Robust Approximation to Chance-Constrained Binary Programming Using Non-Parametric Density Information

Abstract

A chance-constrained binary program (CCBP) is a general optimization problem over binary decision variables restricted by a chance constraint, which ensures that a constraint with uncertain coefficients can be violated only up to a given probability threshold. Despite its wide applications, the CCBP is challenging to solve due to its combinatorial nature and the involvement of its chance constraint. The existing solution methods for the CCBP with tractability guarantees are mainly extended from the methods proposed for problems with continuous decision variables and exploit the parametric information, such as mean and variance, of the uncertain coefficients. In this paper, we propose a general robust optimization framework for the development of solution methods with tractability guarantees for the CCBP, and then follow this framework to develop new solution methods for the CCBP. Unlike the existing solution methods, we exploit the binary characteristic of the decision variables and the non-parametric information about the density function of the uncertain coefficients to devise novel upper bounds on the violation probability of the chance constraint. Based on these upper bounds, we then derive two new robust optimization models to approximate the CCBP. Both models can be decomposed into binary programs, and one of them can be decomposed into nominal problems. Computational results show that our newly proposed solution methods produce significantly better solutions to the CCBP compared with existing methods.