Robustness in Deterministic Vector Optimization

Abstract

In this paper, robust efficient solutions of a vector optimization problem, whose image space is ordered by an arbitrary ordering cone, are defined. This is done from different points of view, including set based and norm based. The relationships between these solution concepts are established. Furthermore, it is shown that, for a general vector optimization problem, each norm-based robust efficient solution is a strictly efficient solution; each isolated efficient solution is a norm-based robust efficient solution; and, under appropriate assumptions, each norm-based robust efficient solution is a Henig properly efficient solution. Various necessary and sufficient conditions for characterizing norm-based robust solutions of a general vector optimization problem, in terms of the tangent and normal cones and the nonascent directions, are presented. An optimization problem for calculating a robustness radius is provided, and then, the largest robustness radius is determined. Moreover, some alterations of objective functions preserving weak/strict/Henig proper/robust efficiency are studied.