Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A three-term Polak-Ribière-Polyak conjugate gradient method for vector optimization

Guangxuan Lin, Shouqiang Du

TL;DR

The paper addresses extending three-term Polak-Ribières-Polyak conjugate gradient methods to vector optimization while guaranteeing descent directions without adjusting conjugate parameters. It introduces the TT-PRP method, a vector extension that leverages a generalized Wolfe line search to prove global convergence under mild assumptions. Key contributions include the first vector-translation of the three-term CG framework, guaranteed descent with a fixed parameter scheme, and convergence proofs without convexity or restarts, complemented by numerical evidence of strong performance on large-scale multiobjective problems. This work provides a robust, scalable first-order method for vector optimization with practical impact on multiobjective applications.

Abstract

A novel three-term Polak-Ribière-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.

A three-term Polak-Ribière-Polyak conjugate gradient method for vector optimization

TL;DR

The paper addresses extending three-term Polak-Ribières-Polyak conjugate gradient methods to vector optimization while guaranteeing descent directions without adjusting conjugate parameters. It introduces the TT-PRP method, a vector extension that leverages a generalized Wolfe line search to prove global convergence under mild assumptions. Key contributions include the first vector-translation of the three-term CG framework, guaranteed descent with a fixed parameter scheme, and convergence proofs without convexity or restarts, complemented by numerical evidence of strong performance on large-scale multiobjective problems. This work provides a robust, scalable first-order method for vector optimization with practical impact on multiobjective applications.

Abstract

A novel three-term Polak-Ribière-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.
Paper Structure (6 sections, 7 theorems, 76 equations, 3 figures, 2 tables)

This paper contains 6 sections, 7 theorems, 76 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Three-term PRP CG method for vector optimization
  4. Global convergence
  5. Numerical experiments
  6. Final remarks and future directions

Key Result

Lemma 2.2

Gon1 Under the function $\Phi:\mathbf{R}^n\to \mathbf{R}^m$ is continuously differentiable, we have that

Figures (3)

  • Figure 1: Performance profiles based on the performance measurements (a)-(d)
  • Figure 2: Approximations of Pareto frontiers generated by the TT-PRP method and corresponding image sets for bicriteria problems.
  • Figure 3: Approximations of Pareto frontiers generated by the TT-PRP method and corresponding image sets for three-criteria problems.

Theorems & Definitions (18)

  • Remark 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Remark 2.4
  • Definition 2.5
  • Definition 2.6
  • Remark 2.7
  • Example 3.1
  • Theorem 3.2
  • proof
  • ...and 8 more