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Remarks on the paper "Treatment of Set-Valued Robustness via Separation and Scalarization"

Abhik Digar, Kuntal Som

TL;DR

The article addresses robustness in uncertain set-valued optimization by scrutinizing earlier results and uncovering inconsistencies in Das2024. It advances the field by introducing new union-based set-order relations, refining scalarization tools, and proposing hat-constructed criteria and robust-solution concepts that do not rely on attainment assumptions. The main contributions include corrected propositions, extended scalarization results, and novel robust-solution notions (via $\hat{B}$ and $\hat{R}$ constructions) for USOP. These developments clarify the theoretical underpinnings and yield more reliable criteria for identifying robust solutions under uncertainty, with potential implications for multi-criteria decision making and robust optimization practice.

Abstract

In this paper, we remark on the published paper "Treatment of Set-Valued Robustness via Separation and Scalarization" [1], which deals with the robust solution to an uncertain constrained set-valued optimization problem via scalarization methods. We show many inconsistencies in the results of the above-mentioned paper. We improve most of these results. In the process, we introduce some new concepts of robust solutions for uncertain set-valued optimization problems. We also improve some results on scalarization methods applicable to set-valued optimization.

Remarks on the paper "Treatment of Set-Valued Robustness via Separation and Scalarization"

TL;DR

The article addresses robustness in uncertain set-valued optimization by scrutinizing earlier results and uncovering inconsistencies in Das2024. It advances the field by introducing new union-based set-order relations, refining scalarization tools, and proposing hat-constructed criteria and robust-solution concepts that do not rely on attainment assumptions. The main contributions include corrected propositions, extended scalarization results, and novel robust-solution notions (via and constructions) for USOP. These developments clarify the theoretical underpinnings and yield more reliable criteria for identifying robust solutions under uncertainty, with potential implications for multi-criteria decision making and robust optimization practice.

Abstract

In this paper, we remark on the published paper "Treatment of Set-Valued Robustness via Separation and Scalarization" [1], which deals with the robust solution to an uncertain constrained set-valued optimization problem via scalarization methods. We show many inconsistencies in the results of the above-mentioned paper. We improve most of these results. In the process, we introduce some new concepts of robust solutions for uncertain set-valued optimization problems. We also improve some results on scalarization methods applicable to set-valued optimization.

Paper Structure

This paper contains 6 sections, 34 theorems, 44 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Scalarization in set-valued optimization
  4. Uncertain Set-Valued Optimization Problem (USOP)
  5. Characterization of Robust Solution for USOP
  6. Conclusion

Key Result

Proposition 2.1

Let $\{P_\gamma: \gamma \in \Gamma\}$ and $\{Q_\lambda: \lambda\in \Lambda\}$ be two collections of non-empty subsets of $Y.$ Set $P=\underset{\gamma\in \Gamma}{\bigcup} P_{\gamma}$ and $Q=\underset{\lambda\in \Lambda}{\bigcup} Q_\lambda.$ If $P\leq_{K}^{L} Q,$ then $P\leq_{K}^{l} Q.$

Figures (2)

  • Figure :
  • Figure :

Theorems & Definitions (75)

  • Definition 2.1: Luc89
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.1
  • proof
  • Example 1
  • Proposition 2.2
  • Example 2
  • Proposition 2.3
  • Example 3
  • ...and 65 more