Recycling Inequalities for Robust Combinatorial Optimization with Budget Uncertainty

Abstract

Robust combinatorial optimization with budget uncertainty is one of the most popular approaches for integrating uncertainty in optimization problems. The existence of a compact reformulation for (mixed-integer) linear programs and positive complexity results give the impression that these problems are relatively easy to solve. However, the practical performance of the reformulation is actually quite poor when solving robust integer problems due to its weak linear relaxation.

To overcome the problems arising from the weak formulation, we propose a procedure to derive new classes of valid inequalities for robust binary optimization problems. For this, we recycle valid inequalities of the underlying deterministic problem such that the additional variables from the robust formulation are incorporated. The valid inequalities to be recycled may either be readily available model constraints or actual cutting planes, where we can benefit from decades of research on valid inequalities for classical optimization problems.

We first demonstrate the strength of the inequalities theoretically, by proving that recycling yields a facet-defining inequality in surprisingly many cases, even if the original valid inequality was not facet-defining. Afterwards, we show in a computational study that using recycled inequalities leads to a significant improvement of the computation time when solving robust optimization problems.