Robust counterparts of inequalities containing sums of maxima of linear functions

TL;DR

This paper develops techniques to accurately reformulate robust optimization problems involving sums of maxima of linear functions, reducing conservatism and improving solution quality across various applications.

Contribution

It introduces general methods for reformulating these problems and compares their performance, providing practical recommendations to reduce conservatism in robust optimization.

Findings

Reformulation techniques can significantly reduce conservatism.

Numerical examples demonstrate improved solutions in inventory, regression, and brachytherapy.

Recommendations help practitioners choose less conservative approaches.

Abstract

This paper addresses the robust counterparts of optimization problems containing sums of maxima of linear functions. These problems include many practical problems, e.g.~problems with sums of absolute values, and arise when taking the robust counterpart of a linear inequality that is affine in the decision variables, affine in a parameter with box uncertainty, and affine in a parameter with general uncertainty. In the literature, often the reformulation is used that is exact when there is no uncertainty. However, in robust optimization this reformulation gives an inferior solution and provides a pessimistic view. We observe that in many papers this conservatism is not mentioned. Some papers have recognized this problem, but existing solutions are either conservative or their performance for different uncertainty regions is not known, a comparison between them is not available, and they are restricted to specific problems. We describe techniques for general problems and compare them with numerical examples in inventory management, regression and brachytherapy. Based on these examples, we give recommendations for reducing the conservatism.