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Stability analysis for set-valued optimization in Geoffroy spaces

James Larrouy

TL;DR

The paper develops a stability framework for set-valued optimization in Geoffroy spaces by introducing $\\Gamma^C$-convergence and its sequential variant $\\Gamma^{C}_{\\text{seq}}$, along with upper/lower convergence of strong level sets. It defines Geoffroy spaces endowed with the set order topology $\\tau^{C}$ and multiple minimality notions, notably Geoffroy minimality, to study perturbations in both the objective and admissible domain. The authors prove external and internal stability results for various minimal solution concepts, leveraging Kuratowski pairs and the sequential lower converse property, and they illustrate the theory with examples and practical implications. The work illuminates how stability in the set-valued, cone-ordered setting can be established via these variational convergences and related topological notions, and it highlights open questions related to existence, hypothesis (H), and potential cone perturbations.

Abstract

In this work, we study the external and internal stability of minimal solutions to set-valued optimization problems in a new functional framework. We consider perturbations on both the objective function and the admissible domain. To address these problems, we introduce two variational convergences for sequences of set-valued maps, namely the Gamma-cone convergence and the sequential Gamma-cone convergence. The upper and the lower convergence of strong level sets are also studied.

Stability analysis for set-valued optimization in Geoffroy spaces

TL;DR

The paper develops a stability framework for set-valued optimization in Geoffroy spaces by introducing -convergence and its sequential variant , along with upper/lower convergence of strong level sets. It defines Geoffroy spaces endowed with the set order topology and multiple minimality notions, notably Geoffroy minimality, to study perturbations in both the objective and admissible domain. The authors prove external and internal stability results for various minimal solution concepts, leveraging Kuratowski pairs and the sequential lower converse property, and they illustrate the theory with examples and practical implications. The work illuminates how stability in the set-valued, cone-ordered setting can be established via these variational convergences and related topological notions, and it highlights open questions related to existence, hypothesis (H), and potential cone perturbations.

Abstract

In this work, we study the external and internal stability of minimal solutions to set-valued optimization problems in a new functional framework. We consider perturbations on both the objective function and the admissible domain. To address these problems, we introduce two variational convergences for sequences of set-valued maps, namely the Gamma-cone convergence and the sequential Gamma-cone convergence. The upper and the lower convergence of strong level sets are also studied.

Paper Structure

This paper contains 10 sections, 25 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and notations
  3. Background materials
  4. On Geoffroy spaces
  5. Minimality concepts
  6. Asymptotic properties of strong level sets
  7. The Gamma-cone convergence
  8. Upper and lower convergence of strong level sets
  9. Stability results for set-valued minimization problems
  10. Conclusion and open questions

Key Result

Theorem 2.2

Let $A, B \in {\mathcal{P}}^\circ(Z)$ be two convex sets. If $A$ is solid, then ${\rm int}(A) + B = {\rm int}(A + B).$

Theorems & Definitions (67)

  • Remark 2.1
  • Theorem 2.2: TKU93
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • proof
  • Proposition 2.5: GLA22
  • Definition 2.6: $\tau^{C}$-continuity of set-valued maps
  • Remark 2.7
  • Definition 2.8: Lower and upper $\tau^{C}$-semicontinuity
  • ...and 57 more