , $k(\pi)$ denotes a minimum-weight 1-tree at $\pi$ , a 1-tree attaining the minimum defining $w(\pi)$ . The ascent iteration (3) is

\pi^{m+1} = \pi^m + t_m\,v_{k(\pi^m)}.

For a target value $\bar w$ , $P_{\bar w}$ is the polyhedron of solutions of $\bar w \le c_k + \pi\cdot v_k$ for all $k$ (system (5)). All norms $\|\cdot\|$ are Euclidean.

Formalization targets

Goal: Theorem 1

With constant step $t_m = \bar t > 0$ , any starting point and any choice of minimum-weight 1-trees,

\sup_m w(\pi^m) \;\ge\; \max_\pi w(\pi) - \tfrac12\,\bar t\,\limsup_{m\to\infty}\|v_{k(\pi^m)}\|^2 .

Milestones

Eq. (2): $C^* \ge w(\pi)$ for every $\pi$ .
Lemma 1: if $w(\bar\pi) \ge w(\pi)$ then $(\bar\pi-\pi)\cdot v_{k(\pi)} \ge w(\bar\pi) - w(\pi) \ge 0$ .
Lemma 2: if $0 < t < 2(w(\bar\pi)-w(\pi))/\|v_{k(\pi)}\|^2$ then $\|\bar\pi - (\pi + t v_{k(\pi)})\| < \|\bar\pi-\pi\|$ .
Féjer-monotone convergence (Motzkin–Schoenberg, quoted in the proof of Lemma 3): a sequence whose distance to every point of a set with nonempty interior is nonincreasing converges.
Lemma 3, Case 1: for $\bar w < \max_\pi w$ and the relaxation iteration

\pi^{m+1} = \pi^m + \lambda_m\,\frac{\bar w - w(\pi^m)}{\|v_{k(\pi^m)}\|^2}\,v_{k(\pi^m)} \qquad (6)

with $0<\varepsilon<\lambda_m\le 2$ , the iterates enter $P_{\bar w}$ or converge to a boundary point of $P_{\bar w}$ . 6. Lemma 3, Case 2: with $\lambda_m = 2$ the iterates enter $P_{\bar w}$ . 7. §3 bound: the restricted minimum $w_{X,Y}(\pi)$ over 1-trees containing the edges $X$ and avoiding the edges $Y$ is a lower bound on every tour of the derived problem.

Milestones 1–5 are the steps of the paper's proof of Theorem 1; 6 and 7 are further results of the paper on the same objects.

Significance

Theorem 1 is the paper's justification of the step rule actually used in its computations: a fixed step $\bar t$ loses at most $\tfrac12\bar t$ times the asymptotic squared deviation of the generated 1-trees from being tours. Since $\|v_k\|^2$ is an even integer that vanishes exactly on tours, and the paper observes it is typically small in practice, the bound explains why the constant-step ascent reaches bounds sharp enough for branch-and-bound. Lemmas 1–3 are the first analysis of a subgradient-type method for a nonsmooth concave function, cast as the relaxation method for the (exponentially large) system (5).

All results are proved in the paper (except the Féjer-monotone convergence and Case 2 of Lemma 3, which it cites from Motzkin and Schoenberg). None is formalized: the platform has the 1-tree lower bound only under a metric assumption on the weights (SupplyChainTheory.held_karp_bound, SupplyChainTheory.one_tree_lower_bound), and the subtour-LP bound MetricTSP.held_karp_le_opt, a different object. This mission produces the bound for arbitrary real weights, the supergradient property of $v_{k(\pi)}$ , and a machine-checked convergence analysis of the relaxation iteration.

Difficulty

Lemmas 1 and 2 and Eq. (2) are short once the finite minimum defining $w$ is handled. The difficulty is in Lemma 3 and Theorem 1. The iteration is not monotone in $w$ , so no descent argument applies; progress is measured by the Euclidean distance to the target polyhedron $P_{\bar w}$ , and turning distance decrease into convergence requires the Féjer-monotonicity theorem, which in turn needs $P_{\bar w}$ to have nonempty interior (from $\bar w < \max_\pi w$ ). In Theorem 1 the step is constant rather than of the relaxation form (6), and the target polyhedron is not given in advance: the relevant relaxation parameters are admissible only eventually and must be kept away from zero, which requires controlling degenerate directions $v_{k(\pi^m)} = 0$ and the behaviour of $w$ along an iteration that is not known a priori to stay bounded. A direct argument that $w(\pi^m)$ increases fails, since single steps can decrease $w$ .

Formalization scope

Vertices are Fin n, the paper's vertex 1 is 0 : Fin n, and graphs are SimpleGraph (Fin n). Weights are c : Sym2 (Fin n) → ℝ, arbitrary reals. A 1-tree is a graph whose restriction to the vertices other than 0 is a tree and in which 0 has degree 2; a tour is a connected graph with all degrees 2. w(π) is the minimum of weight c G + ∑ i, π i * (deg G i − 2) over 1-trees, written as an sInf over a finite set that is nonempty for n ≥ 3; every theorem assumes 3 ≤ n. Euclidean norms and inner products are written as coordinate sums of squares and products, never Mathlib's sup norm on Fin n → ℝ. The minimum-weight 1-tree $k(\pi^m)$ is a hypothesis at every step, and ties may be broken arbitrarily. The goal is stated as "for every $\pi^*$ and $\delta>0$ some iterate has $w(\pi^m) > w(\pi^*) - \tfrac12\bar t L - \delta$ ", which is equivalent to the printed inequality; $L$ is the limsup of a sequence with finitely many values, a genuine real.

The statements exclude trivializing readings: the 1-tree predicate rejects graphs without exactly two edges at vertex 1; w is never a minimum over an empty set under the standing hypothesis 3 ≤ n; no ⨆ of a possibly unbounded family is used; the step size $\bar t$ and the parameter $\varepsilon$ are strictly positive.

A complete development needs finite minima of affine functions (concavity, attainment), 1-tree and tour combinatorics on simple graphs, and Féjer-monotone sequences in $\mathbb R^n$ ; the last two are reusable beyond this mission. Proofs of any milestone, and alternative arguments for Lemma 3, are welcome.

Selected references

M. Held and R. M. Karp, The traveling-salesman problem and minimum spanning trees: Part II, Mathematical Programming 1 (1971) 6–25. https://doi.org/10.1007/BF01584070
M. Held and R. M. Karp, The traveling-salesman problem and minimum spanning trees, Operations Research 18 (1970) 1138–1162. https://doi.org/10.1287/opre.18.6.1138
T. S. Motzkin and I. J. Schoenberg, The relaxation method for linear inequalities, Canadian Journal of Mathematics 6 (1954) 393–404. https://doi.org/10.4153/CJM-1954-038-x
S. Agmon, The relaxation method for linear inequalities, Canadian Journal of Mathematics 6 (1954) 382–392. https://doi.org/10.4153/CJM-1954-037-2
M. Held, P. Wolfe and H. P. Crowder, Validation of subgradient optimization, Mathematical Programming 6 (1974) 62–88. https://doi.org/10.1007/BF01580223

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

Discrete Convex Analysis XXXV: Gross Substitutes and Equilibrium PricesTextbook

Motivation

This mission continues chapter 11's account of the M♮-concave/M♮-convex exchange-economy model begun in mission 14-economic-equilibrium, placing seven of that chunk's own results that were previously left out-of-cone: the two gross-substitutes-style characterizations of M♮-concavity (§11.3), the transfer theorem that lifts an equilibrium of the continuous relaxation to one for indivisible commodities (§11.4), and the explicit polyhedral description of the equilibrium price set together with its feasibility criterion (§11.5).

Setting

Mission 14-economic-equilibrium built the exchange-economy vocabulary this mission redeclares in full (UDom, ArgMaxBot/ArgMinTop, PriceShift/PriceShiftConvex, DemandSet/SupplySet, IsEquilibrium, MNaturalConcave, IsMNaturalConvexSet, the concave/convex closures ConcaveClosureR/ConvexClosureR and their continuous analogues ContDemandSet/ContSupplySet/ IsContEquilibrium) and placed the qualitative structural theorems (Theorems 11.1-11.3, 11.4, 11.16-11.18, 11.23-11.24). This mission adds the gross-substitutes axioms (−M♮-GS[Z], the price-monotonicity property NegGS, and −M♮-SWGS[Z], its one-price-at-a-time refinement NegSWGS), the M♮-convex-set transfer machinery connecting a continuous equilibrium to a discrete one, and the equilibrium price polyhedron built from the three bound families ℓ(j), u(j), u(i,j) (Eqs. (11.40)-(11.42)) that make Theorem 11.16's qualitative L♮-convex-polyhedron fact concrete and linear-programming-checkable.

Formalization targets

Goal: The equilibrium price set is the explicit L♮-convex polyhedron (11.43) (Theorem 11.21)

For a fixed allocation (x,y), the set P* of all equilibrium price vectors is an L♮-convex polyhedron and equals the polyhedron cut out by max{0,ℓ(j)} ≤ p(j) ≤ u(j) and p(j)-p(i) ≤ u(i,j). Chosen as goal: it is the sharpest structural result of chapter 11's computation section, upgrading Theorem 11.16's qualitative fact to a concrete description, and is what Theorem 11.22 (also placed) builds on directly.

Supporting structural targets

Theorem 11.5 and Theorem 11.6 characterize M♮-concavity via the gross-substitutes and stepwise gross-substitutes properties, completing chapter 11's suite of M♮-concavity characterizations begun with Theorem 11.4 (mission 14). Theorem 11.15 is the general transfer theorem (continuous equilibrium ⟹ discrete equilibrium) that mission 14's own Theorem 11.14 invokes as a special case. Theorem 11.22 gives the feasibility criterion for the existence of an equilibrium price vector, the mission's second theorem built on the equilibrium price polyhedron.

Significance

Together with mission 14-economic-equilibrium, this mission completes the book's account of how M♮-concavity/convexity — a purely combinatorial exchange condition — reproduces, and sharpens, the classical gross-substitutes theory of competitive equilibrium for economies with indivisible goods: existence transfers from the continuous relaxation, and the equilibrium price set itself has a description exact enough to reduce to a linear feasibility question. None of these results are open — they are Murota's own account (attributed in the book's own notes to Danilov-Koshevoy- Lang and Murota-Tamura for the gross-substitutes theorems, and to Murota-Tamura for the equilibrium price polyhedron); this mission contributes a faithful, machine-checked formal statement of each (see Formalization scope).

Difficulty

Two of this chunk's seven BRIEF.md results are not drafted this pass, for a disclosed time- budget reason rather than any faithfulness failure: Proposition 11.19 and Theorem 11.20 require the H,L-indexed bipartite MSFP2 flow-network vocabulary (separate vertex sets V+_e, V+_l, V-_h, an M-convex/M-concave-combining flow objective) that neither this mission nor mission 14 builds, and building it in proportion to placing exactly these two results was judged disproportionate to the remaining time in this pass; see HARD.md and STATUS.md. This is explicitly not a hard exclusion — both results are well-posed and provable from the book's own complete proofs — and is recorded as an honest scope limitation for a future pass. Theorem 11.22's own trailing algorithmic remark (that equilibrium prices can be found via a shortest-path computation, yielding a polynomial-time equilibrium-checking algorithm) is a computational/ complexity claim outside this series' propositional-formalization methodology and is omitted; the mathematical "iff feasibility" content is placed in full. See HARD.md.

Formalization scope

Ground set K is a Fintype with DecidableEq; consumer/producer index sets H, L are Fintypes (Nonempty where the price-bound formulas (11.40)-(11.42) need a nonempty sup'/inf' range). All base vocabulary is redeclared fresh from mission 14-economic-equilibrium's own definitions, since this draft cannot import that sibling mission. The gross-substitutes axioms are formalized directly from their defining inequalities (Eqs. preceding (11.19) and following, and p.331); the equilibrium price polyhedron's bound families ℓ(j)/u(j)/u(i,j) are formalized literally from Eqs. (11.40)-(11.42), extracting each WithBot ℝ/WithTop ℝ operand to ℝ before subtracting (since WithBot ℝ carries no subtraction instance). Two results (Proposition 11.19, Theorem 11.20) are not drafted this pass for the disclosed time-budget reason above; one result (Theorem 11.22's trailing algorithmic remark) is scoped out as computational content. Contributions completing any of the five sorrys, or building the MSFP2 vocabulary to place Proposition 11.19/Theorem 11.20 in a follow-up mission, are welcome.

Selected references

K. Murota, Discrete Convex Analysis, SIAM, 2003. DOI: 10.1137/1.9780898718508.
V. Danilov, G. Koshevoy, K. Murota, "Discrete convexity and equilibria in economies with indivisible goods and money," Mathematical Social Sciences, 41 (2001), pp. 251-273 [33] (origin of the gross-substitutes characterization, Theorem 11.6).
K. Murota, A. Tamura, "Application of M-convex submodular flow problem to mathematical economics," Japan Journal of Industrial and Applied Mathematics, 20 (2003), pp. 257-277 [160] (origin of the equilibrium price polyhedron, Theorems 11.20-11.22).

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Combinatorics Operations Research Optimization·Captain: Shuze Chen

Discrete Convex Analysis XIII: Existence of Equilibrium with Indivisible GoodsTextbook

Motivation

Competitive-equilibrium theory for economies of divisible commodities — where consumption and production are real vectors — has rested on a rigorous mathematical foundation since around 1960, built from convexity, compactness, and fixed-point theorems (Debreu 1959; Arrow–Hahn 1971; McKenzie 2002). A large share of real markets, however, trade goods that cannot be split: houses, cars, aircraft, job assignments, radio spectrum licenses. For such economies, no comparably general existence theory existed before the framework this mission formalizes. Kelso and Crawford (1982) and Gul and Stacchetti (1999) had identified the gross substitutes property as the right condition on preferences for equilibrium to exist in labor-market and assignment models; Danilov, Koshevoy, and Murota (1998, 2001) showed that gross substitutes, and several other conditions proposed independently in the economics literature, all coincide with a single combinatorial notion from discrete convex analysis: M-natural-concavity. This mission formalizes the resulting existence theorem for economies with indivisible goods, together with the definitional results that pin down exactly what M-natural-concavity of a utility function means and how it connects to the classical demand-set language of general equilibrium theory.

Setting

Fix a finite set $K$ of indivisible commodity types and a finite set $H$ of consumers ("she"). A consumption bundle is an integer vector $x \in \mathbb Z^K$ , one coordinate per commodity. Consumer $h$ 's preferences over bundles, net of a perfectly divisible numeraire ("money"), are summarized by a utility function $U_h : \mathbb Z^K \to \mathbb R \cup \{-\infty\}$ , where $-\infty$ marks bundles outside her feasible range (her effective domain, $\operatorname{dom} U_h = \{x : U_h(x) \ne -\infty\}$ ). Given a price vector $p \in \mathbb R^K$ (one real price per commodity), consumer $h$ chooses a bundle from her demand set

D_h(p) = \arg\max_{x \in \mathbb Z^K} \big(U_h(x) - \langle p, x\rangle\big),

the bundles that maximize utility net of expenditure; the shorthand $U_h[-p](x) := U_h(x) - \langle p,x\rangle$ is used throughout. For the notion this mission is built around, write $\chi_i \in \mathbb Z^K$ for the $i$ -th unit vector, and for $x, y \in \mathbb Z^K$ let $\operatorname{supp}^+(x-y) = \{k : x(k) > y(k)\}$ and $\operatorname{supp}^-(x-y) = \{k : x(k) < y(k)\}$ . A function $U$ with nonempty effective domain is M-natural-concave if it satisfies the exchange axiom: for $x, y \in \operatorname{dom} U$ and $i \in \operatorname{supp}^+(x-y)$ ,

U(x) + U(y) \le \max\Big(U(x-\chi_i)+U(y+\chi_i),\ \max_{j \in \operatorname{supp}^-(x-y)} \big[U(x-\chi_i+\chi_j) + U(y+\chi_i-\chi_j)\big]\Big),

with the convention that a maximum over the empty set is $-\infty$ . A producer $l$ from a finite set $L$ is symmetric, described by a cost function $C_l : \mathbb Z^K \to \mathbb R \cup \{+\infty\}$ that is M-natural-convex — the mirror-image exchange axiom with $\min$ in place of $\max$ and the inequality reversed — and a supply set $S_l(p) = \arg\max_y(\langle p,y \rangle - C_l(y))$ . An equilibrium for a total initial endowment $x^\circ \in \mathbb Z^K$ is a tuple $((x_h \mid h \in H), (y_l \mid l \in L), p)$ with $x_h \in D_h(p)$ , $y_l \in S_l(p)$ , market clearing $\sum_h x_h = x^\circ + \sum_l y_l$ , and $p \ge 0$ .

Formalization targets

\textbf{Theorem 11.13 (goal).}\quad \text{If every } U_h \text{ is nondecreasing and M-natural-concave with bounded domain, an equilibrium exists for every } x^\circ \in \bigcap_h \operatorname{dom} U_h \text{ in the exchange economy } (L = \emptyset).

This is the weakest form of the existence claim the mission proves in full — no producers, so no interaction between two different M-natural-convexity classes is needed — and is the natural target because it isolates exactly what M-natural-concavity buys on the consumer side alone. Two companion results sharpen the picture: Theorem 11.4 pins down M-natural-concavity through an equivalent single-step ascent property, and Theorem 11.7 restates it through the combinatorial structure (M-natural-convexity) of the demand sets themselves, connecting the definition to the gross-substitutes literature. Proposition 11.12 supplies the structural fact the existence proof turns on (the aggregate excess-cost function inherits M-natural-convexity and has a nonempty subdifferential), Theorem 11.14 extends existence to the general economy with producers by transporting an equilibrium down from a continuous relaxation, and Theorem 11.16 establishes that the set of all equilibrium prices is not merely nonempty but a lattice-structured polyhedron.

Significance

The result itself. Theorem 11.13 is a genuine existence theorem for a discrete general- equilibrium model — not an approximation or a relaxation of the continuous theory, but a free-standing result about the integer lattice. Its consequence set is also constructive by consequence: Theorem 11.16's lattice structure and section 11.5's reduction to submodular-flow computation (not part of this mission) together show that finding an extreme equilibrium price is a polynomial-time problem, not merely a nonempty-existence claim. Before this line of work, economists working with indivisible goods either restricted to special two-sided matching structures or worked with sufficient conditions (such as gross substitutes) whose relationship to each other and to any unifying combinatorial property was not understood; Fujishige–Yang (2003) and Murota–Tamura independently identified the M-natural-concavity connection cited here.

Formalizing it. All of the results in this mission are proved in the source text; nothing here is open. What formalization adds is a machine-checked confirmation that the demand/supply-set and equilibrium definitions, and the exchange-axiom characterization of M-natural-concavity, compose exactly as the informal statements claim — a nontrivial check, since the definitions involve several layers of arg-max/arg-min over integer lattices and price-shifted objectives that are easy to state slightly wrong (e.g. conflating $\arg\max$ and $\arg\min$ , or omitting the extended index $0$ in the single-improvement axiom).

Difficulty

The obvious approach to existence — relax the discrete problem to $\mathbb R^K$ , apply a classical fixed-point argument, and round the resulting continuous equilibrium to the nearest integer point — fails outright, and the book devotes section 11.2 to a two-agent, two-good example that demonstrates this concretely: at certain initial endowments, every candidate integer allocation leaves an unclaimed unit of surplus, so no equilibrium price exists at all, even though the continuous relaxation of the same economy has one. The gap is a failure of convexity in the Minkowski sum $D_1(p) + D_2(p)$ : ordinary discrete demand sets can be "hole-free" individually and still sum to a set with a hole. M-natural-concavity is precisely the condition under which Minkowski sums of demand sets stay hole-free (a consequence of the parallel discrete-convex-set theory this book develops earlier), which is what makes the round-down argument valid after all — but only under this specific hypothesis, not under plain concavity or submodularity.

Formalization scope

Commodities and prices live on a general finite type $K$ (Fintype, DecidableEq), not a fixed $\mathrm{Fin}\ n$ ; consumers and producers are indexed by general finite types $H$ , $L$ , with the pure exchange economy realized as the special case $L = \mathrm{PEmpty}$ . Utility values lie in WithBot ℝ ( $\mathbb R \cup \{-\infty\}$ ) and cost values in WithTop ℝ ( $\mathbb R \cup \{+\infty\}$ ), matching the book's asymmetric conventions for the two families exactly; no constant appears anywhere in this mission's statements (all hypotheses and conclusions are qualitative), so there is no explicit-constant obligation to record. The one hypothesis that must never be silently dropped is "nondecreasing" in Theorem 11.13: it is a real, separate condition from M-natural-concavity (more of a good is always weakly preferred), stated as its own conjunct rather than folded into the concavity predicate. A formalization that replaced M-natural-concavity with ordinary real-valued concavity, or dropped the boundedness hypothesis on the domains, would be a different — and for indivisible goods, false — statement; section 11.2's example is a concrete witness that discreteness together with a weaker structural hypothesis than M-natural-concavity is not enough. This mission's "M-natural-convex set" (used in Theorem 11.7) and "L-natural-convex polyhedron" (used in Theorem 11.16) are each formalized via one of the book's own stated equivalent characterizations (projection of an M-convex set on an extended ground set, and the (SBS-natural[R]) lattice-translation property respectively) rather than reintroduced as new primitives. Reusable beyond this mission: the general subdifferential SubdiffR and the EReal-valued concave/convex closure constructions apply to any discrete convex/concave function, not only to the aggregate cost function of this chapter. Contributions welcome on the sorry'd proofs, and on formalizing section 11.5's computational reduction to the M-convex submodular flow problem (deferred here — see HARD.md — pending chunk 12's flow vocabulary).

Selected references

Murota, K. Discrete Convex Analysis. SIAM, 2003. DOI: 10.1137/1.9780898718508. (Chapter 11.)
Kelso, A. S., Crawford, V. P. "Job Matching, Coalition Formation, and Gross Substitutes." Econometrica 50(6), 1982, 1483–1504.
Gul, F., Stacchetti, E. "Walrasian Equilibrium with Gross Substitutes." Journal of Economic Theory 87(1), 1999, 95–124.
Danilov, V., Koshevoy, G., Murota, K. "Discrete Convexity and Equilibria in Economies with Indivisible Goods and Money." Mathematical Social Sciences 41(3), 2001, 251–273.
Fujishige, S., Yang, Z. "A Note on Kelso and Crawford's Gross Substitutes Condition." Mathematics of Operations Research 28(3), 2003, 463–469.
Debreu, G. Theory of Value: An Axiomatic Analysis of Economic Equilibrium. Yale University Press, 1959.

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Linear Optimization Operations Research Optimization+1·Captain: mikedeng1

Single Machine Scheduling with Release Dates I: The Preemptive Time-Indexed and Mean Busy Time LP Relaxations Have the Same Optimal ValueResearch Paper

Motivation

Minimizing the total weighted completion time $\sum_j w_j C_j$ of jobs with release dates on a single machine, written $1\,|\,r_j\,|\,\sum w_j C_j$ in scheduling notation, is strongly NP-hard. Most constant-factor approximation algorithms for it, and for many related scheduling problems, follow one pattern: solve a linear programming relaxation, which gives a lower bound on the optimum, and round its solution into a schedule whose cost is compared with that bound. The quality of the algorithm is therefore limited by the quality of the relaxation, and the question of which relaxations are equivalent is a basic one for the method.

Two relaxations of $1\,|\,r_j\,|\,\sum w_j C_j$ are central. The time-indexed relaxation of Dyer and Wolsey (doi:10.1016/0166-218X(90)90104-K) has one variable per job and unit time slot, a pseudopolynomial number of variables. The mean busy time relaxation has one variable per job but one constraint per subset of jobs, the shifted parallel inequalities studied by Queyranne and others. Goemans, Queyranne, Schulz, Skutella and Wang (doi:10.1137/S089548019936223X, Section 2) show that both have the same optimal value, and that both are solved by one simple preemptive schedule. This mission formalizes that result, Corollary 2.6, together with the lemmas and theorems its proof uses.

Timeline:

1990: Dyer and Wolsey formulate several time-indexed relaxations of $1\,|\,r_j\,|\,\sum w_j C_j$ , among them the formulation (D) used here.
1993: Queyranne (doi:10.1007/BF01581271) describes the polyhedron of completion-time vectors on one machine without release dates by the parallel inequalities.
1994–1995: Queyranne and Schulz use shifted parallel inequalities in polyhedral approaches to machine scheduling with release dates.
1996: Goemans (IPCO, LNCS 1084) gives a supermodular relaxation for scheduling with release dates, the source of the canonical decompositions used for (R).
2002: Goemans, Queyranne, Schulz, Skutella and Wang prove that (D) and the mean busy time relaxation (R) have equal value, attained by the preemptive "LP schedule", and use this common bound for randomized approximation algorithms with ratios $1.7451$ and $1.6853$ .

Setting

There are $n$ jobs $N = \{1, \dots, n\}$ . Job $j$ has an integral processing time $p_j > 0$ , an integral release date $r_j \ge 0$ and a weight $w_j \ge 0$ . For a nonempty set $S$ of jobs, $p(S) = \sum_{j\in S} p_j$ and $r_{\min}(S) = \min_{j \in S} r_j$ .

A preemptive schedule gives each job a bounded measurable set $A_j \subseteq [r_j, \infty)$ of processing times of Lebesgue measure $p_j$ , the sets pairwise disjoint. The mean busy time of $j$ is $M_j = \frac{1}{p_j}\int_{A_j} t\,dt$ .

The LP schedule processes, at every moment, the available (released, unfinished) job of largest ratio $w_j/p_j$ , ties broken by index. With the jobs indexed so that $w_1/p_1 \ge \cdots \ge w_n/p_n$ , this is the available job of smallest index. As the data are integral, it runs one job or none in each unit slot $[\tau, \tau+1)$ ; $y^{LP}_{j\tau} \in \{0,1\}$ records whether it runs $j$ there, and $M^{LP}_j$ is its mean busy time of $j$ .

The preemptive time-indexed relaxation (D) with horizon $T$ has real variables $y_{j\tau} \ge 0$ for $\tau = r_j, \dots, T-1$ and reads

Z_D = \min \sum_j w_j C_j \quad\text{s.t.}\quad \sum_{j : r_j \le \tau} y_{j\tau} \le 1\ (\tau < T), \qquad \sum_{\tau=r_j}^{T-1} y_{j\tau} = p_j, \qquad C_j = \tfrac12 p_j + \tfrac{1}{p_j}\sum_{\tau=r_j}^{T-1}\big(\tau + \tfrac12\big) y_{j\tau}.

The mean busy time relaxation (R) reads

Z_R = \min \sum_j w_j\big(M_j + \tfrac12 p_j\big) \quad\text{s.t.}\quad \sum_{j\in S} p_j M_j \ge p(S)\big(r_{\min}(S) + \tfrac12 p(S)\big) \quad (\emptyset \ne S \subseteq N).

The horizon $T$ is required to bound the makespan of some feasible nonpreemptive schedule, for instance $T = \max_j r_j + \sum_j p_j$ .

Formalization targets

Goal: Corollary 2.6

For every instance, every weight vector $w \ge 0$ and every admissible horizon $T$ ,

Z_D = Z_R .

No ordering of the jobs and no reference to the LP schedule appear in the goal.

Milestones

Lemma 2.1. (D) has an optimal solution with $y_{j\tau} \in \{0,1\}$ .
Theorem 2.2. With the jobs sorted by $w_j/p_j$ , $y^{LP}$ is an optimal solution to (D).
Lemma 2.4. For every preemptive schedule and nonempty $S$ , $\sum_{j\in S} p_j M_j \ge p(S)\big(r_{\min}(S)+\tfrac12 p(S)\big)$ , with equality if and only if $S$ occupies $[r_{\min}(S), r_{\min}(S)+p(S))$ without interruption.
Theorem 2.5. With the jobs sorted by $w_j/p_j$ , $M^{LP}$ is an optimal solution to (R).
Eq. (2.6). $M^{LP}_j = \frac{1}{p_j}\sum_{\tau=r_j}^{T-1} y^{LP}_{j\tau}\big(\tau + \frac12\big)$ .

Significance

The equality $Z_D = Z_R$ lets one choose, for each purpose, the more convenient of the two relaxations. (D) is intuitive and a transportation problem, but has pseudopolynomially many variables; (R) has $n$ variables and a supermodular right-hand side, which the paper uses to describe its polyhedron. Theorems 2.2 and 2.5 show that the common optimum is attained by the LP schedule, which is computable greedily; the approximation guarantees of Section 3 of the paper, and of later work on $\alpha$ -point scheduling, are all measured against this value.

A formal development adds three things. First, a machine-checked model of preemptive single-machine schedules with release dates, of mean busy times, and of the LP schedule as a concrete recursive object, reusable by any formalization of $\alpha$ -point methods (two companion missions of this series use the same objects). Second, a formal statement of the two relaxations with honest optimal values. Third, verified proofs of results that are proved in the paper by short interchange and averaging arguments, whose measure-theoretic details (integrals over processing sets, null sets in the equality case) the paper leaves implicit. The results are proved in the literature; to our knowledge none of them has been machine-checked.

Difficulty

The paper's arguments are short, and each rests on a step that is informal on the page. Lemma 2.1 cites the integrality of transportation problems, a statement about the vertices of a polytope rather than a one-line fact. Theorem 2.2 ends with the claim that a 0/1 solution admitting no improving exchange "must correspond to the LP schedule", which is a property of the greedy rule that has to be derived from its definition. Theorem 2.5 depends on how the LP schedule arranges the jobs of each prefix $\{1, \dots, i\}$ of the sorted order in time; this is the only place sortedness enters, and it is again a property of the concrete schedule. Lemma 2.4 is an extremal statement about integrals over sets of prescribed measure, and its equality case holds only up to null sets. Finally, the goal concerns arbitrary, unsorted weights, while the two theorems it combines are about sorted indices, so the goal is not a direct conjunction of the milestones.

Formalization scope

Jobs are Fin n, numbered from $0$ ; $p_j$ , $r_j$ and the horizon $T$ are natural numbers and weights are real. A preemptive schedule is a family of processing sets $A_j \subseteq \mathbb R$ , not indicator functions. The LP schedule is defined by recursion on unit slots with the smallest-index rule; the sortedness of $w_j/p_j$ is a hypothesis of Theorems 2.2 and 2.5, not part of the definition. Variables of (D) are functions $y : \text{jobs} \times \mathbb N \to \mathbb R$ required to vanish outside $r_j \le \tau < T$ . $Z_D$ and $Z_R$ are infima of the objective over the feasible sets; under the stated hypotheses the feasible sets are nonempty and the objectives bounded below, so these are the LP values. Optimality in the milestones is stated as attaining the minimum, not through these infima.

The horizon hypothesis rules out the trivializing case in which (D) is infeasible and its infimum takes the junk value $0$ ; (D) is kept a linear program over real $y$ , since restricting to $\{0,1\}$ would make Lemma 2.1 vacuous. The running-time claim of Corollary 2.6 ( $O(n\log n)$ ) is not formalized.

A complete development needs: integrals of the identity over finite unions of intervals; a rearrangement lemma for sets of given measure; basic properties of the LP schedule (it is a preemptive schedule, it is work-conserving and finishes by any admissible $T$ , its blocks are canonical); and a relabelling argument. The schedule model and the LP-schedule lemmas are reusable for the companion missions on $\alpha$ -point scheduling. Contributions of any of these lemmas as separate theorems are welcome.

Selected references

M. X. Goemans, M. Queyranne, A. S. Schulz, M. Skutella, Y. Wang, Single machine scheduling with release dates, SIAM Journal on Discrete Mathematics 15(2):165–192, 2002. doi:10.1137/S089548019936223X
M. E. Dyer, L. A. Wolsey, Formulating the single machine sequencing problem with release dates as a mixed integer program, Discrete Applied Mathematics 26(2–3):255–270, 1990. doi:10.1016/0166-218X(90)90104-K
M. Queyranne, Structure of a simple scheduling polyhedron, Mathematical Programming 58:263–285, 1993. doi:10.1007/BF01581271
M. X. Goemans, Improved approximation algorithms for scheduling with release dates, Proceedings of the 8th ACM-SIAM Symposium on Discrete Algorithms (SODA), 591–598, 1997.

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

Discrete Convex Analysis XXXIII: Near-Optimality for Submodular MinimizationTextbook

Motivation

Chapter 10 turns from structure theory to algorithms: efficient methods for minimizing M-convex functions (via domain reduction) and submodular set functions (via Schrijver's and the Iwata-Fleischer-Fujishige scaling algorithms). Most of chapter 10's numbered results are asymptotic running-time bounds for specific procedural algorithms — a genuinely different kind of claim from the rest of this book (see Formalization scope). This mission places the results of this block that ARE ordinary mathematical propositions: correctness certificates, min-max theorems, and structural facts the algorithms rely on and produce.

Setting

For an M-convex set B ⊆ Z^V, the central part B° (the vectors of B lying away from its boundary, defined via per-coordinate bounds ℓ°_B, u°_B) is what the domain reduction algorithm searches from. For a submodular set function ρ : 2^V → R, the base polyhedron B(ρ) and its extreme bases (one per linear ordering of V, via Eq. (10.12)) let any base be written as a convex combination of finitely many extreme bases (Eq. (10.13)); a candidate minimizer W is certified via the linear orderings representing an optimal base. The Iwata-Fleischer-Fujishige (IFF) scaling algorithm relaxes this problem with a flow-augmentation parameter δ, maintaining a δ-feasible flow φ and vector z = x + ∂φ; near the end of a scaling phase, no augmenting path and no "active triple" together certify near-optimality.

Formalization targets

Goal: Near-optimality from the absence of augmenting paths (Proposition 10.20)

If S ⊆ W ⊆ V∖T, no arc of the auxiliary network leaves W, and no active triple exists, then z⁻(V) ≥ ρ(W)-nδ and x⁻(V) ≥ ρ(W)-n²δ; moreover W exactly minimizes ρ once δ is small enough relative to the smallest positive gap between two values of ρ. Chosen as goal: the book calls this "a key property of the scaling algorithm" and "a relaxation version of the min-max relation in Proposition 10.8", its own proof is the most substantial argument among this chunk's placed results, and Proposition 10.23 is a direct corollary of it.

Supporting structural targets

Proposition 10.8 is the min-max relation underlying the whole of section 10.2 (an Edmonds- intersection-theorem consequence, found by direct reading — the extractor's table missed it). Proposition 10.9 gives the three-part sufficient condition for optimality, in terms of the linear orderings representing an optimal base, that both Schrijver's algorithm and the IFF algorithm use as their termination criterion (also found by direct reading). Propositions 10.5-10.6 establish that the domain reduction algorithm's central part B° is always nonempty, via an explicit vector-extension step (Proposition 10.5 likewise missing from the extractor's table). Proposition 10.23 fixes individual coordinates once a scaling phase ends, and Proposition 10.24 gives the termination certificate for the IFF fixing algorithm's own separate graph-contraction procedure.

Significance

Chapter 10 is where this book cashes out its structure theory as algorithms with provable running times, and this mission places every result of that chapter's first two sections that is a mathematical proposition rather than a runtime bound: two min-max/optimality-certificate theorems (10.8-10.9) that are the combinatorial core making the following two strongly polynomial algorithms (Schrijver's, and Iwata-Fleischer-Fujishige's) correct, one central-part nonemptiness fact (10.5-10.6) underlying the domain reduction algorithm, and the two fixing/termination certificates (10.23-10.24) that let the scaling algorithms actually output a minimizer with a proof of optimality attached, not just a numerical answer.

None of these results are open — they are Murota's own account of submodular-function- minimization algorithms (sections 10.1-10.2). What this mission contributes is a faithful, machine-checked formal statement of each, including two results (Propositions 10.5 and 10.8) the platform's own automated extractor missed; no comparable formalization exists on the platform (see Formalization scope).

Difficulty

Six numbered results in this block (Propositions 10.4, 10.7, 10.17, 10.18, 10.21, 10.22) are excluded as hard: each states a Big-O asymptotic bound on the running time, function-evaluation count, or internal-procedure-call count of a specific iterative algorithm (the domain reduction algorithm, its scaling variant, Schrijver's algorithm, the IFF scaling algorithm). Faithfully stating "this algorithm runs in O(g(n)) time" requires a cost-tracked operational semantics for that specific algorithm — a well-founded recursive procedure with an oracle for evaluating the input function, threading a step/evaluation counter, instantiated over an unbounded family of ground-set sizes n and numeric parameters (K∞, M) — which is a fundamentally different kind of formalization task (computational complexity theory) from every one of the roughly 280 other numbered results in this book, none of which require modeling the cost of computing them. See HARD.md.

Formalization scope

Ground-set elements are a Fintype V with DecidableEq. Base M-convex-set vocabulary is redeclared from prior missions. Linear orderings of V are represented as bijections V ≃ Fin (Fintype.card V) rather than as lists, matching this series' established preference for order-indexed families over sequential data structures. Proposition 10.5's witness vector is stated as an existence claim (the mathematical content of the proposition), rather than by reconstructing the specific recursive modification procedure the book uses to produce it — a choice consistent with how this series has always formalized "the algorithm produces X" claims where X is a mathematical property, by asserting X's existence rather than executing the algorithm (see, e.g., mission 33-ch09c-networkflows's cycle-cancellation theorem). Six numbered results (Propositions 10.4, 10.7, 10.17, 10.18, 10.21, 10.22) are hard; see HARD.md. Contributions completing any of the seven sorrys are welcome; the goal and Proposition 10.9 carry the most independent proof content.

Selected references

K. Murota, Discrete Convex Analysis, SIAM, 2003. DOI: 10.1137/1.9780898718508.
S. Iwata, L. Fleischer, and S. Fujishige, "A combinatorial strongly polynomial algorithm for minimizing submodular functions," Journal of the ACM, 48 (2001), pp. 761-777 [102] (the IFF scaling algorithm this mission's Proposition 10.20 certifies).
A. Schrijver, "A combinatorial algorithm minimizing submodular functions in strongly polynomial time," Journal of Combinatorial Theory, Series B, 80 (2000), pp. 346-355 [182] (Schrijver's algorithm, whose termination criterion is Proposition 10.9).

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Combinatorics Operations Research Optimization·Captain: Shuze Chen

Discrete Convex Analysis XII: Steepest Descent for M-Convex Function MinimizationTextbook

Motivation

Chapters 6 through 9 characterized minimality for M-convex functions structurally (the M-optimality criterion, Theorem 6.26: a point is a global minimizer iff no local swap improves it) without saying how to find one. Chapter 10 turns that structural fact into an algorithm: the local characterization is the termination test of the simplest possible minimization procedure, steepest descent by coordinate swaps. This mission formalizes that algorithm and its two complexity bounds, plus a structural min-max identity for submodular base polyhedra that the chapter's heavier submodular-minimization algorithms build on. It is the first mission in this series whose goal names a method, not just a property of a class of functions — formalizing it faithfully means giving the algorithm itself a Lean representation that the complexity theorem then quantifies over, not just describing its output.

Setting

Let $f : \mathbb Z^V \to \mathbb R \cup \{+\infty\}$ be an M-convex function (MExchangeAxiom, chunk 06), $n = |V|$ . The steepest descent algorithm repeatedly replaces the current point $x$ by $x - \chi_u + \chi_v$ for a pair $u \ne v$ minimizing $f(x - \chi_u + \chi_v)$ , stopping when no such swap improves on $f(x)$ — at which point, by the M-optimality criterion, $x$ is a global minimizer. This mission represents a run of the algorithm as a sequence $x : \mathbb N \to \mathbb Z^V$ satisfying these step and termination relations directly, so that "the number of iterations" is a genuine property of any such run, not an informal gloss. Separately, for a submodular set function $\rho : 2^V \to \mathbb R$ (chunk 04's SubmodularSetFunction), the base polyhedron $B(\rho)$ (chunk 04's BasePolyhedron) is the polytope $\{x \in \mathbb R^V : x(X) \le \rho(X)\ \forall X,\ x(V) = \rho(V)\}$ .

Formalization targets

Goal: Proposition 10.2 (iteration bound with tie-breaking)

For an M-convex function $f$ with finite $\ell^1$ -diameter $K_1 = \max\{\|x-y\|_1 : x,y \in \operatorname{dom} f\}$ (Eq. (10.1)), the number of iterations in the steepest descent algorithm using the tie-breaking rule (10.2) — a fixed lexicographic rule for choosing among tied steepest pairs, based on an arbitrary but fixed ordering $\varphi$ of $V$ — is bounded by $K_1/2$ .

Milestones: Proposition 10.1, Proposition 10.8

Proposition 10.1: an unconditional warm-up — if $f$ has a unique minimizer $x^*$ , any run of the plain (untied) algorithm from $x^0$ terminates within $\|x^0 - x^*\|_1/2$ iterations, with no tie-breaking rule needed. Proposition 10.8: a structural min-max identity, $\max\{x^-(V) : x \in B(\rho)\} = \min\{\rho(X) : X \subseteq V\}$ , for a submodular set function $\rho$ — chosen deliberately as a milestone that names no algorithm at all, in contrast to this mission's other two items.

Significance

The result itself. Proposition 10.2 is the complexity backbone of §10.1: it is what turns "steepest descent terminates" (an easy monotonicity observation) into a genuine polynomial bound, and it is the base case the chapter's more elaborate scaling and domain-reduction algorithms (not drafted here) improve on. Proposition 10.8, though algorithm-free, is the structural fact ("verifying membership in $B(\rho)$ seems to need a submodular minimization procedure — but demonstrating optimality of a cut $X$ only needs a base with $x^-(V) = \rho(X)$ ") that makes Schrijver's and the IFF algorithms' correctness proofs possible, and specializes chunk 04's Edmonds's intersection theorem to the two-function case $\rho_1 = \rho, \rho_2 = 0$ .

Formalizing it. A prior-art search (q=steepest descent, q=submodular minimization, q=base polyhedron) found no existing platform items for any of this chapter's results. This mission gives the first formal statement of an M-convex minimization algorithm's complexity, and directly reuses chunk 06's MExchangeAxiom/ArgMin/CharVec/DomZ and chunk 04's SubmodularSetFunction/BasePolyhedron — genuine cross-chunk substrate reuse spanning two different chapters' worth of prior missions.

Difficulty

The chapter's own framing (quoted in BRIEF.md) is that every other chapter's theorems state a property of a class of functions or sets, while this chapter's theorems state properties of a named algorithm run on such an object. Formalizing "the number of iterations in the steepest descent algorithm is bounded by ..." faithfully means giving the algorithm's steps (S0-S3) and termination test a Lean representation that the bound then quantifies over — stating the bound about "the minimizer" alone, with the algorithm silently dropped, would misrepresent the theorem as a fact about minimizers rather than about a procedure that finds one. This mission represents a run of the algorithm as an abstract sequence satisfying the book's own step-transition relations (documented in full in MODERATION_NOTES.md), letting the complexity theorems quantify over any valid run rather than committing to one executable implementation.

A second difficulty is scope: the chapter's recommended primary goal, Proposition 10.18 (Schrijver's algorithm's complexity), needs a scaling procedure with several auxiliary data structures — a materially larger definitional undertaking than steepest descent's simple greedy-swap loop. Per BRIEF.md's own explicit fallback authorization, this mission takes Proposition 10.2 as its goal instead; see HARD.md and MODERATION_NOTES.md for the full reasoning.

Formalization scope

Runs of the algorithm are represented as x : ℕ → V → ℤ satisfying IsSteepestDescentRun (plain) or IsSteepestDescentRunTieBreak (with the tie-breaking rule) — a step relation plus a termination test, with an explicit iteration count N the theorems bound. The tie-breaking key $\Phi(u,v)$ (Eq. (10.2)) and its lexicographic order are formalized directly (a manual three-way comparison, not Mathlib's default componentwise Prod order). Both iteration bounds are stated as 2 * N ≤ k rather than N ≤ k / 2, avoiding natural-number division. Not drafted: the derived "hence ... in $O(F \cdot n^2 K_1)$ time" corollary of Proposition 10.2 (needs a cost-model primitive for "time" and " $F$ " this mission does not otherwise use — the iteration-count bound itself, this proposition's genuine combinatorial content, is drafted in full); Schrijver's algorithm and everything in §10.2.2 onward; the steepest descent scaling algorithm, the domain reduction algorithm and its scaling variant (§10.1.2-10.1.3, structurally different algorithms); the quasi-M-convex extension mentioned immediately after Proposition 10.2. A trivializing formalization would state the iteration bound as an unconditional fact about "a" minimizer-finding procedure, or would drop the tie-breaking rule from Proposition 10.2 and thereby understate what the bound actually requires; neither is done.

Selected references

K. Murota, Discrete Convex Analysis, SIAM, 2003. DOI: 10.1137/1.9780898718508.

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

Discrete Convex Analysis XXXII: Network DualityTextbook

Motivation

Mission 32-ch09b-networkflows established the potential and negative-cycle optimality criteria for M-convex submodular flow problems. This mission finishes chapter 9 with the two topics that close it out: the constructive engine behind the negative-cycle criterion — cycle cancellation, which actually improves a nonoptimal flow rather than merely detecting suboptimality, resting on a delicate "unique-min condition" for bipartite matchings — and network duality, the chapter's capstone structural theorem showing that M-convexity and L-convexity are preserved (and their conjugacy is preserved) under transformation by an arbitrary network.

Setting

For a feasible integer flow ξ in the M-convex submodular flow problem MSFP2, a negative cycle in the auxiliary network (Gξ,ℓξ) witnesses suboptimality (mission 32's Theorem 9.20); cycle cancellation modifies ξ along a smallest such cycle to produce a strictly better flow ξ̄ (Eq. (9.75)). The unique-min condition for a pair (x,y) of integer vectors with ‖x-y‖∞=1 asks whether the bipartite graph G(x,y) — vertices the positive/negative supports of x-y, weights the M-convex exchange values Δf(x;v,u) — has a unique minimum-weight perfect matching; when it does, the M-convex exchange inequality of Proposition 6.25 becomes an equality. Separately, a network G=(V,A;S,T) with entrance set S and exit set T transforms a pair of functions f,g on Zˢ into induced functions f̃,g̃ on Zᵀ (Eqs. (9.81)-(9.82)), the minimum cost to meet a boundary specification at the exit given a production cost at the entrance and a transportation cost along arcs.

Formalization targets

Goal: Network duality for Z→Z functions (Theorem 9.26)

M-(resp. M $^\natural$ -)convexity and integer-valuedness of f transfer to the induced f̃; L-(resp. L $^\natural$ -)convexity and integer-valuedness of g transfer to g̃; and if f is M $^\natural$ -convex, g is its L $^\natural$ -conjugate, and each arc cost ga is the conjugate of fa, then g̃ is the conjugate of f̃. Chosen as goal: the book calls this "the harmonious relationship between network flow and M-/L-convexity", its own proof runs roughly six pages (the longest argument in this chunk), and it is the general fact from which Theorems 9.27-9.28 (analogues for other type combinations) and Notes 9.29-9.30 (the aggregation and infimal- convolution closure properties of M-convex functions, already placed in mission 22-ch06b-mconvexfunctions's own Theorem 6.13) all descend.

Supporting structural targets

Theorem 9.22 shows cycle cancellation strictly improves the objective; Propositions 9.23-9.25 are "the key ingredient" behind it: Proposition 9.23 shows the unique-min condition upgrades the M-convex exchange inequality to an equality, Proposition 9.24 gives a checkable characterization of when a bipartite weighted graph has a unique minimum-weight perfect matching, and Proposition 9.25 is the fact that makes the machine run — the specific pair (∂ξ,∂ξ̄) arising from cycle cancellation always satisfies the unique-min condition. Theorems 9.27 and 9.28 are the network duality theorem's own analogues for Z→R and R→R functions, the second restoring the conjugacy assertion (missing for Z→R) via the ordinary real Legendre-Fenchel transform.

Significance

Cycle cancellation is this book's constructive answer to the negative-cycle criterion: not just a certificate of suboptimality, but an actual improvement step, the combinatorial core of the cycle-canceling algorithm explained in section 10.4.3 (mission 35-ch10c-algorithms). Its correctness proof is one of the most intricate combinatorial arguments in the entire book — a proof by contradiction using a multiset-union identity (Eq. (9.80)) to derive a smaller negative cycle from an assumed non-uniqueness, itself resting on Proposition 9.24's Monge-like characterization of unique bipartite matchings. Network duality, meanwhile, is the theorem that explains why discrete convex analysis and network flow theory are so tightly intertwined throughout this book: it is the general mechanism (matroid induction, min-max relations, the M-convex aggregation and infimal-convolution closure properties) underlying nearly every construction chapter 2 introduced informally and chapter 6 proved piecemeal.

None of these results are open — they are Murota's own account of cycle cancellation (section 9.5.2) and network duality (section 9.6). What this mission contributes is a faithful, machine-checked formal statement of each, completing the platform's coverage of chapter 9 begun in missions 12-network-flows and 32-ch09b-networkflows; no comparable formalization exists on the platform (see Formalization scope).

Difficulty

Theorem 9.26's own proof needs the full weight of everything chapter 9 has built (Theorem 9.16's potential criterion for integer flows, the conjugacy theorem of chapter 8), which this mission does not re-prove (proofs are sorry throughout, per this pass's scope) but whose statement still needs the induced-function machinery built faithfully: since the general framework's optimal-value-type quantities can genuinely be -∞ (the book's own blanket hypothesis f̃ > -∞ acknowledges this), InducedFTilde/InducedGTilde are EReal-valued, following the same soundness discipline established in mission 31-ch08d-conjugacyduality for Lagrangian duality's derived quantities.

Formalization scope

Ground-set vertices V and arcs A are Fintype with DecidableEq. All base M-/L-convexity vocabulary is redeclared from prior missions. Functions "on Zˢ" for S a proper subset of the ground set are represented as ordinary V→Z functions required to vanish outside S (SupportedOn), rather than as functions on a dependent subset type — a padding-with-zero encoding consistent with this whole series' preference for a single ambient ground-set domain. C[R→R] (univariate real polyhedral convex functions, needed only for Theorem 9.28's arc costs) is formalized as ordinary midpoint-style convexity (IsConvexUnivariateR) rather than the book's own polyhedral characterization, since polyhedrality plays no role in Theorem 9.28's conclusion beyond ensuring the induced functions are well-behaved. All seven numbered results found in this chunk's page range are placed in full, with no partial-coverage scope reduction. Contributions completing any of the seven sorrys are welcome; the goal and Proposition 9.25 carry the most independent proof content.

Selected references

K. Murota, Discrete Convex Analysis, SIAM, 2003. DOI: 10.1137/1.9780898718508.
K. Murota, "Valuated matroid intersection," SIAM Journal on Discrete Mathematics, 9 (1996), pp. 545-561 [135] (the unique-max lemma Proposition 9.23 reformulates, and the proof technique behind Proposition 9.25).
K. Murota, "Discrete convex analysis," Mathematical Programming, 83 (1998), pp. 313-371 [140] (network duality and cycle cancellation for the M-convex submodular flow problem).

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

Discrete Convex Analysis XXXI: The Potential Criterion for Network FlowsTextbook

Motivation

Chapter 9 is where discrete convex analysis meets classical network flow theory: the minimum cost flow problem's three hallmark properties — an optimality criterion by potentials, an optimality criterion by negative cycles, and integrality of optimal solutions — are shown to survive, in a precise and increasingly general form, first for arbitrary polyhedral convex costs (MCFP3), then for the M-convex submodular flow problem (MSFP2/MSFP3), the chapter's own combinatorial generalization of the classical problem. This mission places the potential criterion (Theorem 9.4) and its cascade of six corollaries and generalizations, the block of results this book's own text uses to carry every other result in the chapter.

Setting

A digraph G = (V,A) with tail/head maps ∂⁺,∂⁻ : A → V. A flow ξ : A → R has boundary ∂ξ(v) = Σ{ξ(a) : ∂⁺a=v} − Σ{ξ(a) : ∂⁻a=v}. A potential p : V → R has coboundary δp(a) = p(∂⁺a) − p(∂⁻a). The minimum cost flow problem MCFP3 minimizes Γ₃(ξ) = Σₐ fₐ(ξ(a)) + f(∂ξ) over flows, for polyhedral convex arc costs fₐ : R → R∪{+∞} and boundary cost f : Rⱽ → R∪{+∞}; MCFP0 is its linear-cost, fixed-supply special case. The M-convex submodular flow problem MSFP3 is MCFP3 with f additionally M-convex; MSFP2 is its linear-arc-cost special case.

Formalization targets

Goal: The potential criterion for MCFP3 (Theorem 9.4)

For a feasible flow ξ, ξ is optimal for MCFP3 iff there is a potential p with ξ(a) a minimizer of the reduced arc cost fₐ[δp(a)] for every arc and ∂ξ a minimizer of the reduced boundary cost f[−p]; and any such optimal potential characterizes optimality of every feasible flow. This is the hub result of the whole chunk: the book states Theorem 9.14 is "immediate" from it, and every other placed result either specializes it directly or builds on that specialization.

Supporting structural targets

Theorem 9.5 reformulates MCFP0's optimality as the absence of a negative cycle in an auxiliary network; Theorem 9.6 gives MCFP0's primal and dual integrality, the latter identifying the optimal-potential set as an L-convex polyhedron. Theorem 9.14 specializes the goal to MSFP3; Theorem 9.15 upgrades this to a full polyhedral and integrality structure theorem for MSFP3's optimal-flow-boundary and optimal-potential sets (M2-convex and L-convex polyhedra respectively); Theorem 9.16 is the integer-flow analogue, with the boundary set now literally M2-convex and the integer-optimal-potential set literally L-convex. Theorems 9.18 and 9.20 give the negative-cycle reformulation for MSFP2, real and integer flows respectively, generalizing Theorem 9.5 by admitting a third class of auxiliary arcs governed by the M-convex boundary cost's directional derivative (or its discrete difference, in the integer case).

Significance

This is the chapter's demonstration that M-convexity is not merely an abstract combinatorial axiom but the exact structural hypothesis under which classical network-flow duality survives intact: every one of the four "nice properties" the book opens the chapter with (potentials, negative cycles, integrality, efficient algorithms) is preserved verbatim in the M-convex generalization, and this mission's eight results are the proof of that claim for the first three. The chunk's own internal dependency structure — one foundational theorem (9.4) from which every other placed result descends by specialization or direct generalization — is itself characteristic of how this book organizes its combinatorial machinery around a single convex- analytic core.

None of these results are open — they are Murota's own account of network flow duality under M-convexity (sections 9.1, 9.4, and 9.5). What this mission contributes is a faithful, machine-checked formal statement of each, extending the platform's coverage of chapter 9 begun in mission 12-network-flows (which covered §9.1.1-9.1.2 and §9.3, the feasibility and max-flow min-cut results, deliberately leaving this block for later apparatus); no comparable formalization exists on the platform (see Formalization scope).

Difficulty

The eight results span real- and integer-flow versions of two nested problem hierarchies (MCFP0 ⊂ MCFP3, MSFP2 ⊂ MSFP3) and two distinct optimality certificates (potentials, negative cycles), which this mission handles by building one shared apparatus — FeasibleFlowMCFP3, Gamma3, OptimalFlowMCFP3, IsOptimalPotential — that MCFP0 and MSFP3 both instantiate (MCFP0 literally as the linear-cost/singleton-boundary special case of Eq. (9.11)), and one shared generic cycle/negative-cycle apparatus (IsCycle, CycleLength, HasNegativeCycle) instantiated three times with different auxiliary-arc types (A⊕A for MCFP0, A⊕A⊕(V×V) for MSFP2's extra Cξ arcs governed by the boundary cost's directional derivative). "Primal integral" and "dual integral" polyhedral convex functions (the book's own C[Z|R→R]/C[R→R|Z] notation, used in Theorem 9.15) needed a modeling decision, since the book's own definition of these classes lies outside this chunk's page range; see Formalization scope.

Formalization scope

Ground-set vertices V and arcs A are Fintype with DecidableEq. All base M-/L-convexity vocabulary is redeclared from prior missions in this series. "Primal integral" (C[Z|R→R], M[Z|R→R]) is formalized as integer effective domain (IsDomainIntegerArc/IsDomainIntegerR); "dual integral" (C[R→R|Z], M[R→R|Z]) is formalized as the existence of an integer subgradient at every domain point (IsDualIntegralArc/IsDualIntegralR) — a standard equivalent characterization for polyhedral convex functions, and a deliberate modeling choice recorded in MODERATION_NOTES.md rather than a literal transcription of the book's own (out-of-range) definition of these two notation classes. All eight numbered results found in this chunk's page range are placed in full, with no partial-coverage scope reduction. Contributions completing any of the eight sorrys are welcome; the goal and Theorem 9.15 carry the most independent proof content.

Selected references

K. Murota, Discrete Convex Analysis, SIAM, 2003. DOI: 10.1137/1.9780898718508.
R. T. Rockafellar, Network Flows and Monotropic Optimization, Wiley, 1984 [178] (the classical potential/Fenchel-duality framework this mission's Theorem 9.4 adapts).
K. Murota, "Discrete convex analysis," Mathematical Programming, 83 (1998), pp. 313-371 [140] (the Lagrange duality and negative-cycle theory of section 9.5 this mission's Theorems 9.18 and 9.20 draw from).

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Bandit Algorithms Machine Learning Operations Research+1·Captain: mikedeng1

Optimal Best Arm Identification with Fixed Confidence II: Characterization of the Optimal Proportions of Arm DrawsResearch Paper

Motivation

In best arm identification with fixed confidence, a learner samples $K$ unknown distributions ("arms") sequentially and must name the arm with the largest mean, with error probability at most a prescribed $\delta$ , using as few samples as possible. Garivier and Kaufmann (arXiv:1602.04589, COLT 2016) showed that every $\delta$ -PAC strategy needs, in expectation, at least $T^*(\boldsymbol\mu)\,\mathrm{kl}(\delta,1-\delta)$ samples, where the characteristic time $T^*(\boldsymbol\mu)$ is the value of a max–min optimization problem over the proportions of draws allocated to the arms. The maximizer of that problem, $w^*(\boldsymbol\mu)$ , is the allocation any asymptotically optimal strategy must follow; the Track-and-Stop algorithm of the same paper computes $w^*(\hat{\boldsymbol\mu})$ at plug-in estimates and tracks it.

A strategy can only track $w^*$ if $w^*$ can be computed. This mission formalizes the part of the paper (Section 2.2 and Appendix A) that turns the abstract max–min problem into an explicit recipe: a closed form for the inner infimum, and a characterization of $w^*$ through the root of one increasing scalar function. The problem had been solved in closed form before only for special cases, such as Poisson rewards with all suboptimal arms equal (Vaidhyan and Sundaresan, 2015); the paper's result covers every one-parameter exponential family.

Setting

A canonical one-parameter exponential family is a family of laws $\nu_\theta$ , $\theta\in\Theta$ , on $\mathbb R$ with density $\exp(\theta x-b(\theta))$ with respect to a reference measure $\xi$ . The law $\nu_\theta$ has mean $\dot b(\theta)$ ; the set of attainable means is the mean space $\dot b(\Theta)$ . For means $\mu=\dot b(\theta)$ and $\mu'=\dot b(\theta')$ the Kullback–Leibler divergence is

d(\mu,\mu')=\mathrm{KL}(\nu_\theta,\nu_{\theta'})=b(\theta')-b(\theta)-\dot b(\theta)(\theta'-\theta).

Bernoulli laws and Gaussian laws with known variance are the standard examples.

A bandit model is identified with its vector of means $\boldsymbol\mu=(\mu_1,\dots,\mu_K)\in\dot b(\Theta)^K$ . $\mathcal S$ is the set of models with a unique optimal arm $a^*(\boldsymbol\mu)$ , and $\mathrm{Alt}(\boldsymbol\mu)=\{\boldsymbol\lambda\in\mathcal S:a^*(\boldsymbol\lambda)\ne a^*(\boldsymbol\mu)\}$ is the set of alternatives. $\Sigma_K$ is the probability simplex. The transportation cost of proportions $w\in\Sigma_K$ and the objects of the paper are

c_{\boldsymbol\mu}(w)=\inf_{\boldsymbol\lambda\in\mathrm{Alt}(\boldsymbol\mu)}\sum_{a=1}^Kw_a\,d(\mu_a,\lambda_a),\qquad T^*(\boldsymbol\mu)^{-1}=\sup_{w\in\Sigma_K}c_{\boldsymbol\mu}(w),\qquad w^*(\boldsymbol\mu)=\operatorname*{argmax}_{w\in\Sigma_K}c_{\boldsymbol\mu}(w).

The arms are sorted so that $\mu_1>\mu_2\ge\dots\ge\mu_K$ . The parameterized Jensen–Shannon divergence is, for $\alpha\in[0,1]$ ,

I_\alpha(\mu_1,\mu_2)=\alpha\,d\big(\mu_1,\alpha\mu_1+(1-\alpha)\mu_2\big)+(1-\alpha)\,d\big(\mu_2,\alpha\mu_1+(1-\alpha)\mu_2\big).

For $a\in\{2,\dots,K\}$ let $g_a(x)=(1+x)I_{1/(1+x)}(\mu_1,\mu_a)$ for $x\ge0$ , let $x_a=g_a^{-1}$ , and let $x_1\equiv1$ . Finally

F_{\boldsymbol\mu}(y)=\sum_{a=2}^K\frac{d\big(\mu_1,m_a(y)\big)}{d\big(\mu_a,m_a(y)\big)},\qquad m_a(y)=\frac{\mu_1+x_a(y)\mu_a}{1+x_a(y)}.

Formalization targets

Goal: Theorem 5 (p. 5)

With $D=d(\mu_1,\mu_2)$ : $F_{\boldsymbol\mu}$ is continuous and strictly increasing on $[0,D[$ , $F_{\boldsymbol\mu}(0)=0$ , $F_{\boldsymbol\mu}(y)\to\infty$ as $y\to D$ , the equation $F_{\boldsymbol\mu}(y)=1$ has a unique solution $y^*\in[0,D[$ , and

w\in w^*(\boldsymbol\mu)\iff w_a=\frac{x_a(y^*)}{\sum_{i=1}^Kx_i(y^*)}\quad\text{for every arm }a.

The equivalence says at once that the argmax exists, that it is a single point, and that it is given by eq. (5).

Milestones

Lemma 3 (p. 5): for every $w\in\Sigma_K$ ,

c_{\boldsymbol\mu}(w)=\min_{a\ne1}(w_1+w_a)\,I_{\frac{w_1}{w_1+w_a}}(\mu_1,\mu_a).

Claim after eq. (4) (p. 5): $g_a$ is a strictly increasing one-to-one mapping from $[0,+\infty[$ onto $[0,d(\mu_1,\mu_a)[$ .
Lemma 4 (p. 5): for every maximizer $w^*$ and all $a,b\in\{2,\dots,K\}$ ,

(w^*_1+w^*_a)I_{\frac{w^*_1}{w^*_1+w^*_a}}(\mu_1,\mu_a)=(w^*_1+w^*_b)I_{\frac{w^*_1}{w^*_1+w^*_b}}(\mu_1,\mu_b).

Significance

The result. Theorem 5 reduces a $(K-1)$ -dimensional non-smooth max–min problem to finding the root of one continuous increasing function on a bounded interval, each evaluation of which requires $K-1$ scalar inversions. It gives existence and uniqueness of $w^*(\boldsymbol\mu)$ , which the paper's lower bound only presupposes, and it is the computational core of Track-and-Stop: without an explicit, well-posed $w^*$ the tracking strategy is not defined. Lemma 3 alone gives the closed form of the inner infimum used again in the analysis of the stopping rule.

Formalizing it. The results are proved in the paper; nothing here is open. To the best of our knowledge none of them has a machine-checked proof: the platform's existing best-arm-identification rows concern Gaussian arms and state the characteristic time at the level of measures, without this characterization. The mission produces a checked reduction for general one-parameter exponential families, including the edge cases the text passes over (ties among suboptimal arms, zero weights, the behaviour of $x_a$ near the end of its domain).

Difficulty

The infimum in Lemma 3 ranges over $\mathrm{Alt}(\boldsymbol\mu)$ , a set of models with a unique best arm, so it is an open condition: the minimizing configuration, in which $\lambda_1$ and $\lambda_a$ coincide, lies outside $\mathrm{Alt}(\boldsymbol\mu)$ and is only approached. Other arms may also compete for the best position. A statement in which the infimum is taken over the closed relaxation $\{\lambda_a\ge\lambda_1\}$ is a lemma of the proof, not Lemma 3.

For Theorem 5, the equalization in Lemma 4 needs an argument that holds for every maximizer, not only for one found by a first-order condition, because the objective is a minimum of functions and is not differentiable. The monotonicity of $F_{\boldsymbol\mu}$ needs the monotonicity of each $x_a$ and of each ratio in the moving point $m_a$ , and the limit at $D$ rests on the second-best arm(s) only, which is where the ordering $\mu_1>\mu_2\ge\dots$ enters. Finally the analytic facts about $d$ (continuity, positivity off the diagonal, monotonicity in each argument) must be derived from the exponential family itself.

Formalization scope

Lean represents a model by μ : Fin K → ℝ with 2 ≤ K and every μ a in the mean space deriv F.b '' F.Θ. The paper's arm $1$ is index 0, arm $2$ is index 1. The exponential family is a structure ExpFamily whose parameter set is a nonempty open interval and whose b is twice continuously differentiable with $\ddot b>0$ on $\Theta$ . These two conditions are added to the paper's "convex, twice differentiable" and are disclosed: strict convexity is what makes the mean parameterization unique, and openness is what lets alternatives approach the boundary of $\mathrm{Alt}(\boldsymbol\mu)$ . The reference measure and normalization are part of the structure but unused here.

The transportation cost is an infimum in EReal, so it is the true infimum of the set of values rather than a default 0. $w^*(\boldsymbol\mu)$ is never defined by choice: " $w$ is optimal" is a predicate, and Theorem 5 characterizes the set of such $w$ . The functions $x_a$ are the inverse of $g_a$ on $[0,+\infty[$ and are evaluated only on $[0,d(\mu_1,\mu_2)[$ . "Increasing" in Theorem 5 is stated as strictly increasing, as proved in Appendix A.2. Lemmas 3 and 4 and the claim on $g_a$ assume only that arm $1$ is the unique best arm, which is weaker than the paper's standing ordering.

A formalization that replaces $\mathrm{Alt}(\boldsymbol\mu)$ by $\{\lambda_a\ge\lambda_1\}$ , assumes the maximizer exists and is unique, or asserts only existence of some $y^*$ without the formula for $w^*$ , does not state these results and is ruled out.

A complete development needs: basic calculus of exponential families (the Bregman form of $d$ , its continuity and strict positivity off the diagonal, its monotonicity in the second argument), the inverse function of a continuous strictly monotone map on an interval, and compactness of the simplex. The divergence facts are reusable in every bandit mission built on exponential families. Proofs of the milestones, of auxiliary facts about $d$ and $I_\alpha$ , and a sorry-free instance of ExpFamily (Bernoulli or unit-variance Gaussian) are welcome.

Selected references

A. Garivier, E. Kaufmann, Optimal Best Arm Identification with Fixed Confidence, COLT 2016 (JMLR W&CP 49), arXiv:1602.04589v2. https://arxiv.org/abs/1602.04589
E. Kaufmann, O. Cappé, A. Garivier, On the Complexity of Best-Arm Identification in Multi-Armed Bandit Models, JMLR 17, 2016. https://arxiv.org/abs/1407.4443
N. K. Vaidhyan, R. Sundaresan, Learning to detect an oddball target, arXiv:1508.05572, 2015. https://arxiv.org/abs/1508.05572
O. Cappé, A. Garivier, O.-A. Maillard, R. Munos, G. Stoltz, Kullback–Leibler upper confidence bounds for optimal sequential allocation, Annals of Statistics 41(3), 2013. https://arxiv.org/abs/1210.1136

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Mechanism Design Operations Research Optimization·Captain: mikedeng1

Supply Chain Coordination for False Failure Returns: A Coordinating Target Rebate Helps the Retailer, and the Manufacturer iff Coordinated Effort Is at Least Twice Decentralized EffortResearch Paper

Motivation

A false failure return is a product returned by a consumer as defective although it has no functional or cosmetic defect; managers attribute such returns to installation difficulties, a mismatch with the consumer's preferences, and remorse. Ferguson, Guide and Souza report (pp. 376–377) that false failures account for up to 80% of Hewlett-Packard's inkjet printer returns, roughly 5% of sales, and that the per-unit cost of a false failure return to computer manufacturers is around 25% of the product's price. The manufacturer absorbs most of that cost, while the retailer is the party able to prevent the returns in the short term, by spending time with customers before the sale and supporting them after it. The retailer bears the cost of that effort but captures only part of its benefit, so without an incentive it exerts too little.

The paper (Ferguson, Guide & Souza, MSOM 2006) models this as a single-period manufacturer–retailer problem with non-contractible retailer effort, and asks which contracts restore the supply chain's optimal effort and who gains from them. It belongs to the literature on supply chain coordination with contracts (Cachon 2003) and on channel rebates with sales effort (Taylor 2002); its object is a target rebate, a payment to the retailer for every false failure return below a target.

Setting

A manufacturer with unit cost $c$ sells to a retailer at wholesale price $w$ , who sells at retail price $p$ . Avoiding one false failure return is worth

M_m = m + \delta_m(w - c) \ \text{to the manufacturer},\qquad R_r = r + \delta_r(p - w)\ \text{to the retailer},

where $m$ and $r$ are the parties' return-processing costs and $\delta_m$ , $\delta_r$ are the unit sale impacts of avoiding the return (p. 381). Both are assumed positive.

The retailer chooses an effort $\rho \ge 1$ at cost $a\rho^2/2$ , $a > 0$ . At effort $\rho$ the number of false failures is a nonnegative random variable $X(\rho)$ with mean $\beta/\rho$ , where $\beta > 0$ is the expected number at the minimum effort $\rho = 1$ . The coordinated supply chain earns

\Pi(\rho) = (M_m + R_r)\,\beta\Big(1 - \frac1\rho\Big) - \frac{a\rho^2}{2},

maximized at the coordinated effort $\rho^C = [(M_m + R_r)\beta/a]^{1/3}$ . Without a contract the retailer earns $\pi_R(\rho) = -a\rho^2/2 + R_r\beta(1 - 1/\rho)$ and chooses the decentralized effort $\rho^D = \max\{(R_r\beta/a)^{1/3}, 1\}$ ; the manufacturer then earns $\pi_M(\rho^D) = M_m\beta(1 - 1/\rho^D)$ .

Under a target rebate contract $(u, T)$ the retailer receives $u$ for every false failure below the target $T$ , so the profits become

\pi_R(\rho \mid T, u) = u\,E\{[T - X(\rho)]^+\} - \frac{a\rho^2}{2} + R_r\beta\Big(1 - \frac1\rho\Big),\qquad \pi_M(\rho \mid T, u) = M_m\beta\Big(1 - \frac1\rho\Big) - u\,E\{[T - X(\rho)]^+\}.

The contract coordinates the supply chain when $\rho^C$ maximizes $\pi_R(\cdot \mid T, u)$ over $\rho \ge 1$ . In the uniform case of §3.1, $X(\rho) \sim \mathrm{Uniform}(0, 2\beta/\rho)$ , and the contract must satisfy $T < 2\beta/\rho^C$ .

Formalization targets

Goal: Proposition 2 (p. 383)

Assume $a, \beta, M_m, R_r > 0$ and $(M_m + R_r)\beta > a$ , and let $X(\rho)$ be uniform. For every coordinating contract $(u, T)$ with $u > 0$ , $0 < T < 2\beta/\rho^C$ ,

\pi_R(\rho^C \mid T, u) \ge \pi_R(\rho^D) \qquad\text{and}\qquad \Big(\pi_M(\rho^C \mid T, u) \ge \pi_M(\rho^D) \iff \rho^C \ge 2\rho^D\Big).

Milestones

The milestones follow the paper's §3–§3.1 and the appendix proof, in attack order: concavity of $\Pi$ and optimality of $\rho^C$ (Eqs. (1)–(2)); $\rho^C > 1$ in the interesting case; optimality of $\rho^D$ (Eqs. (3)–(4)); $\rho^C \ge \rho^D$ ; Proposition 1 (concavity of the retailer's rebate profit when $\partial^2 F(x\mid\rho)/\partial\rho^2 \le 0$ ); its uniform instance; the uniform closed form (8); the first-order condition (9); the coordinating target (10) together with the admissibility condition $u > M_m$ ; the manufacturer's profit $M_m\beta(\rho^C - 2)/\rho^C$ under a coordinating contract (25); the retailer's profit (27); and the retailer's gain in the two cases $\rho^D > 1$ (30) and $\rho^D = 1$ (31).

Significance

The result divides the effect of the contract between the two parties. The retailer is always at least as well off as without a contract; the manufacturer, who pays the rebate, gains exactly when the supply chain's optimal effort is at least twice what the retailer would exert alone. When $\rho^D > 1$ this is equivalent to $M_m \ge 7R_r$ (p. 383), so a target rebate pays for the manufacturer only when its own stake in avoiding a false failure dwarfs the retailer's. The companion result (10) shows that for every rebate $u > M_m$ exactly one admissible coordinating target exists, and none for $u \le M_m$ : a coordinating rebate is always larger than the manufacturer's own cost of a return.

The results are proved in the paper by calculus and algebra. None of them has a machine-checked proof that we know of, and nothing on Prove2Me models non-contractible effort or target rebates. The mission produces a checked version of the paper's model with the expectation taken as a genuine integral against the uniform law, a formal notion of coordination as the retailer's optimization, and statements that make explicit which hypotheses each step of the appendix uses. The definitions of effort-dependent profits and coordination are reusable for other effort-inducing contracts in the same paper and in the sales-effort literature.

Difficulty

The algebra of the appendix is short once the first-order condition (9) holds at $\rho^C$ . The substance is getting there. Coordination is defined by optimality of $\rho^C$ for the retailer's profit, and that profit involves the expectation $E\{[T - X(\rho)]^+\}$ , which is piecewise in $\rho$ : it equals $T^2\rho/4\beta$ only while $T \le 2\beta/\rho$ , and $T - \beta/\rho$ beyond. Deriving (9) requires showing that $\rho^C$ is an interior maximizer, that the expectation is differentiable there with the closed-form derivative, and that the side condition $T < 2\beta/\rho^C$ keeps $\rho^C$ in the closed-form region. The converse direction of (10), that the formula for $T$ produces a coordinating contract, needs concavity of the piecewise profit on all of $\rho \ge 1$ , which is where Proposition 1 enters.

Replacing the expectation by the global formula $T^2\rho/4\beta$ is the tempting shortcut and it changes the problem: for $\rho > 2\beta/T$ the formula exceeds the true expectation, and the retailer's maximizer, hence the meaning of "coordinates", changes with it.

Formalization scope

All parameters are real numbers, bundled in a structure Params; $M_m$ and $R_r$ are Params.Mm and Params.Rr. Effort ranges over $\rho \ge 1$ (Set.Ici 1); statements the paper makes for every positive effort (concavity of $\Pi$ , the closed form (8)) are stated on $\rho > 0$ . Cube roots are Real.rpow with exponent $1/3$ on positive bases. The uniform law is Lebesgue measure conditioned on $[0, 2\beta/\rho]$ , and the expectation is the Bochner integral against it. Coordination is IsMaxOn of the retailer's profit on Set.Ici 1 at $\rho^C$ .

Three conventions differ from the printed text, each recorded in the item's formalization note:

The interesting case is printed as $(m + r)\beta > a$ ; the condition equivalent to the stated consequence $\rho^C > 1$ , which the proof uses, is $(M_m + R_r)\beta > a$ . The formalization uses the latter.
The printed evaluation $E_X\{[T - X(\rho)]^+\} = u\int_0^T (T - x)(\rho/2\beta)\,dx$ carries a stray factor $u$ ; the expectation is $T^2\rho/4\beta$ .
Proposition 1 is stated for an arbitrary family of probability laws on $[0,\infty)$ whose distribution functions are $C^2$ in $\rho$ with nonpositive second derivative for $x \in [0,T]$ ; the paper's further assumptions on $F$ (differentiable, strictly increasing in $x$ , mean $\beta/\rho$ ) are not imposed.

The side condition $T < 2\beta/\rho^C$ of §3.1 is a hypothesis of the goal and of the appendix milestones; without it a coordinating contract with $u = M_m$ exists and the "if" direction fails. The unused page assertion $\delta_r < \delta_m < 1$ is not imposed.

The goal is not trivialized by its coordination hypothesis: coordination is the retailer's optimization over the true profit, and milestone (10) shows that coordinating contracts with $T < 2\beta/\rho^C$ exist for every $u > M_m$ , so the hypotheses are satisfiable (Example 1 of the paper, p. 384, is an instance). The retailer half of the goal is comparatively short under this definition of coordination; that is a property of the paper's theorem, not of the encoding. The manufacturer half needs (8), (9) and (25).

The development needs only Mathlib: real calculus (derivatives, concavity, Real.rpow) and Lebesgue integration against a conditioned Lebesgue measure. The model definitions (effort-dependent profits, coordination as the retailer's optimization) are reusable for the paper's other effort-inducing contracts. Contributions welcome: proofs of any milestone, and reusable lemmas on expectations of $[T - X]^+$ under uniform laws.

Selected references

M. Ferguson, V. D. R. Guide Jr., G. C. Souza, Supply Chain Coordination for False Failure Returns, Manufacturing & Service Operations Management 8(4):376–393, 2006. https://doi.org/10.1287/msom.1060.0112
G. P. Cachon, Supply Chain Coordination with Contracts, in Handbooks in Operations Research and Management Science, Vol. 11: Supply Chain Management, Elsevier, 2003. https://doi.org/10.1016/S0927-0507(03)11006-7
T. A. Taylor, Supply Chain Coordination Under Channel Rebates with Sales Effort Effects, Management Science 48(8):992–1007, 2002. https://doi.org/10.1287/mnsc.48.8.992.168

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Combinatorics Graph Theory Operations Research+2·Captain: mikedeng1

Applied Combinatorics VII: Minimum Spanning Trees and Dijkstra's AlgorithmTextbook

Motivation

Two optimization problems on weighted networks sit at the base of operations research and algorithm design. The first asks for the cheapest way to connect every node of a network, such as a cable, pipeline or communication network. The answer is a minimum weight spanning tree. The second asks for the shortest route from a depot to every other node of a road or data network, the single-source shortest path problem. Chapter 12 of Keller and Trotter's Applied Combinatorics (appliedcombinatorics.org, CC BY-SA 4.0) treats both. It proves the structural lemmas behind the greedy spanning tree algorithms of Kruskal (1956) and Prim (1957), and the correctness of the shortest path algorithm of Dijkstra (1959).

The minimum spanning tree problem goes back to Borůvka (1926), who designed an electrical network for Moravia. Kruskal and Prim gave the two greedy algorithms taught today, and Dijkstra's 1959 note treated both problems. Dijkstra's shortest path method, with heap-based refinements such as Fredman and Tarjan (1987), remains the standard solver for non-negative lengths and a building block of routing, scheduling and network flow codes.

Setting

A graph $G = (V, E)$ has a finite vertex set $V$ and a set $E$ of 2-element subsets of $V$ . A weight $w(e) \in \mathbb N_0$ is attached to each edge, and a set $S$ of edges has weight $w(S) = \sum_{e \in S} w(e)$ . A spanning forest of $G$ is an acyclic graph $H = (V, S)$ with $S \subseteq E$ . A spanning tree is a spanning forest that is connected. The weight of a spanning tree is the weight of its edge set. In Lean these are SimpleGraph V with [Fintype V], IsSpanningForest G H ( $H \le G$ and acyclic), IsSpanningTree G T ( $T \le G$ and a tree), and weight w T for a weight w : Sym2 V → ℕ.

A digraph $G = (V, E)$ has $E \subseteq V \times V$ with $x \ne y$ for every directed edge $(x, y)$ . Each directed edge has a length $w(x, y) \in \mathbb N_0$ . The length is extended by $w(x, y) = \infty$ for non-edges. A directed path from $a$ to $b$ is a sequence $(a = u_0, \dots, u_t = b)$ of distinct vertices in which consecutive pairs are directed edges. Its length is $\sum_{i<t} w(u_i, u_{i+1})$ . The distance $\operatorname{dist}(a, b) \in \mathbb N_0 \cup \{\infty\}$ is the minimum length of a directed path from $a$ to $b$ , and is $\infty$ when no such path exists. A shortest path is a directed path attaining it. In Lean this is WeightedDigraph V with ext, IsDirPath, pathLength, dist and IsShortestPath.

Dijkstra's algorithm (Algorithm 12.14) with root $r$ and $n = |V|$ keeps a sequence $\sigma$ of permanent vertices, a value $\delta(x) \in \mathbb N_0 \cup \{\infty\}$ and a sequence $P(x)$ for each vertex. Step 1 sets $\delta(r) = 0$ , $P(r) = (r)$ , $\sigma = (r)$ , and $\delta(x) = w(r, x)$ , $P(x) = (r, x)$ for $x \ne r$ . Step $i$ with $1 < i < n$ scans from the last permanent vertex $v_i$ . For every temporary $x$ it sets $\delta(x) \leftarrow \min\{\delta(x), \delta(v_i) + w(v_i, x)\}$ , and on a strict decrease it replaces $P(x)$ by $P(v_i)$ followed by $x$ . Each step ends by appending to $\sigma$ a temporary vertex of minimum $\delta$ , chosen arbitrarily among ties. The algorithm halts at Step $n$ . DijkstraRun G r i s holds when some sequence of admissible choices leads to state s at the start of Step $i$ .

Formalization targets

Goal: correctness of Dijkstra's algorithm (Theorem 12.18)

For every halted state of every run, and every vertex $x$ ,

\delta(x) = \operatorname{dist}(r, x), \qquad \operatorname{dist}(r, x) < \infty \implies P(x) \text{ is a shortest path from } r \text{ to } x.

Milestones

Proposition 12.3. A spanning forest $H = (V, S)$ of a graph on $n \ge 1$ vertices has $|S| \le n - 1$ and exactly $n - |S|$ components. It is a spanning tree if and only if $|S| = n - 1$ .
Proposition 12.4 (Exchange Principle). Let $T$ be a spanning tree and $xy \in E \setminus T$ . Then $T$ contains a unique path $x = x_0, \dots, x_t = y$ , and replacing any edge $x_i x_{i+1}$ of it by $xy$ gives a spanning tree.
Lemma 12.6. In a connected weighted graph, let $F$ be a spanning forest and $C$ a component of $F$ . A minimum weight edge leaving $C$ lies in some spanning tree that has minimum weight among the spanning trees containing $F$ .
Proposition 12.16. Every prefix and every suffix of a shortest path is a shortest path.
Proposition 12.17. When the algorithm halts, $\delta(v_1) \le \delta(v_2) \le \cdots \le \delta(v_n)$ .

Milestones 4 and 5 are the two statements the book's proof of the goal rests on. Milestones 1–3 are the spanning tree half of the chapter. Lemma 12.6 is the result from which the book derives the correctness of Kruskal's and Prim's algorithms.

Significance

Theorem 12.18 certifies that one pass of $n$ steps computes all distances from $r$ and a shortest path tree, with no condition on the digraph beyond non-negative lengths. Lemma 12.6 is the cut property. Every greedy minimum spanning tree method (Kruskal, Prim, Borůvka) is an instance of it, and the exchange principle is the matroid basis-exchange axiom specialised to the graphic matroid.

All of these results are classical and proved. None is formalized in this form on the platform. Mathlib has spanning trees of connected graphs, uniqueness of paths in acyclic graphs, and the edge count $n - 1$ of a tree. It has no edge–component count for forests, no exchange principle, no weighted spanning trees, and no Dijkstra. On Prove2Me, FamousTheorems.tree_card_edges_6b and ClassicalGaps.isAcyclic_edges_eq_card_sub_one_imp_connected cover only the tree case of Proposition 12.3. KServer.mst_cut_property is a cut property for complete graphs encoded by parent maps, a different statement. The label-correcting algorithm of Dynamic Programming and Optimal Control II (BertsekasDP.label_correcting_*) is a different algorithm: it keeps an open list and scans in arbitrary order, not by minimum label.

Difficulty

The goal is a statement about the final state of a run, but the facts it depends on only become visible across steps: a permanent vertex's $\delta$ and $P$ never change again, and $\delta(x)$ is always the length of the current $P(x)$ . None of this is recorded in the final state itself. An argument over the steps of the run has to show that each $P(x)$ remains a path with distinct vertices, including when edges of length $0$ allow ties. It also has to handle the value $\infty$ , where $\infty + a = \infty$ and a comparison between two infinite values never counts as a decrease. Tie-breaking is arbitrary, so no argument may depend on which minimum is chosen. For Lemma 12.6 the difficulty is the exchange step: removing an edge of a tree path and adding a crossing edge must again give a tree that still contains the forest $F$ , and this is a statement about cycles and components, not about counts.

Formalization scope

Graphs are SimpleGraph V over a Fintype V. Weights are Sym2 V → ℕ (the book's $w : E \to \mathbb N_0$ ; values off $E$ are never used). Acyclic, tree and connected components are Mathlib's. In Proposition 12.3, $|S| = n - k$ is written $|S| + k = n$ and $n \ge 1$ is assumed, which the bound $n - 1$ presupposes.
Lemma 12.6 assumes $G$ connected, the section's standing assumption (p. 239). The page's "to avoid trivialities, we assume $n \ge 3$ " is not imposed, because the statement holds for every $n$ . The crossing edge may have either endpoint in $C$ .
Lengths in the digraph are ℕ, and $\delta$ and distances are ℕ∞, where $\infty$ is ⊤, never a large finite number. A version with real or ℝ≥0 lengths would be a generalization and is not what is asked.
Dijkstra's algorithm is defined step by step exactly as on pp. 246–247, including $\delta(x) = w(r, x) = \infty$ and $P(x) = (r, x)$ for non-neighbours at Step 1. The goal quantifies over every halted state, so it holds for every tie-breaking. A halted state always exists; a sorry-free check of this is in the workspace. For a vertex not reachable from $r$ the book is silent. The distance there is read as $\infty$ , and the shortest-path conclusion is asserted only at finite distance.
A trivializing formalization is ruled out: $\delta$ is computed by the update rule of Algorithm 12.14, not defined as the distance, and the theorem is not stated for an arbitrary procedure satisfying its own conclusion.
The book uses no $O(\cdot)$ bounds or approximate constants in these statements, so there are no constants to instantiate.
Reusable infrastructure: a list-based theory of directed paths and distances in ℕ∞, the invariants of Dijkstra's algorithm, and forest edge counting. Contributions of general lemmas (walks shortcut to paths without increasing length, component counts under edge insertion) are welcome.

Selected references

M. T. Keller and W. T. Trotter, Applied Combinatorics, 2017 Edition, Chapter 12. https://www.appliedcombinatorics.org/
E. W. Dijkstra, "A note on two problems in connexion with graphs", Numerische Mathematik 1 (1959) 269–271. https://doi.org/10.1007/BF01386390
J. B. Kruskal, "On the shortest spanning subtree of a graph and the traveling salesman problem", Proc. AMS 7 (1956) 48–50. https://doi.org/10.1090/S0002-9939-1956-0078686-7
R. C. Prim, "Shortest connection networks and some generalizations", Bell System Technical Journal 36 (1957) 1389–1401. https://doi.org/10.1002/j.1538-7305.1957.tb01515.x
M. L. Fredman and R. E. Tarjan, "Fibonacci heaps and their uses in improved network optimization algorithms", J. ACM 34 (1987) 596–615. https://doi.org/10.1145/28869.28874
O. Borůvka, "O jistém problému minimálním", Práce Moravské přírodovědecké společnosti 3 (1926) 37–58. https://dml.cz/handle/10338.dmlcz/500114

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Linear Optimization Operations Research Optimization·Captain: mikedeng1

Revenue Management Under the Markov Chain Choice Model III: The Reduced Linear Program Is Equivalent to the Choice-Based Linear ProgramResearch Paper

Motivation

Network revenue management decides which products to make available to arriving customers when products share scarce resources: an airline sells itineraries (products) that consume seats on flight legs (resources), and each itinerary has a fare. When customers choose among the offered products, rather than asking for one fixed product, the standard planning tool is a deterministic linear program that replaces random choices by their expected values. Gallego, Iyengar, Phillips and Dubey (2004, Columbia CORC technical report TR-2004-01) and Liu and van Ryzin (2008) formulated this choice-based linear program; its solution drives bid-price and offer-set policies used in practice.

The difficulty is size. The choice-based program has one variable for each subset of products, $2^n$ in all, and is solved by column generation, whose pricing subproblem is itself an assortment problem. Feldman and Topaloglu (Oper. Res. 65(5), 2017) show that when customers choose under the Markov chain choice model of Blanchet, Gallego and Goyal (2016), the choice-based program is equivalent to a linear program with only $2n$ variables and $m+n$ constraints. This mission formalizes that equivalence, Theorem 7 of the paper.

Setting

There are $n$ products, $N=\{1,\dots,n\}$ . Under the Markov chain choice model a customer first visits product $j$ with probability $\lambda_j$ . If the product she visits is offered, she buys it. Otherwise she moves from product $j$ to product $i$ with probability $\rho_{j,i}$ , or leaves without buying with probability $1-\sum_{i\in N}\rho_{j,i}$ . The paper assumes throughout that

\lambda_j>0\quad\text{and}\quad\sum_{i\in N}\rho_{j,i}<1\qquad\text{for all } j\in N.

For an offer set $S\subseteq N$ , $P_{j,S}$ is the expected number of visits to product $j$ while it is offered, which is its purchase probability, and $R_{j,S}$ is the expected number of visits to $j$ while it is not offered. The pair $(P_S,R_S)$ is the solution of the (Balance) equations

P_{j,S}+R_{j,S}=\lambda_j+\sum_{i\in N}\rho_{i,j}R_{i,S}\ \ \forall j\in N,\qquad P_{j,S}=0\ \ \forall j\notin S,\qquad R_{j,S}=0\ \ \forall j\in S .

Dropping the constraints tied to $S$ gives the polyhedron

\mathcal H=\Big\{(x,z)\in\mathbb R^{2n}_+ : x_j+z_j=\lambda_j+\sum_{i\in N}\rho_{i,j}z_i\ \ \forall j\in N\Big\}.

The network has $m$ resources, $M=\{1,\dots,m\}$ , with capacities $c_q$ ; the selling horizon has $T$ periods; product $j$ earns $r_j$ and consumes $a_{q,j}$ units of resource $q$ . With $u_S$ the probability of offering $S$ in a period, the (Choice Based) linear program is

\max_{u\in\mathbb R^{2^n}_+}\Big\{\sum_{S\subseteq N}\sum_{j\in N}T r_jP_{j,S}u_S:\ \sum_{S\subseteq N}\sum_{j\in N}Ta_{q,j}P_{j,S}u_S\le c_q\ \forall q\in M,\ \sum_{S\subseteq N}u_S=1\Big\},

and the (Reduced) linear program is

\max_{(x,z)\in\mathbb R^{2n}_+}\Big\{\sum_{j\in N}T r_jx_j:\ \sum_{j\in N}Ta_{q,j}x_j\le c_q\ \forall q\in M,\ x_j+z_j=\lambda_j+\sum_{i\in N}\rho_{i,j}z_i\ \forall j\in N\Big\}.

In (Reduced), $x_j$ is the expected number of visits to product $j$ while it is available and $z_j$ the expected number of visits while it is not.

Formalization targets

Goal: Theorem 7

Let $(\hat x,\hat z)$ be an optimal solution of (Reduced). Then there are subsets $S^1,\dots,S^K\subseteq N$ and positive scalars $\gamma^1,\dots,\gamma^K$ with $\sum_k\gamma^k=1$ such that

\hat x=\sum_{k=1}^K\gamma^kP_{S^k},\qquad \hat z=\sum_{k=1}^K\gamma^kR_{S^k},

and for any such subsets and scalars the vector $\hat u$ with $\hat u_{S^k}=\gamma^k$ and $\hat u_S=0$ for $S\notin\{S^1,\dots,S^K\}$ is optimal for (Choice Based), with objective value equal to that of $(\hat x,\hat z)$ in (Reduced). In particular the two programs have the same optimal value.

Milestones

(Balance) has a unique and nonnegative solution for every offer set (§2, p. 1325).
Lemma 1: an extreme point $(\hat x,\hat z)$ of $\mathcal H$ equals $(P_{S},R_{S})$ for $S=\{j:\hat x_j>0\}$ (p. 1326).
Lemma 10: $\mathcal H$ is bounded (quoted on p. 1331; proved in the online appendix).
Every point of $\mathcal H$ is a positive convex combination of finitely many extreme points of $\mathcal H$ (proof of Theorem 7, p. 1331).
Every feasible $u$ of (Choice Based) yields the feasible point $\tilde x_j=\sum_S P_{j,S}u_S$ , $\tilde z_j=\sum_S R_{j,S}u_S$ of (Reduced), with the same objective value (proof of Theorem 7, p. 1332).

Significance

The result. Theorem 7 replaces a program with $2^n$ columns by one with $2n$ variables and $m+n$ constraints, solvable directly by any LP solver, and it returns an optimal solution of the original program, not only its value. The optimal value is the standard upper bound on the optimal expected revenue of a network revenue management policy, and the dual variables of the capacity constraints are the bid prices used to control sales. The decomposition of part 1 is what turns the small program's solution back into offer-set frequencies that a policy can implement; Section 7 of the paper makes that decomposition algorithmic (a separate mission of this series).

Formalizing it. The theorem is proved in the paper; to the best of available knowledge no machine-checked version exists. A formal proof checks the link between polyhedral geometry (extreme points of $\mathcal H$ and the solutions of (Balance)) and linear-programming optimality, and records exactly which properties of the Markov chain choice model are used: the standing assumptions enter through uniqueness and nonnegativity of $(P_S,R_S)$ and through boundedness of $\mathcal H$ .

Difficulty

The inequality "(Reduced) $\ge$ (Choice Based)" is a direct computation: averaging the (Balance) equations with weights $u_S$ lands in $\mathcal H$ . The reverse direction is where the obvious argument fails. A point of $\mathcal H$ has no offer set attached to it, and a general polyhedron need not be the convex hull of its extreme points: it can contain lines or rays. The argument requires that $\mathcal H$ is bounded, which depends on the substochasticity $\sum_i\rho_{j,i}<1$ , and that each extreme point is exactly some $(P_S,R_S)$ , which uses the structure of the balance equations and the uniqueness of their solution. Neither follows from general linear-programming facts. In Lean, the finite vertex representation of a bounded polyhedron is also not a one-line consequence of Mathlib's Krein–Milman theorem, which gives only the closure of the convex hull.

Formalization scope

Products are Fin n, offer sets Finset (Fin n), resources Fin m. The model is a structure Model n holding $\lambda$ , $\rho$ (rho j i $=\rho_{j,i}$ , the transition from $j$ to $i$ ) and the standing assumptions $\lambda_j>0$ and $\sum_i\rho_{j,i}<1$ ; the nonnegativity $\rho_{j,i}\ge0$ , implicit in the paper because the $\rho_{j,i}$ are probabilities, is an added field. $(P_S,R_S)$ is a solution of (Balance) chosen by Classical.epsilon, and milestone 1 is what identifies it with the paper's unique solution. $\mathcal H$ is a subset of (Fin n → ℝ) × (Fin n → ℝ), extreme points are Mathlib's Set.extremePoints ℝ, and boundedness is Bornology.IsBounded. $T$ is a natural number entering as a real factor exactly where the paper writes it; $c$ , $a$ , $r$ carry no sign conditions, as in the paper. "Optimal solution" means feasible and at least as good as every feasible point; no supremum is used.

Two conventions are disclosed. The paper defines $\hat u$ by $\hat u_{S^k}=\gamma^k$ ; the formalization sets $\hat u_S=\sum_{k:S^k=S}\gamma^k$ , which agrees when the $S^k$ are distinct and is the only consistent reading otherwise. Milestone 5 is stated for every feasible $u$ of (Choice Based), while the paper applies it to an optimal one; its argument uses only feasibility. No printed statement needed correction.

The goal is not only the decomposition of part 1, which is milestones 2 and 4 combined: it also asserts optimality of $\hat u$ and equality of the optimal values, and a formalization that drops part 2 does not state Theorem 7. Part 2 is required for every decomposition, and part 1 guarantees one exists, so part 2 is not vacuous.

A complete development needs the vertex representation of polytopes (reusable beyond this mission; LinearOptimization.polyhedron_resolution on the platform proves the resolution theorem in another encoding), the theory of substochastic matrices behind (Balance) (invertibility of $I-\bar Q$ with a nonnegative inverse), and finite-sum manipulations over Finset (Fin n). Contributions to any milestone, and bridges to existing polyhedral results, are welcome.

Selected references

J. B. Feldman, H. Topaloglu, Revenue Management Under the Markov Chain Choice Model, Operations Research 65(5):1322–1342, 2017. https://doi.org/10.1287/opre.2017.1628
J. Blanchet, G. Gallego, V. Goyal, A Markov Chain Approximation to Choice Modeling, Operations Research 64(4):886–905, 2016. https://doi.org/10.1287/opre.2016.1505
Q. Liu, G. van Ryzin, On the Choice-Based Linear Programming Model for Network Revenue Management, Manufacturing & Service Operations Management 10(2):288–310, 2008. https://doi.org/10.1287/msom.1070.0172
G. Gallego, G. Iyengar, R. Phillips, A. Dubey, Managing Flexible Products on a Network, Computational Optimization Research Center Technical Report TR-2004-01, Columbia University, 2004 (technical report; no DOI).
M. L. Puterman, Markov Decision Processes: Discrete Stochastic Dynamic Programming, Wiley, 1994. https://doi.org/10.1002/9780470316887

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Dynamic Programming Operations Research Optimization·Captain: mikedeng1

Revenue Management Under the Markov Chain Choice Model II: Optimal Offer Sets Grow with Remaining Capacity and Remaining TimeResearch Paper

Motivation

Airlines, hotels and rental firms sell a fixed stock of capacity (seats on a flight leg, rooms on a night) over a finite selling horizon, and control revenue mainly by deciding which products (fare classes, rate plans) to make available at each moment. When customers substitute between products, the decision in each period is an assortment: a subset of products to offer. Talluri and van Ryzin (Management Science 2004) formulated this single resource revenue management problem as a dynamic program for a general choice model, and showed that structural properties of the optimal policy depend strongly on how customers choose.

The Markov chain choice model of Blanchet, Gallego and Goyal (technical report 2013; Oper. Res. 2016) describes substitution by a Markov chain on the products and approximates a broad class of random-utility models. Feldman and Topaloglu (Oper. Res. 65(5), 2017) study assortment and revenue management problems under this model. This mission formalizes their structural result for the single resource problem (Theorem 5): there is an optimal policy whose offer sets are nested in the remaining capacity and in the remaining time, so that it can be implemented by protection levels, one capacity threshold per product and period.

Setting

There are $n$ products $N=\{1,\dots,n\}$ . A customer arrives to purchase product $j$ with probability $\lambda_j$ . If $j$ is offered she buys it; otherwise she moves to product $i$ with probability $\rho_{j,i}$ , and leaves with probability $1-\sum_{i\in N}\rho_{j,i}$ . The paper assumes $\lambda_j>0$ and $\sum_{i\in N}\rho_{j,i}<1$ for all $j$ . For an offer set $S\subseteq N$ , the purchase probabilities $P_{j,S}$ and the visit counts $R_{j,S}$ of unavailable products solve the (Balance) equations

P_{j,S}+R_{j,S}=\lambda_j+\sum_{i\in N}\rho_{i,j}R_{i,S}\ \ \forall j,\qquad P_{j,S}=0\ \ (j\notin S),\qquad R_{j,S}=0\ \ (j\in S).

With product revenues $r_j$ , the (Assortment) problem is $\max_{S\subseteq N}\sum_{j}P_{j,S}r_j$ , and its (Dual) linear program is

\min_{v\in\mathbb R^n}\Big\{\sum_{j}\lambda_jv_j : v_j\ge r_j,\ v_j\ge\sum_{i}\rho_{j,i}v_i\ \ \forall j\Big\}.

In the single resource problem there are $T$ periods and $c$ units of capacity. In each period at most one customer arrives; a sale of product $j$ earns $r_j$ and uses one unit. The optimal expected revenue $V_t(x)$ from period $t$ on with $x$ units left satisfies the (Single Resource) dynamic program

V_t(x)=\max_{S\subseteq N}\Big\{\sum_{j\in N}P_{j,S}\{r_j+V_{t+1}(x-1)-V_{t+1}(x)\}\Big\}+V_{t+1}(x),

with $V_{T+1}(x)=0$ and $V_t(0)=0$ . An optimal subset $\hat S_t(x)$ is a maximizer of the problem on the right side. The marginal value of capacity is $\Delta V_t(x)=V_t(x)-V_t(x-1)$ .

Formalization targets

Goal: Theorem 5 (p. 1329)

There exists an optimal policy, i.e. a choice of maximizers $\hat S_t(x)$ for all $1\le t\le T$ , $1\le x\le c$ , such that

\hat S_t(x-1)\subseteq \hat S_t(x)\qquad\text{and}\qquad \hat S_{t-1}(x)\subseteq\hat S_t(x).

The offer set shrinks as capacity runs down, and it is smaller when more periods remain.

Milestones

(§2, p. 1325) The (Balance) equations have a unique nonnegative solution for every $S$ .
(Theorem 2, p. 1326) If $\hat v$ is optimal for (Dual), then $\{j:\hat v_j=r_j\}$ is optimal for (Assortment).
(Lemma 3, p. 1327) For $\eta\ge0$ , with $\hat v^\eta$ optimal for (Dual) with revenues $r_j-\eta$ ,

\{j:\hat v^\eta_j=r_j-\eta\}\subseteq\{j:\hat v^0_j=r_j\}.

((Single Resource), pp. 1328–1329) The value functions satisfy the printed recursion $V_t(x)=\max_S\{\sum_jP_{j,S}\{r_j+V_{t+1}(x-1)\}+\{1-\sum_jP_{j,S}\}V_{t+1}(x)\}$ and the boundary conditions.
(Proof of Theorem 5, p. 1329) $\Delta V_{t+1}(x)\le\Delta V_{t+1}(x-1)$ and $\Delta V_{t+1}(x)\le\Delta V_t(x)$ .

Significance

Theorem 5 makes the optimal policy a protection level policy: for each product $j$ and period $t$ there is a threshold $\bar x_{jt}$ such that $j$ is offered exactly when at least $\bar x_{jt}$ units remain. The policy can then be stored as $n\,T$ numbers instead of a table of subsets, and each product can be controlled separately, which is how airline inventory systems are organized. The paper also shows (Table 2, p. 1330) that the products need not be closed in revenue order: a higher-fare product can be closed before a lower-fare one, so the result is genuinely about nested sets, not about nested fare classes. Talluri and van Ryzin (2004) show that under the multinomial logit model an optimal assortment consists of a number of products with the largest revenues; the Markov chain choice model does not have this property (Table 1, p. 1328), so their structure of the optimal policy does not carry over directly.

The result is proved in the paper. No part of it is formalized on Prove2Me. The monotonicity of marginal values for an abstract choice model is published and proved as RevenueManagement.choice_marginal_values, stated over the definitions RevenueManagement_singleResource; milestone 5 is the same statement for this paper's dynamic program and can be bridged to it. A complete development here produces machine-checked Theorem 2 and Lemma 3 for the Markov chain choice model, which are reusable for any assortment problem under this model.

Difficulty

The obvious argument chooses, for each $(t,x)$ , any maximizer of the stage problem. This fails: the stage problems have ties, and an arbitrary choice of maximizers need not be nested, so the theorem is an existence statement about a coordinated choice. The dependence of $P_{j,S}$ on $S$ is through the solution of a linear system, so the effect of adding or removing a product on the other purchase probabilities has no simple sign, and optimal assortments need not be nested by revenue (Table 1, p. 1328). The link between two stage problems is that their revenues differ by the same constant for all products, and what has to be shown is that such a uniform shift moves an optimal assortment in a controlled direction. Revenues in the stage problems, $r_j-\Delta V_{t+1}(x)$ , can be negative.

Formalization scope

Products are Fin n, offer sets Finset (Fin n), the paper's $\subset$ is non-strict inclusion $\subseteq$ . The model is a structure with fields $\lambda$ , $\rho$ and the standing assumptions $\lambda_j>0$ , $\sum_i\rho_{j,i}<1$ ; $\rho_{j,i}\ge0$ is added as a field because the $\rho_{j,i}$ are probabilities. $(P_S,R_S)$ is a solution of (Balance) chosen by Classical.epsilon; milestone 1 states that it is the unique nonnegative one. The value function is defined by recursion on the number of periods to go, and $V_t$ is used only for $1\le t\le T+1$ . Maxima over $S\subseteq N$ are taken over the finite family of all subsets, so they are attained; dual optimality is "feasible and no worse than every feasible point".

One hypothesis is added to Theorem 5 and milestone 5, and disclosed: $\sum_j\lambda_j\le1$ , implied by the paper's description of at most one arrival per period (it makes $1-\sum_jP_{j,S}$ a probability). No sign condition is placed on the revenues, as on the page. Theorem 2 and Lemma 3 are stated for arbitrary real revenues, as the proof of Theorem 5 applies them to $r_j-\Delta V_{t+1}(x)$ . The first inclusion of Theorem 5 is stated for $x\ge2$ : with $x-1=0$ units there is no decision, and the paper reads $\hat S_t(0)$ as $\emptyset$ . The existential requires every $\hat S_t(x)$ in range to be optimal for its stage problem; a statement without that clause would be satisfied by the empty sets and is ruled out.

Theorem 2 and Lemma 3 need LP duality for (Dual) and the (Balance) system; milestone 5 needs the standard induction on the dynamic program, or a bridge to the published choice_marginal_values (with arrival probabilities $1$ and choice model $P_{j,S}$ ). Proofs of any milestone, and bridges to published Mathlib or platform LP duality results, are welcome.

Selected references

J. B. Feldman, H. Topaloglu, Revenue Management Under the Markov Chain Choice Model, Operations Research 65(5):1322–1342, 2017. https://doi.org/10.1287/opre.2017.1628
J. Blanchet, G. Gallego, V. Goyal, A Markov Chain Approximation to Choice Modeling, Operations Research 64(4):886–905, 2016. https://doi.org/10.1287/opre.2016.1505
K. Talluri, G. van Ryzin, Revenue Management Under a General Discrete Choice Model of Consumer Behavior, Management Science 50(1):15–33, 2004. https://doi.org/10.1287/mnsc.1030.0147
K. Talluri, G. van Ryzin, The Theory and Practice of Revenue Management, Springer, 2004. https://doi.org/10.1007/b139000

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Linear Optimization Operations Research Optimization·Captain: mikedeng1

Revenue Management Under the Markov Chain Choice Model I: The Dual Linear Program Yields an Optimal AssortmentResearch Paper

Motivation

A retailer or an airline decides which products to make available, and customers choose among what is offered. When a preferred product is missing, many customers substitute to another product instead of leaving. Assortment optimization asks which subset of products to offer so that the expected revenue from a customer is as large as possible, and its answer depends entirely on the choice model used to describe substitution.

The Markov chain choice model was introduced by Blanchet, Gallego and Goyal (EC 2013; Oper. Res. 64(4), 2016), who showed that it contains the multinomial logit model as a special case and proposed it as an approximation of general random-utility choice models. Feldman and Topaloglu (Oper. Res. 65(5), 2017) study revenue management under this model. Their first result, the subject of this mission, is that the assortment problem, a search over all $2^n$ offer sets, is solved by one linear program with $n$ variables. The same paper uses this result for its dynamic single-resource and network results, which are the subjects of the companion missions II–IV of this series.

Timeline:

2013/2016: Blanchet, Gallego and Goyal introduce the model and give a polynomial-time assortment algorithm.
2017: Feldman and Topaloglu show that the optimal assortment is read off an optimal solution of a dual linear program (their Theorem 2), and derive structural and capacity-control consequences.

Setting

There are $n$ products $N=\{1,\dots,n\}$ . A customer arrives to purchase product $j$ with probability $\lambda_j$ . If the product she visits is offered, she buys it. Otherwise she transitions to product $i$ with probability $\rho_{j,i}$ and checks whether $i$ is offered, or leaves without buying with probability $1-\sum_{i\in N}\rho_{j,i}$ . Throughout, $\lambda_j>0$ , $\rho_{j,i}\ge0$ and $\sum_{i\in N}\rho_{j,i}<1$ for all $j$ .

For an offer set $S\subseteq N$ , let $P_{j,S}$ be the expected number of visits to product $j$ while it is offered (the probability that $j$ is purchased) and $R_{j,S}$ the expected number of visits to $j$ while it is not offered. The pair $(P_S,R_S)$ is the unique solution of the (Balance) equations

P_{j,S}+R_{j,S}=\lambda_j+\sum_{i\in N}\rho_{i,j}R_{i,S}\ \ \forall j\in N,\qquad P_{j,S}=0\ \ \forall j\notin S,\qquad R_{j,S}=0\ \ \forall j\in S .

With revenue $r_j\in\mathbb R$ for product $j$ , the (Assortment) problem is $\max_{S\subseteq N}\sum_{j\in N}P_{j,S}r_j$ . Two linear programs enter: the maximization of $\sum_j r_jx_j$ over the polyhedron

\mathcal H=\Big\{(x,z)\in\mathbb R^{2n}_+:\ x_j+z_j=\lambda_j+\sum_{i\in N}\rho_{i,j}z_i\ \ \forall j\in N\Big\},

and its dual

\min_{v\in\mathbb R^n}\Big\{\sum_{j\in N}\lambda_jv_j:\ v_j\ge r_j\ \ \forall j\in N,\ \ v_j\ge\sum_{i\in N}\rho_{j,i}v_i\ \ \forall j\in N\Big\}.\qquad\text{(Dual)}

Formalization targets

Goal: Theorem 2 (p. 1326)

For every optimal solution $\hat v$ of (Dual), the set $\hat S=\{j\in N:\hat v_j=r_j\}$ is an optimal assortment:

\sum_{j\in N}P_{j,S}\,r_j\ \le\ \sum_{j\in N}P_{j,\hat S}\,r_j\qquad\text{for all } S\subseteq N .

Milestones, in the order the paper uses them

(p. 1325) The (Balance) equations have a unique nonnegative solution for every $S$ .
Lemma 1 (p. 1326): for an extreme point $(\hat x,\hat z)$ of $\mathcal H$ and $S_{\hat x}=\{j:\hat x_j>0\}$ , $P_{j,S_{\hat x}}=\hat x_j$ and $R_{j,S_{\hat x}}=\hat z_j$ for all $j$ .
(p. 1326) The linear program over $\mathcal H$ has an optimal solution, and its optimal value equals the optimal value of (Assortment).
(p. 1326) (Dual) has an optimal solution, and its optimal value equals that of the linear program over $\mathcal H$ .
(p. 1326, proof of Theorem 2) An optimal $\hat v$ satisfies $\hat v_j=r_j$ or $\hat v_j=\sum_i\rho_{j,i}\hat v_i$ for each $j$ .

Significance

Theorem 2 reduces a combinatorial problem over $2^n$ offer sets to a linear program, so the assortment problem under the Markov chain choice model is solvable in polynomial time. The same structure drives the rest of the paper: the dual variables $\hat v_j$ are used to show that optimal offer sets shrink when all revenues fall by a common amount, to show that the optimal offer sets of the single-resource dynamic program are nested in the remaining capacity and time, and to reduce the choice-based network linear program to a compact one.

The result is proved in the paper. This mission produces a machine-checked version of it and of its supporting lemmas; to the knowledge of the mission author no formalization of the Markov chain choice model exists. The definition layer (the model, the (Balance) solution, $\mathcal H$ and (Dual)) is the first formal encoding of this choice model and is shared, with the same encoding, by missions II–IV.

Difficulty

The obvious route, comparing $\sum_jP_{j,S}r_j$ across offer sets directly, fails because $P_{j,S}$ is defined only implicitly through a linear system whose coefficient matrix changes with $S$ ; there is no closed-form expression that can be compared across offer sets, and enumerating the $2^n$ sets is exponential. The connection to linear programming needs an exact correspondence between the vertices of $\mathcal H$ and the (Balance) solutions, which is a statement about polyhedra, not about Markov chains. Even well-posedness is not free: existence, uniqueness and nonnegativity of $(P_S,R_S)$ depend on the row sums $\sum_i\rho_{j,i}$ being strictly below one. Mathlib has extreme points of convex sets but no ready-made theory of vertices of polyhedra or of linear programming duality in this form.

Formalization scope

Products are Fin n (0-based), offer sets are Finset (Fin n) (the paper's $S\subset N$ is non-strict inclusion, so every subset including $\emptyset$ and $N$ is an offer set), and rho j i is $\rho_{j,i}$ , the transition from $j$ to $i$ . The model structure carries the paper's standing assumptions $\lambda_j>0$ and $\sum_i\rho_{j,i}<1$ (p. 1325) and the implicit nonnegativity $\rho_{j,i}\ge0$ . Revenues are arbitrary reals with no sign assumption. $\mathbb R^{2n}$ is (Fin n → ℝ) × (Fin n → ℝ), and extreme points are Mathlib's Set.extremePoints ℝ.

The pair $(P_S,R_S)$ is a solution of (Balance) chosen by Classical.epsilon; milestone 1 states that it satisfies (Balance), is nonnegative and equals every solution, so all statements are about the paper's $(P_S,R_S)$ and not about a junk value. Every optimum (of (Assortment), of the linear program over $\mathcal H$ , of (Dual)) is stated as "feasible and at least as good as every feasible point", never through sSup or sInf. The goal is stated for every optimal $\hat v$ of (Dual), which is what the proof uses; uniqueness of $\hat v$ is neither assumed nor claimed. Replacing the hypothesis " $\hat v$ optimal for (Dual)" by " $\hat v$ feasible for (Dual)" would make the statement false, and an existential over $\hat v$ would weaken it; neither is the target.

No hypothesis beyond the paper's is added. Two printed index slips on pp. 1325–1326 (a garbled sum in the discussion after (Balance), and $\sum_i\rho_{i,j}\hat z_j$ for $\sum_i\rho_{i,j}\hat z_i$ in the proof of Lemma 1) are not part of any statement here.

Welcome contributions: the existence and uniqueness of (Balance) solutions via Neumann series for substochastic matrices, a characterization of vertices of polyhedra given by equality constraints and nonnegativity, and a strong duality statement usable for this primal–dual pair. These are reusable well beyond this mission.

Selected references

J. B. Feldman, H. Topaloglu, Revenue Management Under the Markov Chain Choice Model, Operations Research 65(5):1322–1342, 2017. https://doi.org/10.1287/opre.2017.1628
J. Blanchet, G. Gallego, V. Goyal, A Markov Chain Approximation to Choice Modeling, Operations Research 64(4):886–905, 2016. https://doi.org/10.1287/opre.2016.1505
M. L. Puterman, Markov Decision Processes: Discrete Stochastic Dynamic Programming, Wiley, 1994. https://doi.org/10.1002/9780470316887

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Combinatorics·Captain: mikedeng1

Applied Combinatorics III: Inclusion–Exclusion and the Number of SurjectionsTextbook

Motivation

Counting the objects that avoid every one of a list of conditions is one of the most frequent tasks in enumerative combinatorics: functions that miss no value, permutations that fix no point, integers that share no prime factor with a given number. The Principle of Inclusion-Exclusion converts such a count into an alternating sum of counts of objects that do satisfy prescribed conditions, which are usually far easier to compute. Chapter 7 of Keller and Trotter's Applied Combinatorics (2017 Edition) develops the principle in the abstract language of properties and applies it to three classical questions: the number of surjections, the number of derangements, and Euler's totient function.

This mission formalizes the first two applications, with the chapter's headline result, the closed formula for the number of surjections from an $n$ -element set onto an $m$ -element set, as its goal. The surjection count answers distribution questions of the form "in how many ways can $n$ distinct objects be handed to $m$ distinct recipients so that every recipient receives at least one", which the book illustrates with fifteen lottery tickets shared among four grandchildren ( $S(15,4) = 1016542800$ ).

Setting

Let $X$ be a finite set and let $\mathcal P = \{P_1, \dots, P_m\}$ be a family of properties: for each $x \in X$ and each $i$ , either $x$ satisfies $P_i$ or it does not. For a subset $S \subseteq [m] = \{1, \dots, m\}$ write

N(S) = \bigl|\{x \in X : x \text{ satisfies } P_i \text{ for every } i \in S\}\bigr|,

so that $N(\emptyset) = |X|$ .

Two families of properties are used.

Functions. $X$ is the set of all functions $f : [n] \to [m]$ , and $f$ satisfies $P_i$ when $i$ is not in the range of $f$ , i.e. $f(j) \ne i$ for every $j \in [n]$ . The functions satisfying none of the properties are the surjections, and their number is written $S(n, m)$ .
Permutations. $X$ is the set of all permutations $\sigma$ of $[n]$ , and $\sigma$ satisfies $P_i$ when $\sigma(i) = i$ . The permutations satisfying none of the properties are the derangements, and their number is written $d_n$ .

In Lean, $X$ is a type with [Fintype X], the properties are a decidable predicate family P : Fin m → X → Prop, and $N(S)$ is AppliedComb.InclExcl.N P S. The two families are AppliedComb.InclExcl.NotInRange n m on Fin n → Fin m and AppliedComb.InclExcl.FixesPoint n on Equiv.Perm (Fin n).

Formalization targets

Goal: Theorem 7.9 (number of surjections)

For all $n, m \ge 0$ ,

S(n, m) = \bigl|\{f : [n] \to [m] \text{ surjective}\}\bigr| = \sum_{k=0}^{m} (-1)^k \binom{m}{k} (m-k)^n .

Milestones

Theorem 7.7 (Principle of Inclusion-Exclusion). For every finite $X$ and every family of $m$ properties,

\bigl|\{x \in X : x \text{ satisfies no } P_i\}\bigr| = \sum_{S \subseteq [m]} (-1)^{|S|} N(S).

Lemma 7.8. For the function properties, $N(S)$ depends only on $|S|$ , and $N(S) = (m-k)^n$ when $|S| = k$ .
Lemma 7.10. For the permutation properties, $N(S)$ depends only on $|S|$ , and $N(S) = (n-k)!$ when $|S| = k$ .
Theorem 7.11 (number of derangements). For all $n \ge 0$ ,

d_n = \sum_{k=0}^{n} (-1)^k \binom{n}{k} (n-k)! .

Milestones 1 and 2 are the two ingredients the book names for the goal; milestones 3 and 4 are the parallel derangement application, which shares the definition of $N(S)$ and milestone 1.

Significance

The result itself. The surjection formula is the standard closed form for $m!\,S_2(n,m)$ , where $S_2$ is the Stirling number of the second kind; it gives $S(n, m) = 0$ for $n < m$ and $S(n, n) = n!$ as special cases, and it underlies counts of ordered set partitions, of onto colourings, and of occupancy problems in which every cell is filled. The derangement formula is the standard answer to the hat-check problem and leads directly to the limit $d_n/n! \to 1/e$ (Theorem 7.12 of the book). The Principle of Inclusion-Exclusion in the book's "none of the properties" form is the common tool behind both and behind Möbius inversion in general.

Formalizing it. All results are classical and fully proved in the book. Mathlib contains inclusion-exclusion for unions and intersections of finsets (Finset.inclusion_exclusion_card_biUnion, Finset.inclusion_exclusion_card_inf_compl), the derangement count via its own recurrence (numDerangements, card_derangements_eq_numDerangements, numDerangements_sum), and Stirling numbers of the second kind (Nat.stirlingSecond) with their recurrence, but no closed formula for the number of surjections. The platform has union-form inclusion-exclusion (FamousTheorems.inclusion_exclusion_card_biUnion) and the $1/e$ limit, neither in the property form used here. This mission produces the property-indexed statement of the principle, the two $N(S)$ computations, and the surjection count as a cardinality of surjective functions, in a form that downstream missions of the series (generating functions, Pólya counting) can cite.

Difficulty

The proofs are short on paper, and the work lies in the bookkeeping that the page leaves implicit. Two points stand out. First, the Principle of Inclusion-Exclusion is phrased with properties and $N(S)$ rather than with a union of sets; connecting the two forms, or proving the property form directly, requires handling sums over the power set of $[m]$ together with the sign $(-1)^{|S|}$ in $\mathbb Z$ . Second, the step "the result follows immediately, as there are $\binom{m}{k}$ $k$ -element subsets of $[m]$ " requires regrouping a sum over all subsets $S \subseteq [m]$ by cardinality, which needs Lemma 7.8's statement that $N(S)$ depends only on $|S|$ and a count of the subsets of each size. The book's one-line arguments for Lemmas 7.8 and 7.10 ("a string of length $n$ from an alphabet of $m-k$ letters", "a permutation among the remaining $n-k$ positions") are counting identifications that have to be made precise for subtypes of Fin n → Fin m and of Equiv.Perm (Fin n).

Formalization scope

The book's $[n] = \{1, \dots, n\}$ is represented by Fin n $= \{0, \dots, n-1\}$ ; this is an index shift only.
$S(n, m)$ is Fintype.card {f : Fin n → Fin m // Function.Surjective f} and $d_n$ is Fintype.card {σ : Equiv.Perm (Fin n) // ∀ i, σ i ≠ i}. Defining $S(n, m)$ by the formula, or $d_n$ by the sum, would make the goal hold by rfl; the counts are therefore cardinalities of the sets the book describes. $S(n,m)$ is also not stated through Nat.stirlingSecond, which counts unordered partitions and differs by a factor $m!$ .
All alternating sums are identities in $\mathbb Z$ ; the natural-number counts are cast. The differences $m - k$ and $n - k$ are computed in $\mathbb N$ where $k$ ranges over $0, \dots, m$ (resp. $0, \dots, n$ ) or equals $|S|$ , so they are exact.
The book introduces $S(n,m)$ for positive $n, m$ and states Theorem 7.11 for positive $n$ . The formal statements are made for all $n, m \ge 0$ , where they remain true with Lean's convention $0^0 = 1$ ( $S(0,0) = 1$ , $S(0,m) = 0$ for $m \ge 1$ , $d_0 = 1$ ). No explicit constants arise: the chapter contains no $O(\cdot)$ or approximate statements among the formalized results.
The Principle of Inclusion-Exclusion is stated for an arbitrary finite type and an arbitrary decidable family of $m \ge 0$ properties; for $m = 0$ both sides equal $|X|$ .

A complete development needs only Mathlib's finite-set, power-set and binomial-coefficient API. The property-form inclusion-exclusion (Theorem 7.7) and the two $N(S)$ lemmas are reusable for other sieve-type counts, for example Euler's totient function (Theorem 7.14 of the book), which is not part of this mission. Proofs of any milestone independently of the others are welcome, as are alternative proofs of the goal (for instance via exponential generating functions) that do not pass through Theorem 7.7.

Selected references

Mitchel T. Keller and William T. Trotter, Applied Combinatorics, 2017 Edition, Chapter 7 (Theorems 7.7, 7.9, 7.11; Lemmas 7.8, 7.10), CC BY-SA 4.0. https://www.appliedcombinatorics.org/
Richard P. Stanley, Enumerative Combinatorics, Volume 1, 2nd ed., Cambridge University Press, 2011, Chapter 2 (sieve methods). https://doi.org/10.1017/CBO9781139058520
The Mathlib Community, The Lean Mathematical Library, CPP 2020. https://doi.org/10.1145/3372885.3373824

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Combinatorics Graph Theory Operations Research+2·Captain: mikedeng1

An Analysis of Several Heuristics for the Traveling Salesman Problem IV: Insertion Heuristics Can Return Poor k-Optimal ToursResearch Paper

Motivation

Insertion heuristics build a traveling salesman tour one city at a time: start from a single city, and at each step choose a city not yet on the subtour and splice it into the subtour where it lengthens the subtour least. Local search heuristics start from a tour and repeatedly replace a few of its edges by others while this shortens the tour. Both families are standard in practice, and a natural engineering idea is to combine them: run an insertion heuristic, then polish the result by local search. The question this mission formalizes is whether local optimality of the insertion tour certifies anything about its quality.

Rosenkrantz, Stearns and Lewis (SIAM J. Comput. 6(3), 1977) answered this for graphs satisfying the triangle inequality. Their §4 proves that nearest and cheapest insertion always return a tour of length at most $2(1-1/n)$ times the optimal length, and their Theorem 5 shows this bound is attained. Their §7 then shows that the very tour attaining the bound is $k$ -optimal for every $k\le n/4$ : no exchange of $k$ edges shortens it. So the insertion bound is tight even for tours that local search with $k$ -changes cannot improve.

Timeline, as far as this mission is concerned:

1965: Lin (Bell System Tech. J. 44) defines $k$ -optimal tours and uses 3-optimal local search.
1973: Lin and Kernighan (Oper. Res. 21) generalize the edge-exchange neighbourhoods.
1977: Rosenkrantz, Stearns and Lewis prove the $2(1-1/n)$ upper bound for nearest and cheapest insertion (Theorem 4 and its corollary), its tightness for $n\ge 6$ (Theorem 5), the existence of $k$ -optimal tours with the same ratio (Theorem 6, stated for $n\ge 8$ ), and the Corollary combining the two.

Setting

A traveling salesman graph on $n$ nodes is the node set $N=\{1,\dots,n\}$ with a distance $d(i,j)\ge 0$ that is symmetric and satisfies the triangle inequality $d(i,k)\le d(i,j)+d(j,k)$ . A tour is a Hamiltonian circuit; its length is the sum of its edge lengths; OPTIMAL is the least length of a tour. As in the paper, the identically zero distance is excluded, so OPTIMAL $>0$ .

A subtour is a circuit on a subset of the nodes (a single node is a subtour without edges). For a subtour $T$ and a node $k\notin T$ , TOUR $(T,k)$ is obtained by deleting an edge $(x,y)$ of $T$ minimizing $d(x,k)+d(k,y)-d(x,y)$ and adding $(x,k)$ and $(k,y)$ ; COST $(T,k)$ is the resulting increase in length. An insertion method chooses nodes $a_0,a_1,\dots,a_{n-1}$ , starts from $T_1=\{a_0\}$ and sets $T_{i+1}=\mathrm{TOUR}(T_i,a_i)$ ; INSERT is the length of $T_n$ . Nearest insertion chooses $a_i$ minimizing $d(T_i,x)=\min_{y\in T_i}d(y,x)$ over $x\notin T_i$ ; cheapest insertion chooses $a_i$ minimizing COST $(T_i,x)$ . Ties are broken arbitrarily.

A $k$ -change of a tour deletes $k$ of its edges and adds $k$ other edges so that another tour is obtained. A tour is $k$ -optimal if no $k$ -change produces a strictly shorter tour.

The extremal instance is the circle $(N_n,d_n)$ : $n$ cities equally spaced on a circular road, with $d_n(i,j)$ the smallest $m\ge 0$ with $i-j\equiv m$ or $j-i\equiv m \pmod n$ . The insertion run of Theorem 5 inserts the cities in the order $1,2,\dots,n$ and produces the zig-zag tour $T_n$ : city 1, then the even cities in increasing order, then the odd cities in decreasing order.

Formalization targets

Goal: the Corollary to Theorem 6

For $n\ge 6$ and $4k\le n$ there is a traveling salesman graph with OPTIMAL $>0$ on which some run of nearest insertion, and some run of cheapest insertion, return a $k$ -optimal tour with

\frac{\mathrm{INSERT}}{\mathrm{OPTIMAL}}=2\left(1-\frac1n\right).

Milestones

The insertion run on the circle: the subtours $T_i$ and nodes $a_i=i+1$ form an insertion run that obeys both the nearest and the cheapest rule (proof of Theorem 5).
On the circle, $T_n$ has length $2(n-1)$ and OPTIMAL $=n$ (proof of Theorem 5).
Theorem 5: for $n\ge 6$ there is a graph with INSERT/OPTIMAL $=2(1-1/n)$ for both methods.
Equation (7.4): the length of a tour of the circle is the sum over unit edges $e$ of COUNT $(e,T)$ , the number of times $e$ is traversed when each tour edge is replaced by a shortest arc.
Every tour of the circle is odd or even (all counts of one parity), eq. (7.5).
$T_n$ is the shortest even tour, so every tour shorter than $T_n$ is odd.
$T_n$ is $k$ -optimal for every $k\le n/4$ .
Theorem 6: for $n\ge 8$ there is a graph with a tour that is $k$ -optimal for all $k\le n/4$ and has LOCALOPT/OPTIMAL $=2(1-1/n)$ .

Significance

The result. The Corollary shows that the $2(1-1/n)$ worst-case guarantee of nearest and cheapest insertion cannot be improved by requiring that the returned tour survive $k$ -change local search, for $k$ up to a quarter of the number of cities. Theorem 6 says more generally that $k$ -optimality with $k\le n/4$ does not bound the ratio to the optimum below $2(1-1/n)$ . Together with the paper's upper bound, the insertion guarantee is exact, and it stays exact after local polishing with small neighbourhoods.

Formalizing it. All statements are proved in the paper; none, to our knowledge, has been machine-checked. The platform has an upper bound for nearest insertion (SupplyChainTheory, Theorem 10.7, ratio at most 2) but no tightness example and no notion of $k$ -optimality. This mission produces a reusable definition of $k$ -changes against arbitrary tours, the circle metric, and the parity-counting argument on a cycle, and it supplies the calculations the paper omits ("We omit these calculations but note that they require the assumption $n\ge 6$ ").

Difficulty

Two steps carry the weight. First, the omitted calculations for the insertion run: at each stage one must show that inserting $a_i$ between $i-1$ and $i$ minimizes the insertion increase over every edge of the zig-zag subtour, and that no other outside city can be inserted for less than 2; the claim fails for $n=4$ and $n=5$ , so the verification must use $n\ge 6$ in an essential way. Second, $k$ -optimality is a statement about every tour at edge difference $k$ , not about 2-opt segment reversals or any specific move family. A search over moves of a special form does not establish it; the argument must bound the length of an arbitrary tour at edge difference $k$ from below.

Formalization scope

Nodes are Fin n, so the paper's node $m$ is index $m-1$ and $a_i=i+1$ is index $i$ ; subtour indices stay 1-based ( $T_1=[a_0]$ , approximation $T_n$ ). A distance is d : Fin n → Fin n → ℝ with the structure IsTSPDist (symmetric, nonnegative, triangle inequality, and $d(i,i)=0$ ; the last is a normalization absent from the paper that changes no length). A tour is an Equiv.Perm (Fin n), a subtour a list read cyclically; OPTIMAL is Finset.univ.inf' over permutations, the true minimum. TOUR $(T,k)$ is insertion at a position minimizing the new length; COST is a minimum over positions; the nearest-insertion distance (4.1) takes values in WithTop ℝ, so no junk value arises. $k$ -optimality compares the tour with every permutation whose edge set (unordered pairs) misses exactly $k$ of the tour's edges. Ratios are multiplied out.

COUNT $(e,T)$ needs a choice of shortest arc for antipodal pairs when $n$ is even; the formalization takes the arc through $\min(x,y),\dots,\max(x,y)$ . The paper's argument does not depend on this choice.

Deviations from the printed text: the Corollary is stated for $n\ge 6$ (the printed statement says only $4k\le n$ , but its proof uses the example of Theorem 5, which exists for $n\ge 6$ ; for $k=0$ , $n=3$ the printed statement is false). Theorem 6 keeps its printed $n\ge 8$ . In the proof of Theorem 5 the paper writes "(4.2) holds" where the cheapest-insertion condition (4.3) is meant; the formal statement uses (4.3).

The existence statements carry OPTIMAL $>0$ , the paper's standing assumption (1.1). Without it, the zero distance would make every length zero and every tour $k$ -optimal, which would satisfy the ratio equations trivially; that formalization is ruled out.

Contributions welcome: proofs of any milestone, in particular the omitted insertion calculations and the parity lemma, and reusable lemmas on cyclic lists and edge sets of permutations.

Selected references

D. J. Rosenkrantz, R. E. Stearns, P. M. Lewis II, An Analysis of Several Heuristics for the Traveling Salesman Problem, SIAM J. Comput. 6(3):563–581, 1977. https://doi.org/10.1137/0206041
S. Lin, Computer solutions of the traveling salesman problem, Bell System Tech. J. 44:2245–2269, 1965. https://doi.org/10.1002/j.1538-7305.1965.tb04146.x
S. Lin, B. W. Kernighan, An effective heuristic algorithm for the traveling-salesman problem, Oper. Res. 21(2):498–516, 1973. https://doi.org/10.1287/opre.21.2.498

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Algorithmic Game Theory Operations Research Optimization·Captain: mikedeng1

Approximation Algorithms for Combinatorial Auctions with Complement-Free Bidders III: A Greedy Price-Update Algorithm Is a 2-Approximation for XOS BiddersResearch Paper

Motivation

In a combinatorial auction a set of $m$ items is sold to $n$ bidders who value bundles of items rather than single items. Spectrum auctions, procurement of transportation lanes and the allocation of cloud resources all have this form, and the central algorithmic question is how to allocate the items so as to maximize the social welfare, the sum of the bidders' values for what they receive. Even to describe a general valuation takes $2^m$ numbers, so algorithms access the bidders through queries, and the achievable approximation depends on the class of valuations allowed.

Dobzinski, Nisan and Schapira (Math. Oper. Res. 35(1), 2010) study bidders without complementarities. For the class of XOS valuations, maxima of additive valuations, which strictly contains the submodular valuations, they give LP-based algorithms and, in §3.3, a purely combinatorial algorithm: bidders arrive one at a time, take their demanded bundle at the current item prices, and raise the prices of what they took. Its analysis charges the welfare of any allocation to the item prices the algorithm sets, an argument that uses only the demand and XOS oracles.

Setting

The items are $M=\{1,\dots,m\}$ and the bidders $1,\dots,n$ .

An additive valuation (a clause) $w$ assigns nonnegative values $w(1),\dots,w(m)$ to the items and $w(S)=\sum_{j\in S}w(j)$ to a bundle $S\subseteq M$ .
An XOS valuation $v$ is given by a nonempty finite set of clauses $\{w_1,\dots,w_t\}$ (its XOS expression) through $v(S)=\max_k w_k(S)$ . A clause $w_k$ with $w_k(S)=v(S)$ is a maximizing clause for $S$ . Every XOS valuation is normalized, $v(\emptyset)=0$ , and monotone.
An allocation is a tuple of pairwise disjoint bundles $O_1,\dots,O_n$ ; items may stay unallocated. Its welfare is $\sum_i v_i(O_i)$ .
A demand oracle for bidder $i$ answers, given item prices $p\in\mathbb R^m$ , a bundle maximizing $v_i(T)-\sum_{j\in T}p_j$ . An XOS oracle answers, given a bundle $S$ , a maximizing clause for $S$ in $v_i$ .

The greedy price-update algorithm. Start with all bundles empty and all prices $p_j=0$ . For $i=1,\dots,n$ : let $S_i$ be bidder $i$ 's demand at the current prices; remove the items of $S_i$ from the bundles of the earlier bidders; let $q^i$ be the maximizing clause for $S_i$ in $v_i$ ; set $p_j=q^i_j$ for $j\in S_i$ . Write $p^k$ for the prices after stage $k$ ( $p^0=0$ ), $p^k(T)=\sum_{j\in T}p^k_j$ , and $A_1,\dots,A_n$ for the final bundles.

In the Lean development these objects are XOSExpr, XOSExpr.val, IsAllocation, welfare, IsDemandOracle, IsXOSOracle, greedyState, greedyPrices and greedyAlloc in the namespace ComplementFreeCA.XOSGreedy.

Formalization targets

Goal: Theorem 3.3

For XOS valuations $v_1,\dots,v_n$ , every demand oracle and every XOS oracle, and every allocation $O_1,\dots,O_n$ ,

\sum_{i=1}^n v_i(O_i)\;\le\;2\sum_{i=1}^n v_i(A_i).

The comparison with every allocation is the paper's comparison with the optimal allocation.

Milestones

Lemma 3.4. The final prices are paid for by the algorithm's welfare:

p^n(M)\le\sum_i v_i(A_i).

Lemma 3.5. Prices never decrease: $p^k_j\le p^{k'}_j$ for all items $j$ and stages $k\le k'$ .
Lemma 3.6. For every allocation $O$ ,

\sum_i v_i(O_i)\le 2\,p^n(M).

Three further statements are included without being milestones: the output $A_1,\dots,A_n$ is an allocation (used implicitly by the paper); the first sentence of the proof of Lemma 3.6, that with $\Delta^i=p^i(M)-p^{i-1}(M)$ one has $\Delta^i=\max_{T\subseteq M}\bigl(v_i(T)-p^{i-1}(T)\bigr)$ ; and the factor $2$ is attained on the paper's two-item, two-bidder example (p. 9).

Significance

Theorem 3.3 shows that a factor-2 approximation of the optimal welfare for XOS bidders needs no linear program: one pass over the bidders, one demand query and one XOS query each. The paper's LP-based algorithm of §3.2 attains the better ratio $e/(e-1)$ , and Theorem 4.1 of the paper shows that for the larger class of complement-free bidders no $(2-\epsilon)$ -approximation is possible with polynomial communication.

The result is proved in the paper; to the best of our knowledge it has no machine-checked proof. This mission produces a formal model of XOS valuations, demand and XOS oracles and the greedy run that is reusable for other price-based arguments, and a formal proof of the guarantee for every tie-breaking in both oracles.

Difficulty

The argument is elementary, but its bookkeeping is where a formal proof can go wrong. Items move between bidders: an item taken from an earlier bidder in step (b) is re-priced in step (d), and every priced item lies in exactly one final bundle. Lemma 3.4 needs this invariant across all $n$ stages, together with the fact that a maximizing clause for the demanded set bounds $v_i$ on every subset, which holds because it is a clause of $v_i$ itself, not merely an additive function agreeing with $v_i$ on $S_i$ . Lemma 3.5 is a contradiction argument using optimality of the demand; Lemma 3.6 telescopes the price increases and needs prices to stay nonnegative. The argument must hold for arbitrary oracle answers, so no canonical demand can be assumed.

Formalization scope

Bidders are Fin n, items Fin m, bundles Finset (Fin m), values in ℝ. An XOS valuation is represented by its expression: a nonempty Finset (Fin m → ℝ) of clauses with nonnegative entries, evaluated with Finset.sup'. The paper's standing assumptions, normalized and monotone valuations (p. 1), hold automatically in this representation.
The oracles are function parameters dem : Fin n → (Fin m → ℝ) → Finset (Fin m) and cl : Fin n → Finset (Fin m) → (Fin m → ℝ) with hypotheses IsDemandOracle and IsXOSOracle; every theorem holds for every such pair, so no tie-breaking rule is fixed.
The run is a recursion on the stage: stage $k+1$ processes the 0-based bidder $k$ , i.e. the paper's bidder $k+1$ ; after stage $n$ the state is constant.
The paper's goal compares with the optimal allocation; the Lean goal compares with every allocation, which is equivalent and avoids an argmax. Stating the bound against one particular allocation, or against the algorithm's own output, would be trivial and is excluded.
There are no $O(\cdot)$ constants: the factor $2$ is the paper's.
Printed slip: the model on p. 1 writes an allocation as $S_1,\dots,S_m$ ; it is one bundle per bidder, $S_1,\dots,S_n$ .
Out of scope: running time, the cost of simulating oracles (Proposition 2.1), and communication lower bounds.

Contributions welcome: proofs of the three milestones and of the goal, and reusable lemmas on the invariant that every priced item lies in exactly one final bundle.

Selected references

S. Dobzinski, N. Nisan, M. Schapira, Approximation Algorithms for Combinatorial Auctions with Complement-Free Bidders, Mathematics of Operations Research 35(1):1–13, 2010. https://doi.org/10.1287/moor.1090.0436
B. Lehmann, D. Lehmann, N. Nisan, Combinatorial auctions with decreasing marginal utilities, Games and Economic Behavior 55(2):270–296, 2006. https://doi.org/10.1016/j.geb.2005.02.006

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Algorithmic Game Theory Operations Research Optimization+1·Captain: mikedeng1

Approximation Algorithms for Combinatorial Auctions with Complement-Free Bidders II: Clause-Based Randomized Rounding for XOS BiddersResearch Paper

Motivation

In a combinatorial auction a seller offers $m$ indivisible items to $n$ bidders, each of whom values bundles of items rather than single items. Allocating the items so as to maximize the total value (the social welfare) is the central optimization problem of the area: it models spectrum auctions, procurement and resource allocation, and it is NP-hard and hard to approximate for general valuations. A large literature therefore studies restricted classes of valuations without complementarities. Among them the class XOS (valuations that are a maximum of additive valuations, also called fractionally subadditive) sits strictly between submodular and subadditive valuations and has become a standard benchmark class in algorithmic game theory.

Dobzinski, Nisan and Schapira (Math. Oper. Res. 35(1), 2010) gave, among other results, a randomized algorithm that approximates the optimal welfare for XOS bidders within a factor $1/(1-(1-1/n)^n)$ , which is at most $e/(e-1)\approx 1.582$ . The algorithm rounds the standard LP relaxation and resolves conflicts between bidders using the XOS structure. This mission formalizes that guarantee (Theorem 3.2 of the paper).

A short timeline: Lehmann, Lehmann and Nisan (EC 2001) introduced the XOS terminology and a 2-approximation for submodular bidders; the conference version of the present paper (STOC 2005) gave the $e/(e-1)$ bound for XOS with demand and XOS oracles; Feige (STOC 2006) extended the $e/(e-1)$ ratio to XOS bidders with demand oracles only and gave a 2-approximation for subadditive bidders.

Setting

Items are $M=\{1,\dots,m\}$ and bidders are $N=\{1,\dots,n\}$ with $n\ge1$ . Bidder $i$ has a valuation $v_i$ assigning a real number $v_i(S)$ to every bundle $S\subseteq M$ . An allocation is a tuple $(O_1,\dots,O_n)$ of pairwise disjoint bundles; its welfare is $\sum_i v_i(O_i)$ .

A clause is an additive valuation $w$ given by nonnegative item values $w_1,\dots,w_m$ , with $w(S)=\sum_{j\in S}w_j$ . A valuation $v$ is XOS if there is a nonempty finite set $W$ of clauses with

v(S)=\max_{w\in W}\ \sum_{j\in S}w_j\qquad(S\subseteq M).

A clause of $W$ attaining the maximum for $S$ is a maximizing clause for $S$ in $v$ ; an XOS oracle returns one (arbitrarily, if several attain it).

The LP relaxation has a variable $x_{i,S}$ for every bidder $i$ and bundle $S$ and asks to maximize $\mathrm{OPT}^*=\sum_{i,S}x_{i,S}v_i(S)$ subject to $\sum_{i}\sum_{S\ni j}x_{i,S}\le1$ for each item $j$ , $\sum_S x_{i,S}\le1$ for each bidder $i$ , and $x_{i,S}\ge0$ .

Randomized rounding draws a preallocation $S_1,\dots,S_n$ : independently for each bidder $i$ , bundle $S$ is chosen with probability $x_{i,S}$ and the empty bundle with the remaining probability $1-\sum_S x_{i,S}$ . The preallocation can give an item to several bidders.

The algorithm of §3.2: (i) draw a preallocation from an optimal LP solution $x$ ; (ii) let $p^i=(p^i_1,\dots,p^i_m)$ be the maximizing clause for $S_i$ in $v_i$ ; (iii) give each item $j$ to a bidder $i$ with $p^i_j\ge p^{i'}_j$ for all $i'$ . Write $\mathrm{ALG}$ for the welfare of the resulting allocation.

Formalization targets

Goal: Theorem 3.2

For every XOS profile, every optimal LP solution $x$ , every choice of maximizing clauses, every tie-breaking in step (iii) and every allocation $O$ ,

\Big(1-\Big(1-\frac1n\Big)^n\Big)\sum_i v_i(O_i)\ \le\ \mathbb E[\mathrm{ALG}].

Milestones

$\sum_i v_i(O_i)\le\mathrm{OPT}^*$ for an optimal LP solution (step (i) of the proof of Theorem 3.1).
Pointwise, $\mathrm{ALG}\ge\sum_j Q_j$ with $Q_j=\max_i p^i_j$ .
Eq. (1): for $1\le k\le n$ and $X_1,\dots,X_k\in[0,1]$ with $\sum X_i\le1$ ,

1-\prod_{i\le k}(1-X_i)\ \ge\ 1-\Big(1-\tfrac{\sum X_i}{k}\Big)^k\ \ge\ \Big(1-\big(1-\tfrac1k\big)^k\Big)\sum X_i\ \ge\ \Big(1-\big(1-\tfrac1n\big)^n\Big)\sum X_i .

Lemma 3.3: $\mathbb E[Q_j]\ge(1-(1-1/n)^n)\sum_i\sum_{S\ni j}x_{i,S}\,p^{(i,S)}_j$ for every feasible $x$ .
$\mathbb E[\mathrm{ALG}]\ge(1-(1-1/n)^n)\,\mathrm{OPT}^*(x)$ for every feasible $x$ .

Milestone 5 is stronger than the goal (it compares with the fractional value); the goal is stated against the integral optimum because that is what Theorem 3.2 asserts.

Significance

The bound $1-(1-1/n)^n\ge1-1/e$ is a constant-factor guarantee for a class that includes every submodular valuation, obtained from nothing more than the LP relaxation and the clause structure of XOS. It shows that the integrality gap of the configuration LP for XOS bidders is at most $1/(1-(1-1/n)^n)$ , a fact reused in later work on welfare maximization, online allocation and posted-price mechanisms. The per-item analysis (Lemma 3.3 and Eq. (1)) is the same " $1-1/e$ " correlation-gap argument that recurs in submodular maximization and prophet-inequality proofs.

The result is proved in the paper. To our knowledge it has no machine-checked proof. This mission produces a Lean statement and proof of the guarantee with the randomness made explicit, a reusable model of the configuration LP and of randomized rounding over bundles, and a formal version of the inequality in Eq. (1), which is independently useful.

Difficulty

Each piece of the argument is short; the work is in the bookkeeping. The preallocation is infeasible, so welfare cannot be read off from the rounding directly; the proof reduces it to per-item quantities $Q_j$ and then lower-bounds $\mathbb E[Q_j]$ by comparing with a different assignment of item $j$ that is not the algorithm's. The expectation of that auxiliary assignment involves a product of probabilities over bidders ordered by a conditional expectation, and the final bound needs the calculus inequality of Eq. (1) together with a summation by parts. A naive attempt to bound $\mathbb E[\mathrm{ALG}]$ bidder by bidder fails, because a bidder's received bundle is not contained in its preallocated bundle, and the clause values of other bidders decide what it receives.

Formalization scope

Bidders are Fin n, items Fin m, bundles Finset (Fin m), valuations Finset (Fin m) → ℝ. XOS is stated through its expression: each $v_i$ comes with a nonempty finite clause set $W_i$ of nonnegative clauses, and $v_i(S)$ is attained by a clause of $W_i$ and bounded by all of them. Normalization and monotonicity, the paper's standing assumptions (p. 1), follow from this.
The LP has a variable for every bundle, including $\emptyset$ . "Optimal" is stated as feasible and not beaten by any feasible solution; existence of an optimum is not asserted.
The rounding is a finite product distribution over profiles $\sigma:\mathrm{Fin}\,n\to$ bundles, with bidder $i$ 's law $q_i(S)=x_{i,S}+\mathbf 1[S=\emptyset](1-\sum_T x_{i,T})$ ; expectations are finite sums.
The XOS oracle is a function parameter with its specification, and step (iii) is any rule selecting a bidder with maximal clause value; every statement quantifies over all of them.
"Approximation" in Theorem 3.2 is read in expectation, as its proof establishes. There are no $O(\cdot)$ constants in this mission.
Printed slip: the display of Lemma 3.3 (and the identity for $\mathrm{OPT}^*$ before it) sums $x_{i,S}p^{(i,S)}_j$ over all $(i,S)$ ; the proof's final line restricts to $S\ni j$ , and the printed version is false. The Lean states Lemma 3.3 with $S\ni j$ .
Running time, the ellipsoid method and oracle complexity are out of scope.
Ruled out as trivializing: dropping the LP constraints on $x$ (the item constraint is what makes Eq. (1) apply), stating the bound for a fixed preallocation instead of the expectation, or allowing negative clause values.

A complete development needs finite product distributions over bundles, the AM–GM inequality, monotonicity of $(1-1/k)^k$ , and a summation-by-parts argument. The LP and rounding model are reusable for the other randomized-rounding results of the paper; proofs of any milestone are welcome.

Selected references

S. Dobzinski, N. Nisan, M. Schapira, Approximation Algorithms for Combinatorial Auctions with Complement-Free Bidders, Mathematics of Operations Research 35(1):1–13, 2010. https://doi.org/10.1287/moor.1090.0436
S. Dobzinski, N. Nisan, M. Schapira, Approximation algorithms for combinatorial auctions with complement-free bidders, STOC 2005. https://doi.org/10.1145/1060590.1060681
B. Lehmann, D. Lehmann, N. Nisan, Combinatorial auctions with decreasing marginal utilities, EC 2001. https://doi.org/10.1145/501158.501161
U. Feige, On maximizing welfare when utility functions are subadditive, STOC 2006. https://doi.org/10.1145/1132516.1132540

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Algorithmic Game Theory Operations Research Optimization+1·Captain: mikedeng1

Approximation Algorithms for Combinatorial Auctions with Complement-Free Bidders I: LP Rounding for Subadditive BiddersResearch Paper

Motivation

In a combinatorial auction a seller offers $m$ indivisible items to $n$ bidders, each of whom values bundles of items rather than single items. Allocating the items to maximize total value is the basic welfare problem of spectrum auctions, procurement and resource allocation, and it is the running example of algorithmic mechanism design. For general valuations no polynomial-time algorithm achieves a ratio polynomially better than $\sqrt m$ under standard assumptions, so positive results require restricting the valuations. The most natural restriction is complement freeness (subadditivity): a bundle is never worth more than the sum of its parts.

Dobzinski, Nisan and Schapira (Math. Oper. Res. 35(1), 2010; conference version STOC 2005) gave the first polynomial-time algorithms with sub-polynomial approximation ratios for complement-free bidders given demand oracles. This mission formalizes their Section 3.1 algorithm, which rounds the linear-programming relaxation of the auction and splits the resulting infeasible solution into feasible ones.

Timeline. Lehmann, Lehmann and Nisan (2001) introduced the complement-free hierarchy and treated submodular bidders. The original version of the algorithm formalized here claimed an $O(\log m)$ ratio; Feige observed that the same algorithm achieves $O(\log m/\log\log m)$ and that its ratio is at least $\Omega(\sqrt{\log m/\log\log m})$ (Feige, SIAM J. Comput. 39(1), 2009). Feige then obtained a constant ratio ( $2$ ) for subadditive bidders by a different rounding.

Setting

Items are $M=\{1,\dots,m\}$ and bidders $N=\{1,\dots,n\}$ . Bidder $i$ has a valuation $v_i$ assigning a real value $v_i(S)$ to each bundle $S\subseteq M$ . Every valuation is normalized, $v_i(\emptyset)=0$ , and monotone, $S\subseteq T\Rightarrow v_i(S)\le v_i(T)$ . A valuation is complement free if $v(S\cup T)\le v(S)+v(T)$ for all $S,T$ . An allocation is a tuple $(S_1,\dots,S_n)$ of pairwise disjoint bundles; its welfare is $\sum_i v_i(S_i)$ , and $OPT$ denotes the largest welfare.

The LP relaxation has a variable $x_{i,S}\ge0$ for each bidder and bundle, with $\sum_{i,S\ni j}x_{i,S}\le1$ for each item $j$ and $\sum_S x_{i,S}\le1$ for each bidder $i$ ; its objective is $\sum_{i,S}x_{i,S}v_i(S)$ , with optimum $OPT^*\ge OPT$ . Randomized rounding lets each bidder independently draw bundle $S$ with probability $x_{i,S}$ and $\emptyset$ with the remaining probability. The result, a preallocation, has expected welfare $OPT^*$ but may give an item to several bidders.

The algorithm takes $k=\lfloor 3\log m/\log\log m\rfloor$ and:

rounds until the preallocation $(S_1,\dots,S_n)$ has every item in at most $k$ bundles and $\sum_i v_i(S_i)\ge OPT^*/3$ ;
splits each $S_i$ into layers $S_i^r$ , $r=1,\dots,k$ , where $S_i^r$ holds the items of $S_i$ that appear in exactly $r-1$ of $S_1,\dots,S_{i-1}$ ;
picks the layer index $r$ maximizing $\sum_i v_i(S_i^r)$ and sets $T_i=S_i^r$ ;
if some bidder has $v_i(M)\ge\sum_{i'}v_{i'}(T_{i'})$ , gives that bidder everything instead.

Formalization targets

Goal: Theorem 3.1, with the explicit ratio

For all sufficiently large $m$ , for normalized, monotone, complement-free valuations and an optimal LP solution $x$ with value $OPT^*$ :

if $OPT^*>3\max_i v_i(M)$ , one rounding meets the two conditions of step 1 with probability $>1/6$ ;
from any preallocation meeting them, every admissible run of steps 2–4 outputs an allocation with

\sum_i v_i(\text{output}_i)\ \ge\ \frac{OPT}{3k},\qquad k=\Big\lfloor\frac{3\log m}{\log\log m}\Big\rfloor;

if $OPT^*\le3\max_i v_i(M)$ , the bidder maximizing $v_i(M)$ alone achieves $OPT/3$ .

Milestones

$OPT\le OPT^*$ (proof, step (i)).
The layers of each index form an allocation and partition each $S_i$ (proof, step (ii)).
Complement freeness gives $\sum_r v_i(S_i^r)\ge v_i(S_i)$ , and the best layer has welfare $\ge OPT^*/(3k)$ (proof, step (iii)).
Lemma 3.1: independent Bernoulli variables with $\sum_i p_i\le1$ exceed $3\log m/\log\log m$ with probability $\le1/m^2$ .
Lemma 3.2: for $X$ a sum of independent $[0,1]$ variables with mean $\mu$ , $\Pr[|X-\mu|\ge\alpha]\le\mu/\alpha^2$ .
§3.1.1: some item appears more than $3\log m/\log\log m$ times with probability $\le1/m$ ; the preallocation's welfare falls below $OPT^*/3$ with probability $<3/4$ .

Significance

Theorem 3.1 was among the first polynomial-time approximation guarantees for welfare maximization with general subadditive bidders, and its layering argument is the standard way to turn an LP solution that is feasible up to a factor $k$ into a feasible allocation losing only a factor $k$ for subadditive objectives. The same argument applies to the $k$ -duplicates auction and reappears in later rounding schemes. Lemma 3.1 is the standard balls-in-bins tail bound behind every $\log m/\log\log m$ load estimate.

The results are proved in the paper; none of them is formalized, on this platform or in Mathlib, as far as searches show. The mission produces a machine-checked version of the algorithm's guarantee with an explicit constant $3k$ in place of $O(\cdot)$ , a precise statement of the probabilistic step, and reusable statements of two concentration inequalities for sums of independent bounded variables.

Difficulty

The combinatorial part (steps (ii) and (iii)) is short. The difficulty is in step (i). The rounding is a product distribution over bundles, while the count of an item is a sum over bidders of indicators that depend on each bidder's whole bundle; connecting the finite product law to independent Bernoulli variables, and then to Lemma 3.1, requires building the independence structure explicitly. Lemma 3.1 itself does not follow from a Chernoff bound with a fixed relative deviation: the threshold $3\log m/\log\log m$ grows with $m$ while the mean stays at most $1$ , and the bound must hold uniformly in the number of variables, which a fixed-deviation Chernoff statement does not give. Finally, the event- $B$ bound needs the preallocation's welfare as a sum of independent variables in $[0,1]$ , which requires rescaling by $\max_i v_i(M)$ and monotonicity.

Formalization scope

Bidders are Fin n, items Fin m, bundles Finset (Fin m), valuations Finset (Fin m) → ℝ. Normalization and monotonicity, the paper's standing assumptions (p. 1), are hypotheses of the goal. $\log$ is the natural logarithm; the paper does not fix a base. The rounding law is written as explicit finite sums over profiles $\sigma:\texttt{Fin } n\to\texttt{Finset (Fin } m)$ with product weights, so independence across bidders is literal; Lemmas 3.1 and 3.2 are stated measure-theoretically with Mathlib's iIndepFun.

Explicit constants and conventions that replace the paper's notation:

The ratio $O(k)=O(\log m/\log\log m)$ is stated as $3k$ with $k=\lfloor3\log m/\log\log m\rfloor$ , the constant the proof establishes (step (iii), p. 6). "Sufficiently large $m$ " is an existential $m_0$ .
The w.l.o.g. scaling $\max_i v_i(M)=1$ and the split at $OPT^*=3$ become the scale-free split at $OPT^*=3\max_i v_i(M)$ .
The choice of $r$ in step (iii) and of the bidder in step (iv) are universally quantified over all admissible choices.

Printed slips resolved in the statements:

Lemma 3.1 mixes $n$ and $m$ . Here the number of variables is arbitrary, "sufficiently large" refers to $m$ , and $\sum_ip_i=1$ is relaxed to $\sum_ip_i\le1$ , which is what the application uses.
The display after Lemma 3.1 writes the threshold $\log m/(3\log\log m)$ ; the lemma's $3\log m/\log\log m$ is used.
" $\Pr[\vee_jE_j]<1/n$ " and " $\le 1/n+3/4$ " should read $1/m$ .
The event $B$ is defined without its sum; it means $\sum_iv_i(S_i)<OPT^*/3$ .

Trivializing formalizations are ruled out: the guarantee assumes the item-count bound (without it the layers do not cover the bundles), the probability statements assume an LP-feasible $x$ , neither the layer index nor the step-(iv) bidder is fixed, and the ratio is the explicit $3k$ rather than an unspecified constant. Running time, the ellipsoid method and oracle complexity are out of scope. Contributions of general concentration lemmas for sums of independent bounded variables are welcome and reusable beyond this mission.

Selected references

S. Dobzinski, N. Nisan, M. Schapira, Approximation Algorithms for Combinatorial Auctions with Complement-Free Bidders, Mathematics of Operations Research 35(1):1–13, 2010. https://doi.org/10.1287/moor.1090.0436
U. Feige, On Maximizing Welfare When Utility Functions Are Subadditive, SIAM Journal on Computing 39(1):122–142, 2009. https://doi.org/10.1137/070680977
B. Lehmann, D. Lehmann, N. Nisan, Combinatorial Auctions with Decreasing Marginal Utilities, Games and Economic Behavior 55(2):270–296, 2006. https://doi.org/10.1016/j.geb.2005.02.006
M. Mitzenmacher, E. Upfal, Probability and Computing, Cambridge University Press, 2005. https://doi.org/10.1017/CBO9780511813603

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

Discrete Convex Analysis VII: The L-Optimality Criterion and the Proximity TheoremTextbook

Motivation

Submodularity — the diminishing-returns property $g(p) + g(q) \ge g(p \vee q) + g(p \wedge q)$ on a lattice — is one of the most useful structural hypotheses in combinatorial optimization, underlying efficient algorithms for network flows, matroid theory, and set-function minimization. Chapter 7 studies L-convex functions: functions on the integer lattice $\mathbb Z^V$ that are submodular and linear along the all-ones direction. This is the "dual" notion, under the conjugacy developed later in the book, to chunk 06's M-convex functions, and it inherits the same strong minimization theory — a purely local optimality criterion and a proximity theorem with an explicit distance bound — while additionally supporting a genuinely new characterization with no M-convex counterpart: discrete midpoint convexity, the direct lattice analogue of the classical real-valued midpoint convexity condition. This mission formalizes the chapter's definitional theorem, its midpoint-convexity characterization, the L-optimality criterion, and the L-proximity theorem itself.

Setting

Let $V$ be a finite ground set. A function $g : \mathbb Z^V \to \mathbb R \cup \{+\infty\}$ with nonempty effective domain is an L-convex function if it satisfies (SBF[Z]): $g(p) + g(q) \ge g(p \vee q) + g(p \wedge q)$ for all $p, q$ ( $\vee, \wedge$ componentwise max/min), and (TRF[Z]): there is $r \in \mathbb R$ with $g(p + \mathbf 1) = g(p) + r$ for all $p$ , where $\mathbf 1$ is the all-ones vector. An L $^\natural$ -convex function is one whose lift to the extended ground set $\{0\} \cup V$ is L-convex; equivalently (Theorem 7.1), $g$ satisfies the translation-submodularity axiom (SBF $^\natural$ [Z]): $g(p) + g(q) \ge g((p - \alpha\mathbf 1) \vee q) + g(p \wedge (q + \alpha\mathbf 1))$ for all $p, q$ and all nonnegative integers $\alpha$ . Discrete midpoint convexity asks $g(p) + g(q) \ge g(\lceil (p+q)/2 \rceil) + g(\lfloor (p+q)/2 \rfloor)$ componentwise. For $\alpha$ a positive integer, a point satisfies scaled local optimality if $g(p_\alpha) \le g(p_\alpha \pm \alpha \chi_Y)$ for every $Y \subseteq V$ .

Formalization targets

Goal: Theorem 7.18 (the L-proximity theorem)

Assume $\alpha$ is a positive integer and $n = |V|$ . (1) If $g$ is L-convex with $g(p) = g(p+\mathbf 1)$ for all $p$ , and $p_\alpha \in \operatorname{dom} g$ satisfies $g(p_\alpha) \le g(p_\alpha + \alpha\chi_Y)$ for all $Y \subseteq V$ , then $\arg\min g \ne \emptyset$ and there is $p^* \in \arg\min g$ with the componentwise bound

p_\alpha \le p^* \le p_\alpha + (n-1)(\alpha-1)\mathbf 1.

(2) If $g$ is L $^\natural$ -convex and $p_\alpha$ satisfies the two-sided version, then there is $p^*$ with $p_\alpha - n(\alpha-1)\mathbf 1 \le p^* \le p_\alpha + n(\alpha-1)\mathbf 1$ . The bound is a genuine vector (lattice-order) inequality, not an $\ell^\infty$ -norm bound — the form later chapters' applications need.

Milestones: Theorems 7.1, 7.7, 7.14

Theorem 7.1: L $^\natural$ -convexity (defined via the lift) is equivalent to the direct translation-submodularity axiom. Theorem 7.7: this same class is also characterized by discrete midpoint convexity — a three-way equivalence with the approach property (L $^\natural$ -APR[Z]) as a bridge — giving L-convexity a genuinely different, more geometric face than anything available on the M-convex side. Theorem 7.14 (the L-optimality criterion): global optimality reduces to a purely local check against the sign-pattern neighbors $p \pm \chi_Y$ , mirroring chunk 06's Theorem 6.26 but with the plain L-convex case additionally requiring the periodicity condition $g(p) = g(p+\mathbf 1)$ .

Significance

The result itself. Discrete midpoint convexity (Theorem 7.7) is philosophically important: it shows the lattice-submodularity definition of L-convexity is not an arbitrary discretization choice but coincides exactly with the most direct discrete analogue of ordinary midpoint convexity, the classical characterization of convex functions via $f((p+q)/2) \le (f(p)+f(q))/2$ . The L-optimality criterion and L-proximity theorem give L-convex minimization the same algorithmic footing as M-convex minimization (chunk 06): scaling algorithms for L-convex objectives — which arise naturally from network flow and submodular-function duality — inherit a provable, dimension-and-scale-explicit distance guarantee between a coarse-scale local optimum and the true minimizer.

Formalizing it. No matching item exists on the platform for L-convex functions, discrete midpoint convexity, or the L-optimality/proximity theorems. This mission gives the first formal statement of these results, completing (alongside chunk 06's M-convex-function results) both halves of the exchange-axiom-based theory that chapter 8's conjugacy duality later unifies.

Difficulty

A natural shortcut, given the structural parallel to chunk 06, is to assume the L-proximity theorem's proof is a mechanical relabeling of the M-proximity theorem's proof. It is not: the M-convex proof (chunk 06) crucially uses the exchange axiom's additive four-term inequality to build a chain of strictly improving points, whereas the L-convex proof instead exploits (TRF[Z])'s periodicity directly — it reduces to the case $p_\alpha = 0$ using translation invariance, then constructs a minimal (with respect to the lattice order) point among all sufficiently good solutions and shows this minimality, combined with submodularity (SBF[Z]), forces the componentwise bound. The vector (rather than norm) form of the conclusion is not cosmetic: it is exactly what this lattice-order argument naturally produces, and is the form needed by later chapters' applications.

Formalization scope

The ground set $V$ is a Fintype with DecidableEq; $g : \mathbb Z^V \to \mathbb R \cup \{+\infty\}$ is (V → ℤ) → WithTop ℝ. Unlike chunk 06's M-convex axiom, (SBF[Z]), (TRF[Z]), and (SBF $^\natural$ [Z]) are stated for all of $\mathbb Z^V$ , not restricted to $\operatorname{dom} g$ , so no explicit import of chunk 05's L-convex-set vocabulary was needed for dom g's structure (unlike the corresponding note in chunk 06's BRIEF.md, which flagged the same concern for dom f). L $^\natural$ -convexity is represented via an explicit lift to Option V, matching the book's own primary definition, with the direct axiom (SBF $^\natural$ [Z]) kept as a separate object related to it by Theorem 7.1.

A trivializing formalization of the goal would convert its componentwise vector bound into an $\ell^\infty$ -norm bound (losing the direction-of-approach information the vector form carries) or drop Part (1)'s periodicity hypothesis $g(p) = g(p+\mathbf 1)$ ; neither is done here. Propositions establishing dom g as an L-convex set, the L/L $^\natural$ relationship (Theorem 7.3), the submodular-set-function embedding (Proposition 7.4), and several structural closure properties are cut from this mission's scope (see MODERATION_NOTES.md) but are natural targets for a follow-on mission or for chunk 09, which builds directly on this chunk's exchange-axiom vocabulary, mirroring chunks 06→07.

Selected references

K. Murota, Discrete Convex Analysis, SIAM, 2003. DOI: 10.1137/1.9780898718508.

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Combinatorics Convex Optimization Discrete Geometry+2·Captain: Shuze Chen

Discrete Convex Analysis XXIII: Directional Derivatives and Subdifferentials of M-Convex FunctionsTextbook

Motivation

An M-convex function is defined on the integer lattice, but chapter 6's earlier results (companion missions 06-mconvex-functions-i, 22-ch06b-mconvexfunctions, 23-ch06c-mconvexfunctions) show it always extends to a genuine convex function on real space. Once that extension exists, every tool of classical convex analysis — directional derivatives, subdifferentials, positive homogeneity — becomes available, and the natural question is whether these classical objects remain combinatorially special when applied to an M-convex function's extension. This mission answers that question at its sharpest: the directional derivative of an M-convex function at any point is again a positively homogeneous M-convex function, its subdifferential is exactly the admissible-potential set of a distance function satisfying the triangle inequality, and this correspondence between positively homogeneous M-convex functions and triangle-inequality distance functions is itself a clean one-to-one correspondence. This closes the loop between chapters 4-5 (M-convex and L-convex sets, distance functions) and the continuous convex-analytic machinery chapter 8 needs for its duality theory.

Companion missions 06-mconvex-functions-i, 22-ch06b-mconvexfunctions, and 23-ch06c-mconvexfunctions cover this chapter's optimality theory, algebraic toolkit, and convex-extensibility characterization. This mission builds the vocabulary those results also need (redeclared here, since sibling drafts cannot yet import one another) and proves the chapter's real-variable capstones: the transfer of M-convexity's basic operations, optimality criterion, and supermodularity to the polyhedral (real-variable) setting, the identification of positively homogeneous M-convex functions with distance functions satisfying the triangle inequality, and — this mission's goal — the full directional-derivative/subdifferential correspondence.

Setting

Fix a finite ground set $V$ . A polyhedral convex function $g : \mathbb R^V \to \mathbb R \cup \{+\infty\}$ is (polyhedral) M-convex if it satisfies the real-variable exchange axiom (M-EXC[R]): for $x,y \in \operatorname{dom}_{\mathbb R} g$ and $u \in \operatorname{supp}^+(x-y)$ , some $v \in \operatorname{supp}^-(x-y)$ and $\alpha_0 > 0$ make the exchange inequality hold on $\alpha \in [0,\alpha_0]$ ; M♮-convex if its lift to one extra coordinate is M-convex. The directional derivative of $g$ at $x \in \operatorname{dom}_{\mathbb R} g$ in direction $d$ is $g'(x;d) = \inf_{t>0} (g(x+td) - g(x))/t$ . A function is positively homogeneous if $g(tx) = t \cdot g(x)$ for all $t > 0$ ; write $0M[\mathbb R \to \mathbb R]$ for the positively homogeneous polyhedral M-convex functions. A distance function $\gamma$ satisfying the triangle inequality and its set of admissible potentials $D(\gamma)$ were introduced in chapter 5; the subdifferential $\partial_{\mathbb R} f(x) = \{p : f(y) - f(x) \ge \langle p, y-x \rangle\ \forall y\}$ generalizes this to any function $f$ at a point $x$ in its domain.

Formalization targets

Goal: the directional-derivative/subdifferential correspondence

For $f \in M[\mathbb R \to \mathbb R]$ and $x \in \operatorname{dom}_{\mathbb R} f$ , setting $\gamma_{f,x}(u,v) = f'(x;-\chi_u+\chi_v)$ :

\gamma_{f,x} \text{ satisfies the triangle inequality}, \quad \partial_{\mathbb R} f(x) = D(\gamma_{f,x}) \ne \emptyset, \quad f'(x;\cdot) = \widehat{\gamma_{f,x}}(\cdot),

with the analogous statement for $f \in M[\mathbb Z \to \mathbb R]$ at an integer point $x$ , using $\gamma_{f,x}(u,v) = f(x-\chi_u+\chi_v)-f(x)$ (Theorem 6.61). This is the weakest stable form: it identifies the subdifferential exactly, as a set, rather than bounding its size or complexity, and holds at every point of the domain uniformly.

Supporting structural targets

Ten further results build the real-variable toolkit and the positive-homogeneity correspondence this goal completes: the transfer of M♮-convexity, the basic operations, the optimality criterion, supermodularity, and weighted-minimizer polyhedrality to the real-variable setting (Theorems 6.48-6.52, Proposition 6.53), the identification of the classes $0M[\mathbb Z|\mathbb R \to \mathbb R]$ and $0M[\mathbb R \to \mathbb R]$ and the compatibility of convex extension with positive homogeneity (Proposition 6.56), the two directions of the correspondence between positively homogeneous M-convex functions and triangle-inequality distance functions (Propositions 6.57-6.58, Theorem 6.59), and the fact that a directional derivative of an M-convex function is itself positively homogeneous and M-convex (Proposition 6.60).

Significance

Theorem 6.61 is the technical bridge that lets discrete convex analysis borrow the entire apparatus of classical convex duality: because the subdifferential of an M-convex function is always the admissible-potential set of a chapter-5 distance function, every fact already proved about $D(\gamma)$ (its polyhedral structure, its own L-convexity, its relationship to shortest paths) transfers immediately to subdifferentials of M-convex functions. This is exactly the mechanism the book calls out as essential for Chapter 8's separation theorem for M♮-convex functions. The $0M \leftrightarrow T$ correspondence (Theorem 6.59) is independently significant: it says the positively homogeneous special case of M-convex function theory — which is what directional derivatives of any M-convex function reduce to, by Proposition 6.60 — is exactly as rich as ordinary shortest-path distance function theory, no more and no less, so nothing new needs to be built to understand local behavior at a point.

None of these results are open — they are Murota's account of how the discrete exchange axiom interacts with directional differentiation and subgradients, a bridge chapter between the purely combinatorial theory of chapters 4-6 and the duality theory of chapter 8. What this mission contributes is a faithful, machine-checked formal statement of each, extending the shared Lean vocabulary (MExchangeAxiomR, DirDeriv, GammaHat) the Discrete Convex Analysis series builds on; no comparable formalization exists on the platform (see Formalization scope).

Difficulty

The naive approach to Theorem 6.61 would try to compute $\partial_{\mathbb R} f(x)$ directly from the definition of subgradient and separately verify it happens to equal some $D(\gamma)$ ; the book's actual proof instead derives the equality of sets from the M-optimality criterion (Theorem 6.52) applied pointwise: $p \in \partial_{\mathbb R} f(x)$ is shown, via a chain of logical equivalences, to be exactly the condition defining $D(\gamma_{f,x})$ , so no separate verification of polyhedrality or nonemptiness is needed beyond what Theorem 6.52 and Proposition 6.60 already supply. The genuine difficulty is upstream, in Proposition 6.60 itself: showing a directional derivative is M-convex requires exploiting the local validity of the identity $f(x+d)-f(x) = f'(x;d)$ for small $\|d\|_1$ (Eq. (6.85)) and then extending the exchange property from that neighborhood to all of $\mathbb R^V$ using positive homogeneity — a two-step argument with no single-step shortcut, since the exchange axiom's defining inequality is not obviously homogeneous-invariant on its own.

Formalization scope

Ground-set elements are a Fintype V with DecidableEq; real-domain functions are (V→ℝ)→WithTop ℝ. The directional derivative is built directly as an infimum of difference quotients over $t>0$ , matching the book's own local characterization (Eq. (6.85)) without a separate limit construction. Positive homogeneity and the classes 0M[R→R]/0M[Z→R] are stated exactly as the book defines them (the latter via positive homogeneity of the convex extension, not of f itself, since f is undefined off Zⱽ). Theorems 6.49-6.50 restate 4 of their 8 operations (matching the identical scope decision for chunk 22-ch06b-mconvexfunctions's Theorem 6.13); Theorem 6.61 omits the dual-integral refinement clauses for the M[R→R|Z]/ M[Z→Z] sub-classes. Both reductions are documented, not trivializing omissions — see Difficulty above and HARD.md/MODERATION_NOTES.md. No numeric constants are hard-coded anywhere in this mission. This mission's definitions are redeclared from chunks 06-mconvex-functions-i, 21-ch05b-lconvexsets (for the distance-function/admissible-potential vocabulary), 22-ch06b-mconvexfunctions, and 23-ch06c-mconvexfunctions rather than imported, since sibling drafts in this series cannot yet reference one another. Contributions completing any of the twelve sorrys are welcome; the goal and Proposition 6.60 carry the most independent proof content.

Selected references

K. Murota, Discrete Convex Analysis, SIAM, 2003. DOI: 10.1137/1.9780898718508.
K. Murota and A. Shioura, "M-convex function on generalized polymatroid," Mathematics of Operations Research, 24 (1999), pp. 95-105 (the polyhedral M-convex function theory this mission's real-variable results are drawn from).

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Convex Optimization Numerical Analysis Optimization·Captain: mikedeng1

The Relaxation Method for Linear Inequalities III: Reflexion in a Closed Bounded Convex Set Terminates or Ends in OscillationResearch Paper

Motivation

The relaxation method for a system of linear inequalities, introduced by Agmon and by Motzkin and Schoenberg in back-to-back papers of the Canadian Journal of Mathematics (1954), solves $\sum_j a_{ij}x_j + b_i \ge 0$ by repeatedly moving a point towards, or across, the most violated half-space. It is the ancestor of the perceptron algorithm, of Kaczmarz-type projection methods, and of the method of alternating projections, all of which are still used in feasibility problems, tomography and machine learning.

Motzkin and Schoenberg's paper ends (Part IV, §§9–10) by asking whether the behaviour of the reflexion process — the relaxation step with factor $\lambda = 2$ — survives when the finite family of half-spaces is replaced by an infinite one. They answer this for one natural infinite family: all supporting half-spaces of a closed bounded convex set. This mission formalizes that answer, Theorem 3 of the paper.

Timeline of the thread this mission belongs to:

1922 — Fejér observes that a sequence approaching every point of a set monotonically has useful convergence properties (Fejér, Math. Annalen 85, 1922).
1954 — Agmon proves convergence of the relaxation method for $0 < \lambda < 2$ (Agmon, Canad. J. Math. 6, 1954, pp. 382–392); Motzkin and Schoenberg prove finite termination of the reflexion method ( $\lambda = 2$ ) for full-dimensional solution polytopes (Theorem 1), the oscillation behaviour in lower dimension (Theorem 2), and the convex-body version (Theorem 3) (Motzkin–Schoenberg 1954).

Setting

Let $E_n$ be $n$ -dimensional Euclidean space and let $A \subseteq E_n$ be a nonempty, closed, bounded, convex set. Its dimension $r$ is the dimension of its affine span $L_r$ , the smallest flat containing $A$ .

For $p \notin A$ let $q$ be the point of $A$ nearest to $p$ ; it exists and is unique. The image of $p$ with respect to $A$ is

p_1 = F(p) = p + 2(q - p), \qquad (3.1)

the reflexion of $p$ through $q$ . The reflexion process starts at $p_0 \notin A$ and sets $p_{\nu+1} = F(p_\nu)$ as long as $p_\nu \notin A$ (3.2). Either the process terminates with some $p_N \in A$ , or it produces an infinite sequence of points outside $A$ .

The family $F$ of supporting half-spaces of $A$ consists of the closed half-spaces $H \supseteq A$ whose bounding hyperplane touches $A$ . The paper observes that the half-space $H_0$ through $q$ normal to $pq$ is the member of $F$ farthest from $p$ , so (3.1) is exactly the reflexion step of the relaxation method applied to the infinite family $F$ .

A sequence $\{q_\nu\}$ of points outside $A$ is Fejér-monotone with respect to $A$ if $q_\nu \ne q_{\nu+1}$ and $|q_{\nu+1} - a| \le |q_\nu - a|$ for all $a \in A$ . Two points $u, v$ are symmetric with respect to a flat $L$ if their midpoint lies in $L$ and $u - v$ is orthogonal to $L$ .

Formalization targets

Goal: Theorem 3 (p. 402)

For every nonempty closed bounded convex $A \subseteq E_n$ with affine span $L_r$ , every $p_0 \notin A$ and every run $\{p_\nu\}$ of the reflexion process:

\textbf{Case 1: } r = n \implies \exists N,\ p_N \in A.

\textbf{Case 2: } r < n \implies \begin{cases} p_0 \in L_r \implies \exists N,\ p_N \in A,\\[2pt] p_0 \notin L_r \implies p_\nu \notin A\ \forall \nu, \text{ and } \exists \nu_0, u \ne v \text{ symmetric w.r.t. } L_r:\ \{p_\nu, p_{\nu+1}\} = \{u, v\}\ \forall \nu > \nu_0. \end{cases}

No number of steps is fixed: termination is finite but not uniformly bounded.

Milestones (attack order)

§9 — the half-space $H_0$ belongs to $F$ , maximizes $\operatorname{dist}(p, H)$ over $F$ , and every farthest member of $F$ yields the step (3.1).
§10 — an infinite run of the reflexion process is Fejér-monotone with respect to $A$ .
Lemma 1, Case 1 — a sequence Fejér-monotone with respect to a set of dimension $n$ converges to a point.
§10 — the limit of an infinite run lies in $A$ and on its boundary.
Theorem 3, Case 1 — if $r = n$ the process always terminates.
§10 — orthogonal projection on a flat $L \supseteq A$ commutes with the image map, and each step keeps the distance to $L$ while switching sides.

Significance

The result. Theorem 3 shows that the dichotomy proved in the paper for finitely many half-spaces — finite termination when the target is full-dimensional, eventual oscillation otherwise — holds for the infinite family of all supporting half-spaces of a convex body. Case 1 says that reflecting through the nearest point, a method that uses no information about $A$ beyond metric projection, reaches a full-dimensional convex body in finitely many steps from any start. Case 2 says that for a lower-dimensional body the process detects this: the iterates settle into a two-cycle whose midpoint is a point of $A$ , so a solution can be read off.

Formalizing it. The theorem is proved in the paper (1954), with a short proof of Case 1 that relies on geometric intuition about normal cones near a boundary point. To our knowledge no machine-checked proof exists. A complete development produces a verified finite-termination theorem for a projection method on general convex bodies, together with reusable facts about Fejér-monotone sequences and metric projections that recur throughout the analysis of projection algorithms.

Difficulty

The obvious argument for Case 1 is: the run is Fejér-monotone, hence converges (Lemma 1), and its limit $a$ lies on the boundary of $A$ ; then derive a contradiction. For a polytope (Theorem 1) the contradiction comes from finiteness: eventually every reflexion is in one of finitely many hyperplanes through $a$ , which keeps the iterates on a sphere around $a$ . For a convex body there are infinitely many supporting hyperplanes near $a$ , and the iterates are reflected in a different one at every step; no finiteness argument is available. What replaces finiteness is an argument about how the supporting hyperplanes of $A$ at boundary points near $a$ are oriented, and the paper gives it only as an informal geometric sketch. Making this step rigorous is the main work of the mission.

Numerical experiments show that termination can take tens of thousands of steps near sharp corners of a polygon, so no bound on the number of steps in terms of the distance of $p_0$ to $A$ alone can be expected.

Formalization scope

Space. $E_n$ is EuclideanSpace ℝ (Fin n). $A$ is a Set with four explicit hypotheses: A.Nonempty, IsClosed A, Convex ℝ A, Bornology.IsBounded A. Nonemptiness is implicit in the paper ("of dimension $r$ ", "the point of $A$ nearest to $p$ ").
The image as a relation. IsImage A p p₁ holds when $p_1 = p + 2(q - p)$ for a nearest point $q$ of $A$ ; the nearest point is not chosen by a function. For closed convex nonempty $A$ this relation is a function on $E_n$ . A run is any sequence with IsImage A (p ν) (p (ν+1)) whenever p ν ∉ A; values after entering $A$ are unconstrained. Every statement quantifies over every run from every $p_0 \notin A$ ; a formalization that only asserts the existence of some terminating run is ruled out.
Dimension. $L_r$ is affineSpan ℝ A; $r = n$ is affineSpan ℝ A = ⊤, $r < n$ is affineSpan ℝ A ≠ ⊤. No separate natural number $r$ is introduced.
Symmetry. IsSymmetricWrt L u v: midpoint ℝ u v ∈ L and u - v orthogonal to L.direction. For $r < n - 1$ this is a point reflection through the foot of the perpendicular, not a reflection in a hyperplane. The goal also requires $u \ne v$ and strict alternation $p_\nu \leftrightarrow p_{\nu+1}$ for $\nu > \nu_0$ (strict inequality, as printed).
Boundary is frontier A; distances to sets are Metric.infDist.
Generalizations recorded. Lemma 1, Case 1 is stated for an arbitrary set $A \subseteq E_n$ with full affine span (the paper states it for the polytope of (1.4) and applies it in §10 to a convex body). The projection milestone is stated for any flat $L \supseteq A$ , not only after the paper's reduction to $r = n - 1$ . The §9 milestone expresses "the reflexion process with respect to $F$ amounts to (3.1)" as: every farthest member of $F$ , with any nearest point on it, gives the same step.
Duplication. Case 1 appears both as a milestone and inside the goal, because the paper's proof of Case 2 cites Case 1. Solvers will need to transfer Case 1 from $E_n$ to the flat $L_r$ (an isometric copy of $E_r$ ).
Infrastructure. Mathlib's metric projection onto complete convex sets (exists_norm_eq_iInf_of_complete_convex, norm_eq_iInf_iff_real_inner_le_zero) and EuclideanGeometry.orthogonalProjection cover the basic geometry. Normal cones of convex sets and their upper semicontinuity are not in Mathlib; a contribution there is reusable well beyond this mission. Proofs of any milestone, and alternative proofs of Case 1, are welcome.

Selected references

T. S. Motzkin and I. J. Schoenberg, The relaxation method for linear inequalities, Canadian Journal of Mathematics 6 (1954), 393–404. https://doi.org/10.4153/CJM-1954-038-x
S. Agmon, The relaxation method for linear inequalities, Canadian Journal of Mathematics 6 (1954), 382–392 (the companion paper in the same issue).
L. Fejér, Über die Lage der Nullstellen von Polynomen, die aus Minimumforderungen gewisser Art entspringen, Mathematische Annalen 85 (1922), 41–48.

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