The discipline of applying mathematical analysis to complex decision problems in operations: allocating scarce resources, scheduling, routing, inventory, and the design of service and production systems. Drawing on mathematical programming, stochastic modeling, queueing, simulation, and game-theoretic reasoning, it seeks policies that perform provably well in systems shaped by constraints, congestion, and uncertainty.
Missions
🏆Completed
Captain: Shuze Chen
Bandit Algorithms III: Asymptotic and Minimax Optimality of UCBTextbook
The basic UCB regret bound of Mission II is logarithmic but not tight: its leading constant $16/\Delta_i$ is eight times the information-theoretic limit, and its worst-case rate carries a spurious $\sqrt{\log n}$. Chapters 8–9 of Lattimore–Szepesvári close both gaps. A refined confidence schedule $f(t) = 1 + t\log^2 t$ yields the asymptotically optimal $\limsup_{n\to\infty} R_n/\log n \le \sum_{i:\Delta_i>0} 2/\Delta_i$ — exactly matching the instance-dependent lower bound of Mission VII for Gaussian noise. The MOSS index $\hat\mu_i + \sqrt{\tfrac{4}{T_i}\log^+\!\big(\tfrac{n}{k T_i}\big)}$ achieves minimax regret $R_n \le 39\sqrt{kn} + \sum_i \Delta_i$, matching the $\Omega(\sqrt{kn})$ lower bound up to a constant. These two theorems are the gold standard for finite-armed stochastic bandits.
24 thms8 active usersReviewed
🏆Completed
Captain: Shuze Chen
Bandit Algorithms II: Stochastic Bandits and the UCB AlgorithmTextbook
A learner repeatedly chooses one of $k$ slot machines, observes only the reward of the chosen arm, and wants to earn almost as much as the best arm in hindsight. This is the stochastic multi-armed bandit, the canonical model of the exploration–exploitation dilemma. This mission formalizes the model (environments, policies, regret, and the regret decomposition $R_n = \sum_i \Delta_i\,\mathbb{E}[T_i(n)]$) and the two classical algorithms of Chapters 6–7 of Lattimore–Szepesvári: Explore-Then-Commit and the Upper Confidence Bound algorithm built on the optimism principle. The goal theorem is the instance-dependent UCB regret bound $R_n \le 3\sum_i \Delta_i + \sum_{i:\Delta_i>0} 16\log(n)/\Delta_i$ — logarithmic regret with explicit constants, the single most cited result of bandit theory — together with its distribution-free companion $R_n \le 8\sqrt{nk\log n} + 3\sum_i \Delta_i$.
27 thms8 active usersReviewed
🏆Completed
Captain: Shuze Chen
Bandit Algorithms V: Adversarial Bandits and Exp3Textbook
What if the rewards are not random at all, but chosen by an adversary who knows your algorithm? Remarkably, a randomized learner can still compete with the best fixed arm in hindsight. Chapters 11–12 of Lattimore–Szepesvári develop the adversarial $k$-armed bandit: rewards $x_{ti} \in [0,1]$ are an arbitrary fixed matrix, the learner samples $A_t \sim P_t$, and regret is measured against $\max_i \sum_t x_{ti}$. The exponential-weights algorithm Exp3, fed by importance-weighted loss estimates $\hat X_{ti} = 1 - \mathbb{1}\{A_t = i\}(1 - X_t)/P_{ti}$, achieves $R_n \le \sqrt{2nk\log k}$ — the goal theorem. The companion Exp3-IX, which deliberately biases its estimator, upgrades this to a bound holding with high probability rather than only in expectation. These results are the foundation of all adversarial online learning with partial feedback.
12 thms6 active usersReviewed
🏆Completed
Captain: Shuze Chen
Convex Optimization VI: Self-Concordance and the Barrier MethodTextbook
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
$$\Bigl\lceil \sqrt{m}\,\log_2\bigl(m/(t^{(0)}\varepsilon)\bigr)\Bigr\rceil$$
centering steps, each of uniformly bounded Newton cost.
19 thms5 active users
🏆Completed
Captain: Shuze Chen
Convex Optimization III: Conic Duality and the S-procedureTextbook
Two quadratic functions can be compared losslessly. The **S-procedure** says that, when the constraint is strictly feasible, the implication
$$q_1(x) \le 0 \;\Longrightarrow\; q_2(x) \le 0, \qquad q_k(x) = x^{T}F_k x + 2g_k^{T}x + h_k,$$
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.
17 thms5 active usersReviewed
🏆Completed
Captain: Shuze Chen
Convex Optimization I: Prékopa's TheoremTextbook
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.
29 thms5 active usersReviewed
🏆Completed
Captain: Shuze Chen
Bandit Algorithms XIV: Bayesian Bandits, the Gittins Index and Thompson SamplingTextbook
The oldest bandit algorithm (Thompson, 1933) is also the most modern: sample a parameter from the posterior and act greedily. Chapters 34–36 of Lattimore–Szepesvári develop the Bayesian view in two crowning results. The Gittins index theorem: for infinite-horizon discounted Markov bandits, the seemingly intractable dynamic program is solved *exactly* by an index policy — each arm gets a retirement-value index computable arm-by-arm, and playing the largest index is Bayesian optimal. And the frequentist analysis of Thompson sampling — the goal theorem: with Gaussian posteriors, Thompson sampling on 1-subgaussian bandits achieves $\lim_{n\to\infty} R_n/\log n = \sum_{i:\Delta_i>0} 2/\Delta_i$, exactly asymptotically optimal, alongside the minimax-grade $R_n \le C\sqrt{kn\log n}$. Together they explain why posterior sampling is both principled and practically dominant.
88 thms5 active usersReviewed
🏆Completed
Captain: Shuze Chen
Bandit Algorithms VII: Lower Bounds for Finite-Armed BanditsTextbook
How well can any algorithm possibly do? Chapters 13–17 of Lattimore–Szepesvári answer with three matching impossibility results. The divergence decomposition identifies the information a policy collects: $D(\mathbb{P}_{\nu\pi}, \mathbb{P}_{\nu'\pi}) = \sum_i \mathbb{E}[T_i(n)] D(P_i, P_i')$. Feeding it into the Bretagnolle–Huber inequality of Mission VI yields the goal theorem — the minimax lower bound $R_n \ge \frac{1}{27}\sqrt{(k-1)n}$ over Gaussian bandits, showing MOSS (Mission III) is optimal up to a constant. The same machinery gives the instance-dependent bound of Lai–Robbins type: every consistent policy suffers $\liminf_n R_n/\log n \ge \sum_{i:\Delta_i>0} \Delta_i / d_{\inf}(P_i, \mu^*, \mathcal{M}_i)$, certifying the asymptotic optimality of the UCB of Mission III and KL-UCB of Mission IV, and a high-probability lower bound showing the Exp3-IX guarantees of Mission V cannot be improved.
Every lower bound in bandit theory rests on one question: how hard is it to tell two probability measures apart from a sample? The answer is quantified by the relative entropy $D(P,Q)$, and the sharpest elementary tool is the Bretagnolle–Huber inequality: for any event $A$, $P(A) + Q(A^c) \ge \frac{1}{2}\exp(-D(P,Q))$ — no test can distinguish $P$ from $Q$ with total error probability below $\frac{1}{2}e^{-D(P,Q)}$. This mission formalizes Chapter 14 of Lattimore–Szepesvári: the Bretagnolle–Huber inequality (the goal theorem, proved via Le Cam's inequality $\int p \wedge q \ge \frac{1}{2}(\int\sqrt{pq})^2$), Pinsker's inequality $\delta(P,Q) \le \sqrt{D(P,Q)/2}$, and the closed-form divergences between Gaussians and Bernoullis. These half-page inequalities power every impossibility result in Missions VII, XI and beyond.
5 thms5 active usersReviewed
🏆Completed
Captain: Shuze Chen
Convex Optimization V: Newton's MethodTextbook
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),
$$\#\text{iterations} \;\le\; \frac{f(x^{(0)}) - p^{\star}}{\gamma} \;+\; \log_2\log_2(\varepsilon_0/\varepsilon), \qquad \gamma = \frac{\alpha\beta\eta^2 m}{M^2}, \quad \varepsilon_0 = \frac{2m^3}{L^2},$$
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.
12 thms4 active usersReviewed
🏆Completed
Captain: Shuze Chen
Convex Optimization II: KKT ConditionsTextbook
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.
15 thms4 active usersReviewed
🏆Completed
Captain: Shuze Chen
Introduction to Linear Optimization X: Max-Flow Min-CutTextbook
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 $$C(S)=\sum_{\{(i,j)\in\mathcal{A}\mid i\in S,\,j\notin S\}}u_{ij}$$ — the archetypal combinatorial min-max theorem, which the book notes can also be read as LP duality (pp. 311-312).
14 thms4 active usersReviewed
🏆Completed
Captain: Shuze Chen
Matrix Completion has No Spurious Local MinimumResearch Paper
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 $$f(X)=\min_X\frac12\|P_\Omega(M-XX^\top)\|_F^2+\lambda R(X)$$
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 $XX^\top = ZZ^\top$.
14 thms4 active users
🏆Completed
Captain: Shuze Chen
Bandit Algorithms XIII: Pure Exploration and Best-Arm IdentificationTextbook
Sometimes reward during learning is irrelevant — a pharmaceutical company running phase-II trials only cares about identifying the best treatment, as quickly and as reliably as possible. Chapter 33 of Lattimore–Szepesvári formalizes fixed-confidence best-arm identification: a policy together with a stopping time $\tau$ and a recommendation must be *sound* (wrong with probability at most $\delta$) while minimizing $\mathbb{E}[\tau]$. The information-theoretic complexity is $c^*(\nu)^{-1} = \sup_{\alpha\in\mathcal{P}_{k-1}} \inf_{\nu'\in\mathcal{E}_{alt}(\nu)} \sum_i \alpha_i D(\nu_i, \nu_i')$: every sound strategy needs $\mathbb{E}[\tau] \ge c^*(\nu)\log\frac{1}{4\delta}$, and the Track-and-Stop algorithm — the goal theorem — achieves $\lim_{\delta\to 0} \mathbb{E}[\tau]/\log(1/\delta) = c^*(\nu)$ exactly. The mission also covers the fixed-budget counterpart, sequential halving.
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,
$$x_0 + \tfrac{1}{n}\,(\mathcal{E} - x_0) \;\subseteq\; C \;\subseteq\; \mathcal{E},$$
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.
8 thms3 active usersReviewed
🏆Completed
Captain: Shuze Chen
Introduction to Linear Optimization XIII: Lagrangean Duality and Integer ProgrammingTextbook
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 $$Z_{IP} = \min\{c'x : Ax \ge b,\ Dx \ge d,\ x \text{ integer}\}$$ 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 $$Z(p) = \min_{x \in X}\big(c'x + p'(b - Ax)\big)$$ 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 $$\min\{c'x : Ax \ge b,\ x \in CH(X)\}$$. 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.
18 thms3 active usersReviewed
🏆Completed
Captain: tianyipeng
Markov Entanglement: Value Decomposition Error in Multi-agent MDPsResearch Paper
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.
25 thms3 active usersReviewed
🏆Completed
Captain: Shuze Chen
Introduction to Linear Optimization XII: Interior Point Methods and Path FollowingTextbook
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 $$B_\mu(\mathbf{x}) = \mathbf{c}'\mathbf{x} - \mu\sum_{j=1}^n \log x_j$$ 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 $$\|\frac{1}{\mu}XS\mathbf{e} - \mathbf{e}\| \le \beta$$. 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 $K$.
9 thms3 active usersReviewed
🏆Completed
Captain: Shuze Chen
Introduction to Linear Optimization XI: The Ellipsoid MethodTextbook
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 $$E(\mathbf{z}, D) = \{\mathbf{x} \in \mathbb{R}^n \mid (\mathbf{x}-\mathbf{z})'D^{-1}(\mathbf{x}-\mathbf{z}) \le 1\}$$ 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 $E' = E(\bar{\mathbf{z}}, \bar{D})$, $$\bar{\mathbf{z}} = \mathbf{z} + \frac{1}{n+1}\frac{D\mathbf{a}}{\sqrt{\mathbf{a}'D\mathbf{a}}},$$ $$\bar{D} = \frac{n^2}{n^2-1}\big(D - \frac{2}{n+1}\frac{D\mathbf{a}\mathbf{a}'D}{\mathbf{a}'D\mathbf{a}}\big),$$ and the volume contracts: $$\mathrm{Vol}(E') < e^{-1/(2(n+1))}\,\mathrm{Vol}(E)$$. 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.
14 thms3 active usersReviewed
🏆Completed
Captain: Shuze Chen
Introduction to Linear Optimization VIII: Sensitivity Analysis and Subgradients of the Optimal CostTextbook
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.
11 thms3 active usersReviewed
🏆Completed
Captain: Shuze Chen
Introduction to Linear Optimization VII: Cones, Extreme Rays, and the Resolution TheoremTextbook
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.
21 thms3 active usersReviewed
🏆Completed
Captain: Shuze Chen
Introduction to Linear Optimization IV: The Simplex MethodTextbook
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 $$\bar{c}_j = c_j - c_B'B^{-1}A_j$$ 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.
16 thms3 active usersReviewed
🏆Completed
Captain: Shuze Chen
Introduction to Linear Optimization I: Polyhedra and Basic Feasible SolutionsTextbook
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.
9 thms3 active usersReviewed
🏆Completed
Captain: Shuze Chen
Bandit Algorithms XVI: Markov Decision Processes and UCRL2Textbook
The final step from bandits to reinforcement learning: actions now change the state of the world. Chapter 38 of Lattimore–Szepesvári studies online learning in an unknown Markov decision process with $S$ states, $A$ actions and rewards in $[0,1]$. The optimism principle of Mission II scales up: UCRL2 maintains confidence sets over transition kernels, solves an *extended* MDP by extended value iteration, and recomputes only when a state-action count doubles. The goal theorem: with probability $1-\delta$, $\hat R_n < C\,D(M)\,S\sqrt{An\log(nSA/\delta)}$, where $D(M)$ is the diameter of the MDP — sublinear regret with no prior knowledge of the dynamics. The matching lower bound $\mathbb{E}[\hat R_n] \ge C'\sqrt{DSAn}$ brackets the true complexity of tabular reinforcement learning up to $\sqrt{DS}$.
32 thms3 active users
🏆Completed
Captain: Shuze Chen
Bandit Algorithms IV: Bernoulli Bandits and KL-UCBTextbook
When rewards are binary — a click or no click, a cure or no cure — the subgaussian machinery of Missions I–III is not tight: the variance of a Bernoulli arm degrades near the boundary of $[0,1]$, and the correct exponential rate is governed by the binary relative entropy $d(p,q) = p\log\frac{p}{q} + (1-p)\log\frac{1-p}{1-q}$ rather than a squared distance. Chapter 10 of Lattimore–Szepesvári develops Chernoff's tail bound in its information-theoretic form and the KL-UCB algorithm, whose upper confidence bounds are level sets of $d$. The goal theorem shows KL-UCB attains $$\limsup_{n\to\infty} R_n/\log n = \sum_{i:\Delta_i>0} \Delta_i/d(\mu_i, \mu^*)$$ — asymptotic optimality with exactly the constant demanded by the lower bound of Mission VII, strictly improving subgaussian UCB on every Bernoulli instance.