The point-based robustness gap for uncertain multiobjective optimization

Abstract

In robust single-objective optimization, the robustness gap is a measure of the distance between the robust optimal objective value and the optimal objective values of the scenarios. While robust multiobjective optimization is a growing field of study, no notion of a robustness gap has been proposed. A concept of a point-based robustness gap for uncertain multiobjective optimization problems is introduced. The gap is defined as the minimal distance between the robust Pareto set and the Pareto sets of the scenarios. It is shown that the gap is zero whenever the uncertainty is constraint-wise and objective-wise, supplementing a major result about the single-objective robustness gap. Because the distance between Pareto sets is hard to compute, lower and upper bounds on the gap are constructed for convex problems. Specific results about the zero gap and the bounds are presented for linear problems. Numerical examples are included.

Keywords:

• Multi-criteria programming
• robust optimization
• efficient solution
• pareto outcome
• minmax optimization

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The first author has been supported by grant DFG RTG 1703 ‘Resource Efficiency in Interorganizational Networks’; the fourth author has partially been supported by grant N00014-16-1-2725 funded by the United States Office of Naval Research.