Journal Article

Convergence Analysis for Distributionally Robust Optimization and Equilibrium Problems

Xu, Huifu, Sun, Hailin

2016

Overview

In this paper, we study distributionally robust optimization approaches for a one-stage stochastic minimization problem, where the true distribution of the underlying random variables is unknown but it is possible to construct a set of probability distributions, which contains the true distribution and optimal decision is taken on the basis of the worst-possible distribution from that set. We consider the case when the distributional set (which is also known as the ambiguity set) varies and its impact on the optimal value and the optimal solutions. A typical example is when the ambiguity set is constructed through samples and we need to look into the impact of increasing the sample size. The analysis provides a unified framework for convergence of some problems where the ambiguity set is approximated in a process with increasing information on uncertainty and extends the classical convergence analysis in stochastic programming. The discussion is extended briefly to a stochastic Nash equilibrium problem where each player takes a robust action on the basis of the worst subjective expected objective values.

• Publisher: INFORMS,Institute for Operations Research and the Management Sciences
• Subject: Convergence (Mathematics) / convergence analysis / distributionally robust minimization / Game theory / Hoffman’s lemma / Mathematical optimization / Mathematical research / Nash equilibrium / Optimization techniques / Probability distribution / pseudometric / Random variables / robust Nash equilibrium / Stochastic models / Studies / total variation metric