Convergence Analysis for Distributionally Robust Optimization and Equilibrium Problems

Abstract

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.

A distributionally robust perspective on uncertainty quantification and chance constrained programming

“…Ambiguity sets of special interest include the Markov ambiguity set containing all distributions with known mean and support [48], the Chebyshev ambiguity set containing all distributions with known bounds on the first and second-order moments [12,14,22,31,39,46,49,51,52], the Gauss ambiguity set containing all unimodal distributions from within the Chebyshev ambiguity set [38,41], various generalized Chebyshev ambiguity sets that specify asymmetric moments [12,13,35], higher-order moments [7,30,45] or marginal moments [17,18], the median-absolute deviation ambiguity set containing all symmetric distributions with known median and mean absolute deviation [24], the Huber ambiguity set containing all distributions with known upper bound on the expected Huber loss function [15,48], the Wasserstein ambiguity set containing all distributions that are close to the empirical distribution with respect to the Wasserstein metric [19,34,40], the KullbackLeibler divergence ambiguity set and likelihood ratio ambiguity set [10,26,27,31,47] containing all distributions that are sufficiently likely to have generated a given data set, the Hoeffding ambiguity set containing all component-wise independent distributions with a box support [3,8,10], the Bernstein ambiguity set containing all distributions from within the Hoeffding ambiguity set subject to marginal moment bounds [36], several φ-divergence-based ambiguity sets [2,…”

Convergence Analysis for Mathematical Programs with Distributionally Robust Chance Constraint

“…Lemma 2.1 is given in [42,Lemma 1]. It is noted that the conclusion is drawn from [7, Theorem 3.5] (or Theorem 5.4 in an earlier version of the book).…”

“…It is noted that the conclusion is drawn from [7, Theorem 3.5] (or Theorem 5.4 in an earlier version of the book). Here we provide a proof in the appendix in that no proof is given in [42] and there are indeed some subtle arguments behind the claim.…”

“…By Prokhorov's theorem, P is weakly compact (see [4] and [42,Proposition 7] for a more recent discussions in this regard).…”

“…Indeed, when P reduces to a singleton, MPDRCC collapses to MPCC and our analysis coincides with classical stability analysis of MPCC. The research can also be viewed as an extension of convergence analysis of distributionally robust formulation of a one stage stochastic program by Sun and Xu [42] where the impact of the optimal value and the optimal solutions is investigated against variation of the underlying ambiguity set.…”