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Newton Method for Multiobjective Optimization Problems of Interval-Valued Maps

Tapas Mondal, Debdas Ghosh, Do Sang Kim

TL;DR

A Newton-based method for solving multiobjective interval optimization problems (MIOPs) is proposed and it is proved that the sequence generated by the proposed algorithm converges to a Pareto critical point.

Abstract

In this article, we propose a Newton-based method for solving multiobjective interval optimization problems (MIOPs). We first provide a connection between weakly Pareto optimal points and Pareto critical points in the context of MIOPs. Introducing this relationship, we develop an algorithm aimed at computing a Pareto critical point. The algorithm incorporates the computation of a descent direction at a non-Pareto critical point and employs an Armijo-like line search strategy to ensure sufficient decrease. Under suitable assumptions, we prove that the sequence generated by our proposed algorithm converges to a Pareto critical point. The effectiveness and performance of the proposed method are demonstrated through a series of numerical experiments on some test problems. Finally, we apply our proposed algorithm in a portfolio optimization problem with interval uncertainty.

Newton Method for Multiobjective Optimization Problems of Interval-Valued Maps

TL;DR

A Newton-based method for solving multiobjective interval optimization problems (MIOPs) is proposed and it is proved that the sequence generated by the proposed algorithm converges to a Pareto critical point.

Abstract

In this article, we propose a Newton-based method for solving multiobjective interval optimization problems (MIOPs). We first provide a connection between weakly Pareto optimal points and Pareto critical points in the context of MIOPs. Introducing this relationship, we develop an algorithm aimed at computing a Pareto critical point. The algorithm incorporates the computation of a descent direction at a non-Pareto critical point and employs an Armijo-like line search strategy to ensure sufficient decrease. Under suitable assumptions, we prove that the sequence generated by our proposed algorithm converges to a Pareto critical point. The effectiveness and performance of the proposed method are demonstrated through a series of numerical experiments on some test problems. Finally, we apply our proposed algorithm in a portfolio optimization problem with interval uncertainty.
Paper Structure (15 sections, 9 theorems, 83 equations, 4 figures, 4 tables, 1 algorithm)

This paper contains 15 sections, 9 theorems, 83 equations, 4 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Literature Survey
  3. Motivation and Work Done
  4. Delineation
  5. Preliminaries
  6. Interval Analysis
  7. Multiobjective Interval Optimization Problem
  8. Newton Method
  9. Computing Descent Direction
  10. Step Length
  11. Convergence Analysis
  12. Numerical Experiments
  13. Application
  14. Conclusion and Future Directions
  15. List of Test Problems

Key Result

Lemma 2.1

Let $H:{\mathbb{R}}^n\rightarrow {\it I}({\mathbb{R}})$ be an IVM given by $H:=[\underline{H},\overline{H}]$. Then, the following results hold.

Figures (4)

  • Figure 1: I-BK1.
  • Figure 2: Performance profile.
  • Figure 3: For five randomly chosen initial points, the locations of $G\left(x^\star\right)$ in the objective feasible region of biobjective test problems given in Appendix \ref{['Test problems']}.
  • Figure 4: For a randomly chosen initial point, the location of $G\left(x^0\right)$ and $G\left(x^\star\right)$ of the triobjective test problems given in Appendix \ref{['Test problems']}.

Theorems & Definitions (37)

  • Definition 2.1: $gH$-difference stefanini2008generalization
  • Definition 2.2: Dominance relation of intervals chauhan2021generalized
  • Definition 2.3: Norm on ${{\it I}({\mathbb{R}})}$ moore1966interval
  • Definition 2.4: Norm on ${{\it I}({\mathbb{R}})^n}$ ghosh2022generalized
  • Definition 2.5: $gH$-continuity ghosh2017newton
  • Definition 2.6: $gH$-Lipschitz continuity ghosh2022generalized
  • Lemma 2.1
  • Definition 2.7: $gH$-derivative debnath2022generalized
  • Remark 2.1
  • Definition 2.8: $gH$-partial derivative debnath2022generalized
  • ...and 27 more