We consider the situation that the grand set of all players, in a cooperative game in characteristic function form with a nonempty core, consists of recognizable subsets of players. The underlying assumption is that the players of each subset have something in common between themselves which is not shared by the players of other subsets. … Read more
Kuhn-Tucker (KT) demand models characterize consumers’ budget allocation through the Karush-Kuhn-Tucker conditions of utility maximization. Classical specifications, including quadratic utility, are widely used to study pricing, assortment, and competition, but remain difficult to estimate from consumer data. By contrast, models designed for tractable estimation often impose restrictive assumptions on demand behavior. We address this gap … Read more
The infimal displacement vector of the nonexpansive operator underlying the primal-dual hybrid gradient algorithm (PDHG) is known to encode useful information for infeasible optimization problems. In particular, for the ellipsoid separation problem (ESP), where the goal is to find a separating hyperplane between two collections of ellipsoids, the infimal displacement vector corresponding to an infeasible … Read more
We consider a network of finitely many agents aiming to cooperatively solve a convex-concave min-max problem combining nonsmooth terms with a smooth saddle function whose partial gradients are only locally Lipschitz continuous. Without requiring prior knowledge of the local Lipschitz moduli of the partial gradients, we propose a distributed proximal-reflected-gradient algorithm which incorporates a distributed … Read more
We study decentralized minimization of sums of smooth strongly convex functions, one per agent, with communication restricted to neighbors in a graph. Existing decentralized algorithms typically emulate a {\it centralized} method; {\it the network serves only to exchange information}, from which agents approximate the global quantities that the method needs. This paper breaks with this … Read more
We introduce a framework that transforms discrete optimization into continuous optimization over parameterized probability distributions. The framework reveals a common distributional structure behind several methods that have largely developed independently, including continuous reformulations for multilinear binary optimization, Markov decision processes and reinforcement learning, the cross-entropy method, mean-field variational optimization, and the quantum approximate optimization algorithm. … Read more
Stochastic Gradient Descent (\(\textsf{SGD}\)) is one of the most classical optimization algorithms with favorable theoretical guarantees, yet its practical implementation differs subtly from its well-known form and is often referred to as Shuffling Stochastic Gradient Descent (\(\textsf{Shuffling SGD}\)). A particularly popular strategy in \(\textsf{Shuffling SGD}\) is Random Reshuffling (\(\textsf{RR}\)), which has achieved great empirical success. … Read more
We show that the Bregman Douglas–Rachford splitting method (BDRS) can fail for matrix scaling and unregularized optimal transport. For matrix scaling, we construct a strictly positive \(36\times6\) integer matrix, positive rational marginals of equal mass, and a positive auxiliary initialization for which the matrix iterates enter a nonconstant six-cycle after one iteration. The construction prescribes … Read more
This paper presents Peppy, an AI-assisted workflow for discovering tight, analytic convergence proofs for first-order optimization algorithms. Generic approaches to using LLMs to conduct mathematical research target an unspecified, broad spectrum of problems and sometimes use the Lean 4 proof assistant for formalization. On the other hand, Peppy leverages domain-specific knowledge more heavily and is … Read more
We study nonsmooth convex-concave saddle-point problems over compact convex sets, assuming access to stochastic subgradients of the payoff function. We develop single-loop projection-free algorithms that use linear minimization oracles over the primal and dual domains. Unlike prior projection-free approaches that rely on smoothing, our methods are purely subgradient-based and handle nonsmoothness directly. This design makes … Read more