Integer Set Reduction for Stochastic Mixed-Integer Programming

Two-stage stochastic mixed-integer programming (SMIP) problems with general integer variables in the second-stage are generally difficult to solve. This paper develops the theory of integer set reduction for characterizing the subset of the convex hull of feasible integer points of the second-stage subproblem which can be used for solving the SMIP. The basic idea is to consider a small enough subset of feasible integer points that is necessary for generating a valid inequality for the integer subproblem. An algorithm for obtaining such a subset based on the solution of the subproblem LP-relaxation is then devised and incorporated into the Fenchel decomposition method for SMIP. To demonstrate the performance of the new integer set reduction methodology, a computational study based on randomly generated test instances was performed. The results of the study show that integer set reduction provides significant gains in terms of generating cuts faster leading to better bounds in solving SMIPs than using a direct solver.