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Nonlinear Conjugate Gradient Method for Multiobjective Optimization Problems of Interval-Valued Maps

Tapas Mondal, Debdas Ghosh, Jingxin Liu, Jie Li

Abstract

In this article, we propose an algorithm for the nonlinear conjugate gradient method to find a Pareto critical point of unconstrained multiobjective interval optimization problems. In this algorithm, we use the Wolfe line search procedure to find the step length. After defining the standard Wolfe conditions and the strong Wolfe conditions, we prove that there exists an interval of the step length that satisfies the standard Wolfe conditions and the strong Wolfe conditions. Further, to study the convergence analysis of our proposed algorithm, we derive the result related to the Zoutendijk condition. In the convergence analysis, first, we prove the global convergence property of our proposed algorithm for a general conjugate gradient algorithmic parameter. Further, we consider four variants of the conjugate gradient algorithmic parameter, such as Fletcher-Reeves, conjugate descent, Dai-Yuan, and modified Dai-Yuan. For each variant of the algorithmic parameter, we prove the global convergence results of our proposed algorithm. Finally, we test our algorithm on some test problems and make a performance profile.

Nonlinear Conjugate Gradient Method for Multiobjective Optimization Problems of Interval-Valued Maps

Abstract

In this article, we propose an algorithm for the nonlinear conjugate gradient method to find a Pareto critical point of unconstrained multiobjective interval optimization problems. In this algorithm, we use the Wolfe line search procedure to find the step length. After defining the standard Wolfe conditions and the strong Wolfe conditions, we prove that there exists an interval of the step length that satisfies the standard Wolfe conditions and the strong Wolfe conditions. Further, to study the convergence analysis of our proposed algorithm, we derive the result related to the Zoutendijk condition. In the convergence analysis, first, we prove the global convergence property of our proposed algorithm for a general conjugate gradient algorithmic parameter. Further, we consider four variants of the conjugate gradient algorithmic parameter, such as Fletcher-Reeves, conjugate descent, Dai-Yuan, and modified Dai-Yuan. For each variant of the algorithmic parameter, we prove the global convergence results of our proposed algorithm. Finally, we test our algorithm on some test problems and make a performance profile.
Paper Structure (14 sections, 15 theorems, 146 equations, 3 figures, 1 algorithm)

This paper contains 14 sections, 15 theorems, 146 equations, 3 figures, 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. Nonlinear Conjugate Gradient Method
  9. Convergence Analysis
  10. Fletcher-Reeves
  11. Conjugate Descent
  12. Dai-Yuan
  13. Numerical Experiments
  14. Conclusion and Future Directions

Key Result

Lemma 2.1

Let $U\subseteq {\mathbb{R}}^n$ be an open, and consider an IVM $\Gamma:U\to\mathcal{ I}\left(\mathbb{R}\right)$. Assume that $\Gamma$ is $gH$-differentiable at a point $x^\diamond$. Then there exists a $\delta>0$ such that for every $w\in{\mathbb{R}}^n$ and for every scalar $\alpha$ satisfying $\le where $L_{x^\diamond}\left(w\right)$ is the linear IVM. Furthermore, if the $gH$-gradient of $\Gamm

Figures (3)

  • Figure 1: Performance profile.
  • Figure 2: For five randomly selected starting points, the locations of $G\left(x^\star\right)$ within the attainable objective region for each biobjective test problem.
  • Figure 3: For a randomly selected starting point, the location of $G\left(x^0\right)$ and $G\left(x^\star\right)$ of the triobjective test problems.

Theorems & Definitions (47)

  • Definition 2.1: $gH$-difference stefanini2008generalization
  • Definition 2.2: Dominance relation of intervals chauhan2021generalized
  • Definition 2.3: Norm on ${\mathcal{I}({\mathbb{R}})}$ moore1966interval
  • Definition 2.4: $gH$-continuity ghosh2017newton
  • Definition 2.5: $gH$-Lipschitz continuity ghosh2022generalized
  • Remark 2.1
  • Definition 2.6: $gH$-derivative debnath2022generalized
  • Remark 2.2
  • Definition 2.7: $gH$-partial derivative debnath2022generalized
  • Definition 2.8: $gH$-gradient debnath2022generalized
  • ...and 37 more