Second-Order Composed Radial Derivatives of the Benson Proper Perturbation Map for Parametric Multi-Objective Optimization Problems

In this paper, we introduce second-order composed radial derivatives of set-valued maps and establish some of its properties. By applying this second-order derivative, we obtain second-order sensitivity results for parametric multi-objective optimization problems under the Benson proper efficiency without assumptions of cone-convexity and Lipschitz continuity. Some of our results improve and derive the recent corresponding ones in the literature.

Characterizations of Benson proper efficiency of set-valued optimization in real linear spaces

A new class of generalized convex set-valued maps termed relatively solid generalized cone-subconvexlike maps is introduced in real linear spaces not equipped with any topology. This class is a generalization of generalized cone-subconvexlike maps and relatively solid cone-subconvexlike maps. Necessary and sufficient conditions for Benson proper efficiency of set-valued optimization problem are established by means of scalarization, Lagrange multipliers, saddle points and duality. The results generalize and improve some corresponding ones in the literature. Some examples are afforded to illustrate our results.

Approximate proper efficiency for multiobjective optimization problems

This paper is devoted to the study of a new kind of approximate proper efficiency in terms of proximal normal cone and co-radiant set for multiobjective optimization problem. We derive some properties of the new approximate proper efficiency and discuss the relations with the existing approximate concepts, such as approximate efficiency and approximate Benson proper efficiency. At last, we study the linear scalarizations for the new approximate proper efficiency under the generalized convexity assumption and give some examples to illustrate the main results.