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

TL;DR

This paper introduces a novel, faster column-and-constraint generation approach for two-stage robust mixed-integer programs with finite uncertainty sets, improving efficiency especially on computationally challenging second-stage problems.

Contribution

It combines existing speed-up techniques with new methods like dual bounds, adaptive time limits, and gap propagation to enhance solution speed and scalability.

Findings

Successfully solves larger instances within time limits.

Outperforms recent methods on a hard routing problem.

Proves applicability to problems with easier second stages.

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.