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Characterization of Highly Robust Solutions in Multi-Objective Programming in Banach Spaces

Morteza Rahimi, Majid Soleimani-damaneh

Abstract

This paper delves into the challenging issues in uncertain multi-objective optimization, where uncertainty permeates nonsmooth nonconvex objective and constraint functions. In this context, we investigate highly robust (weakly efficient) solutions, a solution concept defined by efficiency across all scenarios. Our exploration reveals important relationships between highly robust solutions and other robustness notions, including set-based and worst-case notions, as well as connections with proper and isolated efficiency. Leveraging modern techniques from variational analysis, we establish necessary and sufficient optimality conditions for these solutions. Moreover, we explore the robustness of multi-objective optimization problems in the face of various uncertain sets, such as ball, ellipsoidal, and polyhedral sets.

Characterization of Highly Robust Solutions in Multi-Objective Programming in Banach Spaces

Abstract

This paper delves into the challenging issues in uncertain multi-objective optimization, where uncertainty permeates nonsmooth nonconvex objective and constraint functions. In this context, we investigate highly robust (weakly efficient) solutions, a solution concept defined by efficiency across all scenarios. Our exploration reveals important relationships between highly robust solutions and other robustness notions, including set-based and worst-case notions, as well as connections with proper and isolated efficiency. Leveraging modern techniques from variational analysis, we establish necessary and sufficient optimality conditions for these solutions. Moreover, we explore the robustness of multi-objective optimization problems in the face of various uncertain sets, such as ball, ellipsoidal, and polyhedral sets.
Paper Structure (7 sections, 22 theorems, 109 equations)

This paper contains 7 sections, 22 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Highly Robust Solutions in Multi-Objective Optimization
  4. Characterization of Highly Robustness
  5. Highly Robustness With Special Uncertain Sets
  6. Ellipsoidal And Ball Uncertain Sets
  7. Polyhedral Uncertain Sets

Key Result

Lemma 1

Let $S\subseteq Y$ be compact and the set-valued mapping $F:Y\rightrightarrows X^*$ be weak$^*$ closed at each $\bar{y}\in S$. Then, $F(S):=\bigcup_{y\in S}F(y)$ is weak$^*$ closed.

Theorems & Definitions (54)

  • Lemma 1
  • proof
  • Definition 1
  • Example 1
  • Proposition 1
  • proof
  • Theorem 2
  • proof
  • Proposition 3
  • proof
  • ...and 44 more