Integer set reduction for stochastic mixed-integer programming

Abstract

Two-stage stochastic mixed-integer programs (SMIPs) with general integer variables in the second-stage are generally difficult to solve. This paper develops the theory of integer set reduction for characterizing a subset of the convex hull of feasible integer points of the second-stage subproblem which can be used for solving the SMIP with pure integer recourse. The basic idea is to use the smallest possible subset of the subproblem feasible integer set to generate a valid inequality like Fenchel decomposition cuts with a goal of reducing computation time. An algorithm for obtaining such a subset based on the solution of the subproblem linear programming relaxation is devised and incorporated into a decomposition method for SMIP. To demonstrate the performance of the new integer set reduction methodology, a computational study based on randomly generated knapsack test instances was performed. The results of the study show that integer set reduction aids in speeding up cut generation, leading to better bounds in solving SMIPs with pure integer recourse than using a direct solver.