Conjugate gradient methods without line search for multiobjective optimization

TL;DR

This paper introduces a line search-free conjugate gradient method for unconstrained multiobjective optimization, reducing computational costs and maintaining convergence, with numerical validation against existing methods.

Contribution

It proposes a no-line-search conjugate gradient approach for multiobjective optimization, avoiding Wolfe-type line search and demonstrating convergence and practical efficiency.

Findings

The no-line-search method achieves comparable convergence to Wolfe-type methods.

Numerical experiments show improved computational efficiency.

The approach is effective for various multiobjective problems.

Abstract

This paper addresses unconstrained multiobjective optimization problems where two or more continuously differentiable functions have to be minimized. We delve into the conjugate gradient methods proposed by Lucambio P\'{e}rez and Prudente (SIAM J Optim, 28(3): 2690--2720, 2018) for such problems. Instead of the Wolfe-type line search procedure used in their work, we employ a fixed stepsize formula (or no-line-search scheme), which can mitigate the pressure of choosing stepsize caused by multiple inequalities and avoid the computational cost associated with function evaluations in specific applications. The no-line-search scheme is utilized to derive the condition of Zoutendijk's type. Global convergence encompasses the vector extensions of Fletcher--Reeves, conjugate descent, Dai--Yuan, Polak--Ribi\`{e}re--Polyak and Hestenes--Stiefel parameters, subject to certain mild assumptions. Additionally, numerical experiments are conducted to demonstrate the practical performance of the proposed stepsize rule, and comparative analyses are made with the multiobjective steepest descent methods using the Armijo line search and the multiobjective conjugate gradient methods using the Wolfe-type line search.