A fast approximate column-and-constraint generation method for two-stage robust mixed-integer programs

Abstract

This paper presents a new column-and-constraint generation method for two-stage robust mixed-integer programs with finite uncertainty sets. Our method combines and extends speed-up techniques used in previous column-and-constraint generation methods and introduces several new techniques. In particular, it uses dual bounds for second-stage problems in order to allow a faster identification of the next promising scenario to be added to the master problem. Moreover, adaptive time limits are imposed to avoid getting stuck on particularly hard second-stage problems, and a gap propagation between master problem and second-stage problems is used to stop solving them earlier if only a given non-zero optimality gap is to be reached overall. This makes our method particularly effective for problems where solving the second-stage problem is computationally challenging. To evaluate the method's performance, we compare it to two recent column-and-constraint generation methods from the literature on two applications: a robust capacitated location routing problem and a robust integrated berth allocation and quay crane assignment and scheduling problem. The first problem features a particularly hard second stage, and we show that our method is able to solve considerably more and larger instances in a given time limit. Using the second problem, we verify the general applicability of our method, even for problems where the second stage is relatively easy.