Necessary and sufficient conditions for Pareto efficiency in robust multiobjective optimization

TL;DR

This paper establishes precise conditions under which solutions are robustly Pareto efficient in multiobjective optimization with uncertainty, linking scalarizing functions' properties to solution optimality.

Contribution

It provides necessary and sufficient conditions for robust efficiency based on scalarizing functions, extending deterministic efficiency concepts to uncertain multiobjective problems.

Findings

Robust efficiency corresponds to optimality in strongly increasing scalarizing functions.

Optimality in strictly increasing scalarizing functions is necessary for robustness.

The curvature of scalarizing functions affects solution conservatism in uncertain optimization.

Abstract

We provide necessary and sufficient conditions for robust efficiency (in the sense of Ehrgott et al. (2014)) to multiobjective optimization problems that depend on uncertain parameters. These conditions state that a solution is robust efficient (under minimization) if it is optimal to a strongly increasing scalarizing function, and only if it is optimal to a strictly increasing scalarizing function. By counterexample, we show that the necessary condition cannot be strengthened to convex scalarizing functions, even for convex problems. We therefore define and characterize a subset of the robust efficient solutions for which an analogous necessary condition holds with respect to convex scalarizing functions. This result parallels the deterministic case where optimality to a convex and strictly increasing scalarizing function constitutes a necessary condition for efficiency. By a numerical example from the field of radiation therapy treatment plan optimization, we illustrate that the curvature of the scalarizing function influences the conservatism of an optimized solution in the uncertain case.