# A Vectorization for Nonconvex Set-valued Optimization

## TL;DR

This paper introduces a new vectorization method for nonconvex set-valued optimization problems using an extended Gerstewitz function, enabling analysis without convexity assumptions.

## Contribution

It develops a novel vectorizing function based on Gerstewitz extension and studies its properties, establishing links between set-valued and vector optimization without convexity.

## Key findings

- Properties of the new vectorizing function are characterized.
- Relationships between set-valued and vector optimization problems are established.
- Necessary and sufficient optimality conditions are derived without convexity assumptions.

## 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.