A three-term Polak-Ribi\`{e}re-Polyak conjugate gradient method for vector optimization

TL;DR

This paper introduces a novel three-term Polak-Ribière-Polyak conjugate gradient method for vector optimization, extending scalar methods to vector problems with proven convergence and demonstrated superior performance through numerical experiments.

Contribution

It is the first to extend three-term conjugate gradient methods from scalar to vector optimization, with a new Wolfe-type line search ensuring global convergence.

Findings

Method generates sufficient descent directions independently of line search.

Global convergence is established without convexity or restart restrictions.

Numerical experiments show the method's favorable performance.

Abstract

A novel three-term Polak-Ribi\`{e}re-Polyak conjugate gradient method is proposed for solving vector optimization problems. It should be emphasized that this is the first extension of three-term conjugate gradient methods from scalar optimization to vector optimization. The method can consistently generate a sufficient descent direction independent of line search procedures and without modifying the conjugate parameters. This result improves upon the corresponding conclusions in SIAM J. Optim. 28, 2690-2720 (2018), J. Optim. Theory Appl. 204,13 (2025) and Optim. Methods Softw. 28, 725-754 (2025). Based on a new Wolfe-type line search, the global convergence of the proposed scheme is established without imposing restrictions such as self-adjusting strategies, regular restarts and convexity assumptions. Numerical experiments demonstrate the favourable performance of the proposed method.