Optimality Conditions for Approximate Solutions of Set Optimization Problems with the Minkowski Difference

Abstract

In this paper, we study the optimality conditions for set optimization problems with set criterion. Firstly, we establish a few important properties of the Minkowski difference for sets. Then, we introduce the generalized second-order lower radial epiderivative for a set-valued maps by Minkowski difference, and discuss some of its properties. Finally, by virtue of the generalized second-order lower radial epiderivatives and the generalized second-order radial epiderivatives, we establish the necessary optimality conditions and sufficient optimality conditions of approximate Benson proper efficient solutions and approximate weakly minimal solutions of unconstrained set optimization problems without convexity conditions, respectively. Some examples are provided to illustrate the main results obtained.

Keywords

• set optimization problems
• optimality conditions
• generalized second-order lower radial epiderivatives
• minkowski difference