Scalarization in vector optimization by functions with uniform sublevel sets

TL;DR

This paper explores scalarization methods in vector optimization using functions with uniform sublevel sets, providing new conditions for efficiency and weak efficiency in decision-making contexts.

Contribution

It introduces a novel scalarization approach with functions having uniform sublevel sets, offering both necessary and sufficient conditions for optimality in vector optimization.

Findings

Established basic properties of efficient points in vector optimization.

Derived sufficient conditions for solutions using minimal solutions of functionals.

Showed that functions with uniform sublevel sets can be continuous, Lipschitz, convex, or sublinear.

Abstract

In this paper, vector optimization is considered in the framework of decision making and optimization in general spaces. Interdependencies between domination structures in decision making and domination sets in vector optimization are given. We prove some basic properties of efficient and of weakly efficient points in vector optimization. Sufficient conditions for solutions to vector optimization problems are shown using minimal solutions of functionals. We focus on the scalarization by functions with uniform sublevel sets, which also delivers necessary conditions for efficiency and weak efficiency. The functions with uniform sublevel sets may be, e.g., continuous or even Lipschitz continuous, convex, strictly quasiconcave or sublinear. They can coincide with an order unit norm on a subset of the space.