Functions with uniform sublevel sets and scalarization in linear spaces

TL;DR

This paper studies functions with uniform sublevel sets in linear spaces, exploring their properties and applications in scalarization for vector optimization without relying on topological assumptions.

Contribution

It characterizes real-valued and extended functions with uniform sublevel sets in linear spaces, including convex and sublinear cases, and applies them to scalarize vector optimization problems.

Findings

Functions can separate non-convex sets without topology.

Characterization of functions as Minkowski functionals or order unit norms.

Application to scalarization of vector optimization problems.

Abstract

Functions with uniform sublevel sets can represent orders, preference relations or other binary relations and thus turn out to be a tool for scalarization that can be used in multicriteria optimization, decision theory, mathematical finance, production theory and operator theory. Sets which are not necessarily convex can be separated by functions with uniform sublevel sets. This report focuses on properties of real-valued and extended-real-valued functions with uniform sublevel sets which are defined on a linear space without assuming topological properties. The functions may be convex or sublinear. They can coincide with a Minkowski functional or with an order unit norm on a subset of the space. The considered functionals are applied to the scalarization of vector optimization problems. These vector optimization problems refer to arbitrary domination sets. The consideration of such sets is motivated by their relationship to domination structures in decision making.