A Unified Characterization of Nonlinear Scalarizing Functionals in Optimization

TL;DR

This paper provides a comprehensive analysis of various scalarizing functionals in optimization, establishing their relationships, and introduces a new class based on quasidifferentiability, linking these concepts to set optimization.

Contribution

It fully characterizes the inclusion relations among existing scalarizing functionals and introduces a broader class based on quasidifferentiability, connecting scalarization with set optimization.

Findings

Gerstewitz functionals form a minimal class.

A new quasidifferentiable, positively homogeneous class is introduced.

Results connect scalarization with set optimization theories.

Abstract

Over the years, several classes of scalarization techniques in optimization have been introduced and employed in deriving separation theorems, optimality conditions and algorithms. In this paper, we study the relationships between some of those classes in the sense of inclusion. We focus on three types of scalarizing functionals defined by Hiriart-Urruty, Drummond and Svaiter, and Gerstewitz. We completely determine their relationships. In particular, it is shown that the class of the functionals by Gerstewitz is minimal in this sense. Furthermore, we define a new (and larger) class of scalarizing functionals that are not necessarily convex, but rather quasidifferentiable and positively homogeneous. We show that our results are connected with some of the set relations in set optimization.