Scalarizations and Optimality of Constrained Set-Valued Optimization Using Improvement Sets and Image Space Analysis

Abstract

In this paper, we aim at applying improvement sets and image space analysis to investigate scalarizations and optimality conditions of the constrained set-valued optimization problem. Firstly, we use the improvement set to introduce a new class of generalized convex set-valued maps. Secondly, under suitable assumptions, some scalarization results of the constrained set-valued optimization problem are obtained in the sense of (weak) optimal solution characterized by the improvement set. Finally, by considering two classes of nonlinear separation functions, we present the separation between two suitable sets in image space and derive some optimality conditions for the constrained set-valued optimization problem. It shows that the existence of a nonlinear separation is equivalent to a saddle point condition of the generalized Lagrangian set-valued functions.

Conditional gradient method for vector optimization

“…where the first equality holds in view of (5). According to (32), there are two options: Case 1. Let t k = 1.…”

“…We now recall the concept of oriented distance function (also called assigned distance function or Hiriart-Urruty function), which was proposed by Hiriart-Urruty [28] to investigate optimality conditions of nonsmooth optimization problems from the geometric point of view. The oriented distance function has been extensively used in several works, such as scalarization for vector optimization [29,30], optimality conditions for vector optimization [31], optimality conditions for set-valued optimization problems [32], etc. Herein, we consider the oriented distance function in R m .…”

A scalarization scheme for binary relations with applications to set-valued and robust optimization

“…For example, property (12) is true if C is an improvement set with respect to a convex cone D ⊂ Y (i.e., C + D = C and 0 / ∈ C, see [3,8,31] and the references therein), e ∈ D and C is pointed (i.e., C ∩ (−C) = ∅). This particular case follows by applying [9, Lemma 2.3(c)].…”

“…The first characterization is based on -representing and strictly -preserving mappings, and it is a direct consequence of Propositions 1 and 4. The second one is based on the order preserving and strict order representing properties introduced in statements ( 14) and (31), respectively, and follows as a consequence of Corollaries 1 and 7 and extends [15, Theorem 3.2(i)] to arbitrary binary relations. The third one considers strictly -representing and -preserving mappings and follows by Propositions 2 and 3, statement (30) and Remark 6.…”

The connectedness of the solutions set for set-valued vector equilibrium problems under improvement sets

“…Gutiérrez et al [2] extended the notion of improvement set and E-optimal solution to a locally convex topological vector spaces. Much follow-up work about the improvement set E one finds in [3][4][5][6][7][8][9][10][11][12]. Chen et al [13] introduced a new vector equilibrium problem based on improvement set E named the unified vector equilibrium problem (UVEP), linear scalarization characterizations of the efficient solutions, weak efficient solutions, Benson proper efficient solutions for (UVEP) were established, and some continuity results of parametric (UVEP) were obtained by applying scalarization method.…”