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
We study nonconvex stochastic functions that admit a structured second-order representation, a generalization of lower- and upper-C² functions to normal integrands. Similarly to the deterministic case, lower-C² normal integrands are defined as the pointwise maxima over a compact random index set of functions that are twice-continuously differentiable in the decision variable with an integrable bounded … 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
A stochastic proximal-gradient method is proposed and analyzed for solving regularized optimization problems subject to nonlinear equality constraints and nonnegativity bounds. The objective function’s values and gradients are available only through inexact stochastic oracles. The method decomposes each trial step into one component that reduces linearized infeasibility and a second component that seeks optimality while … 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
This paper proposes a simple facet enumeration algorithm for convex polytopes. The complexity of the algorithm is analyzed. The algorithm is implemented in Matlab. Some polytopes with known H-representations and V-representations are used as test examples. The preliminary numerical test shows the effectiveness and efficiency of the proposed algorithm. Due to the duality between the … Read more
We propose robust and distributionally robust formulations of the risk budgeting problem under the mean absolute deviation (MAD) risk measure. The proposed models address uncertainty in asset returns and their distribution while allocating portfolio risk according to prescribed risk budgets. We first develop a robust model in which historical return observations are perturbed within a … Read more