Quantitative Stability Analysis for Distributionally Robust Optimization with Moment Constraints

In this paper we consider a broad class of distributionally robust optimization (DRO for short) problems where the probability of the underlying random variables depends on the decision variables and the ambiguity set is defined through parametric moment conditions with generic cone constraints. Under some moderate conditions including Slater type conditions of cone constrained moment system and Hölder continuity of the underlying random functions in the objective and moment conditions, we show local Hölder continuity of the optimal value function of the inner maximization problem w.r.t. the decision vector and other parameters in moment conditions, local Hölder continuity of the optimal value of the whole minimax DRO w.r.t the parameter. Moreover, under the second order growth condition of the Lagrange dual of the inner maximization problem, we demonstrate and quantify the outer semicontinuity of the set of optimal solutions of the minimax DRO w.r.t variation of the parameter. Finally we apply the established stability results to two particular class of DRO problems.