Continuity and Stability of a Quadratic Mixed-Integer Stochastic Program

Abstract

For the two-stage quadratic stochastic program where the second-stage problem is a general mixed-integer quadratic program with a random linear term in the objective function and random right-hand sides in constraints, we study continuity properties of the second-stage optimal value as a function of both the first-stage policy and the random parameter vector. We also present sufficient conditions for lower or upper semicontinuity, continuity, and Lipschitz continuity of the second-stage problem's optimal value function and the upper semicontinuity of the optimal solution set mapping with respect to the first-stage variables and/or the random parameter vector. These results then enable us to establish conclusions on the stability of optimal value and optimal solutions when the underlying probability distribution is perturbed with respect to the weak convergence of probability measures.