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

TL;DR

This paper introduces a nonlinear conjugate gradient algorithm for multiobjective interval optimization, employing Wolfe line search and convergence proofs, including for various algorithmic variants, with empirical testing on test problems.

Contribution

It develops a novel conjugate gradient method for multiobjective interval problems with convergence analysis and empirical validation, extending existing optimization techniques.

Findings

Algorithm converges globally under various parameters.

Proven existence of Wolfe line search step lengths.

Performance tested on multiple benchmark problems.

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.