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Set optimization - a rather short introduction

Andreas H. Hamel, Frank Heyde, Andreas Löhne, Birgit Rudloff, Carola Schrage

TL;DR

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.

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.

Set optimization - a rather short introduction

TL;DR

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.

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.

Paper Structure

This paper contains 30 sections, 51 theorems, 235 equations.

Table of Contents

  1. Introduction
  2. Set relations and lattices of sets
  3. Extending pre-orders from linear spaces to their power sets
  4. Comments on set relations
  5. Inf-residuated conlinear spaces of sets
  6. Comments on conlinear spaces and residuation
  7. Minimality notions
  8. Basic concepts
  9. Comments on solution concepts in set optimization
  10. Set-valued functions
  11. Basic concepts
  12. Scalarization of $\mathcal{G}\left( Z, C \right)$-valued functions
  13. Comments on convexity, semicontinuity and scalarization
  14. Set-valued convex analysis
  15. Conlinear functions
  16. ...and 15 more sections

Key Result

Proposition 2.1

(a) Both $\preccurlyeq_C$ and $\curlyeqprec_C$ are reflexive and transitive relations on $\mathcal{P}\left( Z \right)$. Moreover, they are not antisymmetric in general, and they do not coincide. (b) $A \preccurlyeq_C B$$\Leftrightarrow$$-B \curlyeqprec_C -A$$\Leftrightarrow$$B \preccurlyeq_{-C} A$.

Theorems & Definitions (88)

  • Proposition 2.1
  • Proposition 2.2
  • Remark 2.3
  • Definition 2.4
  • Example 2.5
  • Proposition 2.6
  • Example 2.7
  • Proposition 2.8
  • Proposition 2.9
  • Remark 2.10
  • ...and 78 more