Robust Mixed 0-1 Programming and Submodularity

Abstract

The paper studies a robust mixed integer program with a single unrestricted continuous variable. The purpose of the paper is the polyhedral study of the robust solution set using submodularity. A submodular function is a set function with a diminishing returns property, and little work has been studied on the utilization of submodularity in the study of optimization problems considering data uncertainty. In this paper, we propose valid inequalities using submodularity. Valid inequalities for the robust mixed integer program are defined. A polynomial separation algorithm is proposed, and we show that the convex hull of the problem can be completely described using the proposed inequalities. In computational tests, we showed the proposed cuts are effective when they are applied to general robust discrete optimization problems with one or multiple constraints.