Set optimization - a rather short introduction

TL;DR

This paper provides a concise overview of recent advances in set optimization, covering set relations, convex analysis, duality, applications, and algorithms, highlighting the current state of research and practical applications.

Contribution

It offers a comprehensive survey and extension of set optimization theories, including new set relations, convex analysis, duality, and algorithmic approaches, with applications to finance and vector optimization.

Findings

Summarizes recent developments in set relations and convex analysis.

Discusses duality theory and its applications in set optimization.

Reviews algorithmic methods and practical applications in finance.

Abstract

Recent developments in set optimization are surveyed and extended including various set relations as well as fundamental constructions of a convex analysis for set- and vector-valued functions, and duality for set optimization problems. Extensive sections with bibliographical comments summarize the state of the art. Applications to vector optimization and financial risk measures are discussed along with algorithmic approaches to set optimization problems.