Dominance for Multi-Objective Robust Optimization Concepts
In robust optimization, the parameters of an optimization problem are not deterministic but uncertain. Their values depend on the scenarios which may occur. Single-objective robust optimization has been studied extensively. Since 2012, researchers have been looking at robustness concepts for multi-objective optimization problems as well. In another line of research, single-objective uncertain optimization problems are transformed to deterministic multi-objective problems by treating every scenario as an objective function. In this paper we combine these two points of view. We treat every scenario as an objective function also in uncertain multi-objectiveoptimization, and we define a corresponding concept of dominance which we call multi-scenario efficiency. We sketch this idea for finite uncertainty sets and extend it to the general case of infinite uncertainty sets. We then investigate the relation between this dominance and the concepts of highly, locally highly, flimsily, and different versions of minmax robust efficiency. For all these concepts we prove that every strictly robust efficient solution is multi-scenario efficient. On the other hand, under a compactness condition, the set of multi-scenario efficient solutions contains a robust efficient solution for all these concepts which generalizes the Pareto robustly optimal(PRO) solutions from single-objective optimization to Pareto robust efficient (PRE) solutions in the multi-objective case. We furthermore present two results on reducing an infinite uncertainty set to a finite one which are a basis for computing multi-scenario efficient solutions.