A vectorization for nonconvex set-valued optimization

Abstract

Vectorization is a technique that replaces a set-valued optimization problem with a vector optimization problem. In this work, by using an extension of Gerstewitz function [1], a vectorizing function is defined to replace a given set-valued optimization problem with respect to set less order relation. Some properties of this function are studied. Also, relationships between a set-valued optimization problem and a vector optimization problem, derived via vectorization of this set-valued optimization problem, are examined. Furthermore, necessary and sufficient optimality conditions are presented without any convexity assumption.

Characterizations of the Set Less Order Relation in Nonconvex Set Optimization

“…But for nonconvex sets one needs more elaborate approaches, which are sometimes named scalarization (e.g. see [13] for early results and also [5,9,10,12,21,22] together with [20] for related results). These scalarization approaches work with appropriate sup inf problems.…”

Optimality Conditions of Interval-Valued Optimization Problems by Using Subdifferantials

“…The first vectorization in the optimization theory was used by Küçük et al [4,5]. Jahn [6] and Karaman et al [7] obtained some optimality conditions for set optimization by using vectorization. Jahn [8] and Karaman et al [9] studied on optimality conditions for set optimization by using directional derivative.…”

Nonsmooth set variational inequality problems and optimality criteria for set optimization

“…Naturally, it has been attracted the attention of scientists, who have been working in mathematics, engineering, economics, management, economic equilibria, optimal control, nonlinear optimization transportation, and many other disciplines. There are many methods to solve and obtain optimality conditions of the optimization problems such as scalarization [14], vectorization [10,12], directional derivative [13], subdifferential [9,11], embedding space [15,18], variational inequality problems [1, 2, 4-8, 19, 21].…”