On proper minimality in set optimization

Abstract

The aim of this paper is to extend some notions of proper minimality from vector optimization to set optimization. In particular, we focus our attention on the concepts of Henig and Geoffrion proper minimality, which are well-known in vector optimization. We introduce a generalization of both of them in set optimization with finite dimensional spaces, by considering also a special class of polyhedral ordering cone. In this framework, we prove that these two notions are equivalent, as it happens in the vector optimization context, where this property is well-known. Then, we study a characterization of these proper minimal points through nonlinear scalarization, without considering convexity hypotheses.

Weak Henig proper solution sets for set optimization problems

“…As pointed out by Khan et al [7], since set-valued mappings appear naturally in many practical problems, set optimization problems will remain an important and active research topic in both the near and foreseeable future. To the best of our knowledge, there are only three papers [8][9][10] considering proper efficiency for set optimization problems. Huerga et al [9] introduced two notions of proper efficiency in the sense of Henig (named Henig proper solution and weak Henig proper solution) for set optimization problems.…”