Robust Counterparts of Inequalities Containing Sums of Maxima of Linear Functions

CentER Discussion Paper Series No. 2011-115

28 Pages Posted: 26 Oct 2011 Last revised: 8 Dec 2011

See all articles by Bram Gorissen

Bram Gorissen

Tilburg University - Tilburg University School of Economics and Management

Dick den Hertog

Tilburg University - Department of Econometrics & Operations Research

Date Written: October 25, 2011

Abstract

This paper addresses the robust counterparts of optimization problems containing sums of maxima of linear functions and proposes several reformulations. 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 that is exact when there is no uncertainty is used. 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 too conservative or their performance for diff erent uncertainty regions is not known, a comparison between them is not available, and they are restricted to specfi c problems. We provide techniques for general problems and compare them with numerical examples in inventory management, regression and brachytherapy. Based on these examples, we give tractable recommendations for reducing the conservatism.

Keywords: robust optimization, sum of maxima of linear functions, biffine uncertainty

JEL Classification: C61

Suggested Citation

Gorissen, Bram and den Hertog, Dick, Robust Counterparts of Inequalities Containing Sums of Maxima of Linear Functions (October 25, 2011). CentER Discussion Paper Series No. 2011-115. Available at SSRN: https://ssrn.com/abstract=1949752 or http://dx.doi.org/10.2139/ssrn.1949752

Bram Gorissen (Contact Author)

Tilburg University - Tilburg University School of Economics and Management ( email )

P.O. Box 90153
Tilburg, 5000 LE
Netherlands

Dick Den Hertog

Tilburg University - Department of Econometrics & Operations Research ( email )

Tilburg, 5000 LE
Netherlands

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