The OR Formalization Drive

Every linear programming problem asks to minimize a linear cost $c'x$ over a polyhedron — a set of the form $P = \{x \in \mathbb{R}^n \mid Ax \ge b\}$, or in standard form $\{x \mid Ax = b,\ x \ge 0\}$. Chapter 2 of Bertsimas–Tsitsiklis develops the geometry of these feasible sets, and its central achievement is making the intuitive notion of a "corner point" rigorous. There are three natural candidates: the extreme point — a point of $P$ that cannot be written as a convex combination of two other points of $P$ (purely geometric, representation-independent); the vertex — the unique minimizer of some linear cost $c'y$ over $P$ (geometric, via supporting hyperplanes); and the basic feasible solution — a feasible point at which $n$ linearly independent constraints are active (algebraic, the object the simplex method actually computes with). This mission formalizes polyhedra, active constraints, vertices and basic (feasible) solutions, and proves the fundamental Theorem 2.3: for a nonempty polyhedron all three notions coincide. Around the capstone sit the supporting pillars: polyhedra are convex (Theorem 2.1), the characterization of points pinned down by $n$ linearly independent active constraints (Theorem 2.2), finiteness of the set of basic solutions (Corollary 2.1), and the basis-column characterization of basic solutions in standard form (Theorem 2.4) — the combinatorial engine behind the simplex method of Chapter 3 and the root of the entire series.

Where should one look for the optimum of a linear programming problem? Chapter 1 of Bertsimas–Tsitsiklis suggests that optima "tend to occur at corners" of the feasible polyhedron; §§2.5–2.6 turn this intuition into theorems. Not every polyhedron has a corner — a halfspace in $\mathbb{R}^n$ ($n > 1$) has none — and the exact dividing line is the presence of an infinite line: a nonempty polyhedron

has an extreme point if and only if it does not contain a line, if and only if $n$ of the vectors $a_1, \dots, a_m$ are linearly independent (Theorem 2.6). In particular every nonempty bounded polyhedron and every nonempty standard-form polyhedron has a basic feasible solution (Corollary 2.2). The capstone, Theorem 2.8, is the sharpest form of the corner principle: if $P$ has at least one extreme point, then for any cost vector $c$ either the optimal cost is $-\infty$, or there is an extreme point of $P$ that is optimal — existence of an optimal solution comes for free once the cost is bounded below. Its companion Theorem 2.7 places an optimal extreme point under the weaker assumption that an optimal solution exists, and Corollary 2.3 — the fundamental theorem of linear programming — concludes that every feasible LP either has optimal cost $-\infty$ or attains an optimal solution, in stark contrast with nonlinear problems such as minimizing $1/x$ over $x \ge 1$. These results license the extreme-point search that the simplex method (Mission IV) performs.

Matrix completion — recovering a low-rank matrix $M = ZZ^\top$ from a small random subset of its entries — powers recommender systems and collaborative filtering. In practice it is solved by running (stochastic) gradient descent on the non-convex objective

where $\Omega=\{(i,j)|M_{i,j} \text{ is observed}\}$ and $R(X)$ is a certain regularizer. from a random starting point, and it just works.

Ge, Lee and Ma (NeurIPS 2016 Best student paper award) explained why: the regularized objective has no spurious local minima — every local minimum is global and exactly recovers $M$. This mission formalizes that landmark theorem in Lean 4, in its strongest known form and along its simplest known proof: the unified landscape analysis of Ge–Jin–Zheng (ICML 2017) and an improved sampling bound in Chen–Li (JMLR 2019). Conditional on an explicit good-sample predicate (which holds with high probability under Bernoulli sampling), every local minimum $X$ of $f$ satisfies $XX^\top = ZZ^\top$.

Chapter One of Jim Hefferon's Linear Algebra develops Gauss's method and asks what row reduction actually preserves. The answer arrives as the Linear Combination Lemma: row operations change the rows of a matrix but never the subspace those rows span, and that invariant is complete. The goal theorem is that completeness — two matrices are row equivalent exactly when they have the same row space — which is what makes reduced echelon form a genuine canonical form. The milestones are the two results the chapter builds on the way: that row operations leave a system's solution set alone, and that a solution set is always one particular solution translated by the solutions of the associated homogeneous system.

How does one actually solve a linear program? Chapter 2 showed that if a standard-form problem $\min c'x$ subject to $Ax = b$, $x \ge 0$ has an optimal solution, it has an optimal basic feasible solution; the simplex method searches among basic feasible solutions, moving along edges of the feasible set in cost-reducing directions. This mission formalizes the mathematics of Chapter 3 of Bertsimas–Tsitsiklis: feasible directions, the reduced costs

measuring the cost rate along the basic directions, the optimality conditions of Theorem 3.1 ($\bar{c} \ge 0$ implies optimality, and conversely at nondegenerate optima), the basis change of Theorem 3.2, and the pivot iteration itself — encoded as a predicate relating a basis/BFS pair to its successor, so that every theorem covers every pivoting rule. The goal theorem is Theorem 3.3: if the feasible set is nonempty and every basic feasible solution is nondegenerate, the simplex method terminates after a finite number of iterations, ending either with an optimal basis and an associated optimal basic feasible solution, or with a direction $d$ satisfying $Ad = 0$, $d \ge 0$, $c'd < 0$ certifying optimal cost $-\infty$. The secondary capstone, Theorem 3.4, removes the nondegeneracy assumption: under the lexicographic pivoting rule every tableau row other than the zeroth stays lexicographically positive, the zeroth row strictly increases lexicographically, and the simplex method terminates on every problem — the anticycling guarantee that also supplies the optimal-basis existence used by the strong duality theorem of Mission V.

Every linear programming problem has a shadow. To the primal $\min c'x$ we associate the dual $\max p'b$, whose variables price the primal constraints: one dual variable per primal constraint and one dual constraint per primal variable, with signs governed by the correspondence of Table 4.1. This mission formalizes §4.1–4.5 of Bertsimas–Tsitsiklis: the dual of a general-form linear program, the involution "the dual of the dual is the primal" (Theorem 4.1), and weak duality $p'b \le c'x$ for any primal-feasible $x$ and dual-feasible $p$ (Theorem 4.3) with its two corollaries — an unbounded primal forces an infeasible dual (Corollary 4.1), and feasible $x, p$ with $p'b = c'x$ are automatically both optimal (Corollary 4.2). The goal theorem is strong duality (Theorem 4.4): if a linear programming problem has an optimal solution, so does its dual, and the respective optimal costs are equal — proved in the book by running the simplex method with the lexicographic pivoting rule of Mission IV on a standard-form transform. The statement is deliberately the book's attainment form: by Table 4.2 the primal and the dual can be simultaneously infeasible (Example 4.5), so an unguarded equality of optimal values is false. The mission closes with complementary slackness (Theorem 4.5): feasible $x$ and $p$ are simultaneously optimal if and only if $p_i(a_i'x - b_i) = 0$ for all $i$ and $(c_j - p'A_j)x_j = 0$ for all $j$ — the certificate structure behind the dual simplex method and every LP optimality check.

When is a system of linear constraints infeasible? Sections 4.6-4.7 of Bertsimas-Tsitsiklis answer with the archetypal theorem of the alternative. The capstone is Farkas' lemma (Theorem 4.6): for an $m \times n$ matrix $A$ and $b \in \mathbb{R}^m$, exactly one of the following holds — (a) some $x \ge 0$ satisfies $Ax = b$, or (b) some $p$ satisfies $p'A \ge 0'$ and $p'b < 0$; such a $p$ is a certificate of infeasibility, geometrically a hyperplane separating $b$ from the cone of the columns of $A$. The mission also carries the cone-membership restatement (Corollary 4.3), the inequality form (Theorem 4.7: every solution of $Ax \le b$ satisfies $c'x \le d$ iff some $p \ge 0$ has $p'A = c'$ and $p'b \le d$), and the application to asset pricing (Theorem 4.8: a market's prices admit no arbitrage iff there is a nonnegative state-price vector $q$ with $p_i = \sum_s q_s r_{si}$). The book proves Farkas' lemma from LP strong duality; Section 4.7 then reverses the arrow from first principles: every polyhedron is closed (Theorem 4.9), Weierstrass' theorem (Theorem 4.10, already in Mathlib), and the separating hyperplane theorem (Theorem 4.11: for nonempty closed convex $S$ and $x^* \notin S$ there exists $c$ with $c'x^* < c'x$ for all $x \in S$), from which Farkas' lemma — and hence the duality theorem itself — follows geometrically.

How can an unbounded polyhedron be described by finitely many geometric objects? Sections 4.8-4.9 of Bertsimas-Tsitsiklis build the cone machinery: recession cones $\{d \mid Ad \ge 0\}$ and their rays, extreme rays (defined, like basic solutions, by $n-1$ linearly independent active constraints), the pointedness criterion (Theorem 4.12: $0$ is an extreme point of a polyhedral cone iff the cone contains no line iff $n$ of the constraint vectors are linearly independent), and the characterization of unbounded linear programs (Theorems 4.13-4.14: over a pointed polyhedral cone, and then over any polyhedron with an extreme point, the optimal cost is $-\infty$ iff some extreme ray $d$ has $c'd < 0$). The capstone is the resolution theorem (Theorem 4.15): a nonempty polyhedron $P$ with at least one extreme point equals $Q = \{\sum_i \lambda_i x^i + \sum_j \theta_j w^j \mid \lambda_i \ge 0, \theta_j \ge 0, \sum_i \lambda_i = 1\}$ — the convex hull of its extreme points plus the cone generated by a complete set of its extreme rays. It specializes to Theorem 2.9 / Corollary 4.4 (a nonempty bounded polyhedron is the convex hull of its extreme points) and Corollary 4.5 (a pointed polyhedral cone is generated by its extreme rays). The converse, Theorem 4.16, states that every finitely generated set is a polyhedron — in particular the convex hull of finitely many vectors is a polyhedron. Together these form the Minkowski-Weyl equivalence of the two representations of polyhedra, verified absent from Mathlib and the genuine content of this mission.

How does the optimal cost of a linear program respond when the problem data change? Chapter 5 of Bertsimas-Tsitsiklis studies the standard form problem $\min\{c'x \mid Ax = b,\ x \ge 0\}$ (rows of $A$ linearly independent) as the requirement vector $b$ and the cost vector $c$ vary. On the convex set $S = \{b \mid P(b) \neq \emptyset\}$ of feasible right-hand sides, and under the standing assumption that the dual feasible set is nonempty, the optimal cost $F(b)$ is finite and convex (Theorem 5.1) — indeed $F(b) = \max_{i} (p^i)'b$ over the extreme points $p^1, \dots, p^N$ of the dual feasible set, a piecewise linear convex function whose breakpoints are exactly where the dual optimum is non-unique. The capstone (Theorem 5.2) identifies the generalized gradients of $F$: if the primal at $b^*$ is feasible with finite optimal cost, then $p$ is an optimal solution of the dual if and only if $p$ is a subgradient of $F$ at $b^*$ (Definition 5.1: $F(b^*) + p'(b - b^*) \le F(b)$ for all $b \in S$) — the precise sense in which dual variables are marginal costs. Dually (Theorem 5.3), the set $T$ of cost vectors with finite optimal cost is convex, the optimal cost $G(c)$ is concave on $T$, and near any $c$ with a unique primal optimum $x^*$, $G$ is linear with gradient $x^*$. Local ranging (Section 5.1) and parametric programming (Section 5.5) are the procedural companions, folded into the design notes.

How much flow can be sent from a source $s$ to a sink $t$ through a network with arc capacities $u_{ij}\in(0,\infty]$ — and what certifies that no more is possible? This mission formalizes §7.4-7.5 of Bertsimas & Tsitsiklis. The circulation calculus of §7.4 supplies the two structural tools: the flow decomposition theorem (Lemma 7.1 — every nonzero nonnegative circulation is a positive combination $\mathbf{f}=\sum_i a_i\mathbf{f}^i$ of simple circulations with only forward arcs, with integer $a_i$ when $\mathbf{f}$ is integer) and the optimality criterion for the minimum cost network flow problem (Theorem 7.6 — a feasible flow is optimal if and only if there is no unsaturated cycle with negative cost). Section 7.5 then formulates the maximum flow problem ($\max b_s$ s.t. $\mathbf{A}\mathbf{f}=\mathbf{b}$, $b_t=-b_s$, $b_i=0$ for $i\ne s,t$, $0\le\mathbf{f}\le\mathbf{u}$), defines augmenting paths (Definition 7.2: $f_{ij}<u_{ij}$ on forward arcs, $f_{ij}>0$ on backward arcs) and the Ford–Fulkerson algorithm, and proves integer invariance and finite termination for integer capacities (Theorem 7.8). The goal is Theorem 7.10:

(a) if the Ford–Fulkerson algorithm terminates because no augmenting path can be found, the current flow is optimal;

(b) the value of the maximum flow equals the minimum cut capacity

— the archetypal combinatorial min-max theorem, which the book notes can also be read as LP duality (pp. 311-312).

Can the feasibility of a system of linear inequalities be decided in a provably small number of iterations? The ellipsoid method — the algorithm with which Khachiyan showed in 1979 that linear programming is polynomially solvable — answers this with pure convex geometry. This mission formalizes Chapter 8 of Bertsimas–Tsitsiklis. An ellipsoid is

with $D$ symmetric positive definite. The geometric engine is Theorem 8.1: the half-ellipsoid $E \cap \{\mathbf{x} \mid \mathbf{a}'\mathbf{x} \ge \mathbf{a}'\mathbf{z}\}$ is contained in the explicitly constructed ellipsoid $E' = E(\bar{\mathbf{z}}, \bar{D})$,

and the volume contracts:

. Two integer-data estimates make the contraction decisive: every extreme point of $P = \{\mathbf{x} \mid A\mathbf{x} \ge \mathbf{b}\}$ with entries bounded by $U$ has coordinates in $[-(nU)^n, (nU)^n]$ (Lemma 8.2), and a full-dimensional bounded such polyhedron has $\mathrm{Vol}(P) > n^{-n}(nU)^{-n^2(n+1)}$ (Lemma 8.4). The goal theorem is Theorem 8.2: started on a ball $E(\mathbf{x}_0, r^2 I)$ of volume at most $V$ containing $P$, with $v$ a lower bound on $\mathrm{Vol}(P)$ when $P$ is nonempty, the ellipsoid method correctly decides whether $P$ is empty within $t^* = \lceil 2(n+1)\log(V/v) \rceil$ iterations — the explicit iteration count behind the polynomial-time headline.

Interior point methods solve linear programs by moving through the interior of the feasible set instead of along its edges — the approach that turned Karmarkar's 1984 breakthrough into today's practical large-scale solvers. This mission formalizes the primal path following algorithm of Chapter 9 of Bertsimas–Tsitsiklis. For $\mu > 0$ the logarithmic barrier

replaces the constraint $\mathbf{x} \ge \mathbf{0}$; the minimizers $\mathbf{x}(\mu)$ of $B_\mu$ over $\{A\mathbf{x} = \mathbf{b}\}$ trace the central path, characterized by the KKT conditions (9.17): $A\mathbf{x} = \mathbf{b}$, $\mathbf{x} \ge \mathbf{0}$, $A'\mathbf{p} + \mathbf{s} = \mathbf{c}$, $\mathbf{s} \ge \mathbf{0}$, $XS\mathbf{e} = \mu\mathbf{e}$ (Lemma 9.5). The algorithm follows the path with one Newton step of the barrier problem per shrink $\mu^{k+1} = \alpha\mu^k$, maintaining the proximity invariant

. The goal theorem is Theorem 9.7: with $\alpha = 1 - \frac{\sqrt{\beta}-\beta}{\sqrt{\beta}+\sqrt{n}}$ and a $\beta$-close start, after $K = \Big\lceil \frac{\sqrt{\beta}+\sqrt{n}}{\sqrt{\beta}-\beta}\,\log\frac{(\mathbf{s}^0)'\mathbf{x}^0(1+\beta)}{\varepsilon(1-\beta)} \Big\rceil$ iterations the algorithm reaches primal and dual feasible solutions with duality gap $(\mathbf{s}^K)'\mathbf{x}^K \le \varepsilon$ — the explicit form of the celebrated $O(\sqrt{n}\log(1/\varepsilon))$ iteration bound. Alongside it we formalize the generic potential-reduction scheme (Theorem 9.4): any algorithm cutting $G(\mathbf{x},\mathbf{s}) = q\log\mathbf{s}'\mathbf{x} - \sum_j \log x_j - \sum_j \log s_j$ by $\delta$ per step reaches gap $\varepsilon$ within an explicit $K$.

Value decomposition — approximating the value of a joint state by a sum of per-agent local values — is a staple of multi-agent dynamic programming and reinforcement learning, from index policies for restless bandits to modern MARL architectures, yet it is normally used without justification. Chen and Peng (arXiv:2506.02385) supply one. They show a multi-agent MDP admits an exact value decomposition precisely when its transition matrix is not entangled — a notion built in direct analogy with quantum entanglement — and then turn that qualitative characterisation into a quantitative one: a measure of Markov entanglement bounds the decomposition error in general. This mission formalizes that core theory. The goal is Theorem 6, the general N-agent bound in the occupancy-weighted norm; the milestones are the equivalence between separability and exact decomposition, the perturbation machinery that carries a one-step transition error into a value-function error, and the extensions to shared global state and shared rewards. The paper's restless-bandit application, which needs mean-field machinery of its own, is left to a second mission in the series.

Linear programming has a complete duality theory; integer programming does not — and the Lagrangean dual measures exactly how far duality reaches. This mission formalizes the duality theory of integer programming from Section 11.4 of Bertsimas–Tsitsiklis, built on the general linear programming duality of Section 4.10. For the integer program

with integer data, the complicating constraints $Ax \ge b$ are dualized with multipliers $p \ge 0$ over the tractable set $X = \{x \text{ integer} \mid Dx \ge d\}$: the dual function is

and the Lagrangean dual is $Z_D = \max_{p \ge 0} Z(p)$. Weak duality $Z_D \le Z_{IP}$ (Theorem 11.2) always holds, but strong duality can fail. The convex hull $CH(X)$ of the integer points of a polyhedron with integer data is itself a polyhedron (Theorem 11.3, Meyer's theorem), and the capstone — Theorem 11.4, the central result of Section 11.4 — identifies the Lagrangean dual exactly: $Z_D$ equals the optimal cost of the linear program

. This is the geometric explanation of the strength of Lagrangean relaxation, yields the bound ordering $Z_{LP} \le Z_D \le Z_{IP}$, and Corollary 11.1 characterizes exactly when the bounds collapse. The polyhedral engine is the general weak/strong duality pair (Theorems 4.17/4.18) over a primal $\min c'x$ s.t. $Ax \ge b$, $x \in P = \{x \mid Dx \ge d\}$, and the formulation-strength comparison $P_{sub} \subseteq P_{cut}$ of Theorem 10.1 supplies the motivating principle that tighter relaxations of the same integer set give sharper bounds.

Log-concave functions are the meeting point of convex analysis and probability: densities of Gaussian, exponential, uniform and Wishart distributions are all log-concave, and countless facts of applied probability flow from one structural theorem — integrating out variables preserves log-concavity. This mission builds the convex-analysis spine of Boyd & Vandenberghe's Convex Optimization (Chapters 2–3) — separation and supporting hyperplanes, dual cones, the first- and second-order differential characterizations of convexity, Fenchel conjugacy — and climbs to Prékopa's theorem via the Prékopa–Leindler inequality, a landmark of Brunn–Minkowski theory absent from Mathlib.

The Karush–Kuhn–Tucker conditions are the central result of convex optimization: for a convex differentiable problem satisfying Slater's condition, a point is optimal exactly when primal feasibility, dual feasibility, complementary slackness and Lagrangian stationarity hold. This mission formalizes Chapters 4–5 of Boyd & Vandenberghe end to end — the first-order optimality criterion, concavity of the Lagrange dual, weak duality, Slater's strong-duality theorem with dual attainment (via the separating-hyperplane argument of §5.3.2), the saddle-point characterization, sensitivity bounds and Pareto scalarization — culminating in the full KKT characterization.

Every full-dimensional convex body is sandwiched between an ellipsoid and its $n$-fold dilation: shrinking the minimum-volume covering (Löwner–John) ellipsoid $\mathcal{E}$ about its centre $x_0$ by the factor $1/n$ lands inside the body,

and the factor $n$ is tight on simplices. This rounding theorem underlies the ellipsoid method, John's theorem on the Banach–Mazur distance to the Euclidean ball, and much of modern convex geometry. The mission formalizes §8.4 of Boyd & Vandenberghe for polytopes $C = \operatorname{conv}\{x_1,\dots,x_m\}$, exactly as the book proves it: existence and uniqueness of the extremal ellipsoid, the KKT identities at the normalized optimum ($\sum_i \lambda_i x_i x_i^{T} = I$, $\sum_i \lambda_i x_i = 0$, $\sum_i \lambda_i = n$), the convex-combination step that produces the $1/n$ ball, and affine invariance.

The classical convergence theory of smooth convex minimization. For a function that is $m$-strongly convex and $M$-smooth ($mI \preceq \nabla^2 f(x) \preceq MI$), gradient descent converges linearly, while Newton's method exhibits its famous two phases: a damped phase in which every backtracking step decreases the objective by a fixed amount $\gamma$, and a quadratically convergent phase in which the scaled gradient norm squares at each step, $\tfrac{L}{2m^2}\lVert \nabla f(x^{+})\rVert_2 \le \bigl(\tfrac{L}{2m^2}\lVert \nabla f(x)\rVert_2\bigr)^2$. Together they give the iteration count of B&V (9.36),

with $L$ the Lipschitz constant of the Hessian and $\alpha,\beta$ the backtracking parameters. This mission formalizes Chapters 9–10 of Boyd & Vandenberghe with every constant exactly as printed — a quantitative theory entirely absent from Mathlib.

Two quadratic functions can be compared losslessly. The S-procedure says that, when the constraint is strictly feasible, the implication

holds if and only if a single nonnegative multiplier certifies it as a matrix inequality, $\lambda \begin{bmatrix} F_1 & g_1 \\ g_1^{T} & h_1\end{bmatrix} \succeq \begin{bmatrix} F_2 & g_2 \\ g_2^{T} & h_2\end{bmatrix}$ for some $\lambda \ge 0$. It is a cornerstone of control theory, trust-region methods and robust optimization, and a rare case in which a nonconvex problem has zero duality gap. The route runs through the theory this mission builds from Boyd & Vandenberghe §5.8–5.9 and Appendix B: strong alternatives for convex inequality systems, cone-program strong duality under a generalized Slater condition, semidefinite programming duality, the LMI theorems of alternatives, and the hidden convexity of the joint range of two quadratic forms.

Why do interior-point methods solve convex programs in $O(\sqrt{m}\log(1/\varepsilon))$ Newton steps? Nesterov and Nemirovskii's answer is self-concordance: a convex function whose third derivative is controlled by its second, $|\varphi'''(t)| \le 2\,\varphi''(t)^{3/2}$ along every line, admits a Newton analysis with absolute constants and no condition number — and the logarithmic barrier is self-concordant. This mission formalizes §9.6 and Chapter 11 of Boyd & Vandenberghe: the self-concordance calculus, the Newton-decrement analysis, the duality gap $m/t$ along the central path, the per-centering work bound $m(\mu - 1 - \log\mu)/\gamma + c$, and the crown result — with the aggressive schedule $\mu = 1 + 1/\sqrt{m}$ the barrier method reaches duality gap $\varepsilon$ after

centering steps, each of uniformly bounded Newton cost.