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A self-adaptive subgradient extragradient method with conjugate gradient-type direction for pseudomonotone variational inequalities

Ibrahim Arzuka, Parin Chaipunya, Poom Kumam

TL;DR

The paper addresses solving pseudomonotone variational inequalities in Hilbert spaces by combining a self-adaptive subgradient extragradient framework with a conjugate gradient-type direction and Halpern-type perturbations. The method eliminates the need for a priori Lipschitz information, uses a nonincreasing step-size sequence, and yields strong convergence to the VIP solution via a key descent framework. Theoretical results are supported by numerical experiments showing faster convergence and robustness compared with existing SEGM variants, including high-dimensional finite problems and matrix-game VIPs. This approach broadens the applicability of accelerated, adaptive projection methods for variational inequalities, with practical impact in computation-heavy VIPs where Lipschitz constants are unknown or costly to estimate.

Abstract

This paper introduces a subgradient extragradient algorithm with a conjugate gradient-type direction to solve pseudomonotone variational inequality problems in Hilbert spaces. The algorithm features a self-adaptive strategy that eliminates the need for prior knowledge of the Lipschitz constant and incorporate a conjugate gradient-type direction to enhance convergence speed. We establish a result describing the behavior generated therefrom toward the solution set. Using this result, we prove the strong convergence of the proposed method and provide numerical experiments to demonstrate its computational efficacy and robustness.

A self-adaptive subgradient extragradient method with conjugate gradient-type direction for pseudomonotone variational inequalities

TL;DR

The paper addresses solving pseudomonotone variational inequalities in Hilbert spaces by combining a self-adaptive subgradient extragradient framework with a conjugate gradient-type direction and Halpern-type perturbations. The method eliminates the need for a priori Lipschitz information, uses a nonincreasing step-size sequence, and yields strong convergence to the VIP solution via a key descent framework. Theoretical results are supported by numerical experiments showing faster convergence and robustness compared with existing SEGM variants, including high-dimensional finite problems and matrix-game VIPs. This approach broadens the applicability of accelerated, adaptive projection methods for variational inequalities, with practical impact in computation-heavy VIPs where Lipschitz constants are unknown or costly to estimate.

Abstract

This paper introduces a subgradient extragradient algorithm with a conjugate gradient-type direction to solve pseudomonotone variational inequality problems in Hilbert spaces. The algorithm features a self-adaptive strategy that eliminates the need for prior knowledge of the Lipschitz constant and incorporate a conjugate gradient-type direction to enhance convergence speed. We establish a result describing the behavior generated therefrom toward the solution set. Using this result, we prove the strong convergence of the proposed method and provide numerical experiments to demonstrate its computational efficacy and robustness.
Paper Structure (5 sections, 7 theorems, 49 equations, 3 figures, 1 algorithm)

This paper contains 5 sections, 7 theorems, 49 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Algorithm
  4. Numerical Example
  5. Conclusion

Key Result

Lemma 2.3

Let $x,y\in H$ and $\alpha \in \mathbb{R}$. Then

Figures (3)

  • Figure 4.1: Convergence plots for Example \ref{['EEe1']} with dimension d=100
  • Figure 4.2: Convergence plots for \ref{['EE22']} with dimension n=100
  • Figure 4.3: Convergence plots for Example \ref{['EE33']} with $w_{j}=50$

Theorems & Definitions (16)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Lemma 2.4: he2013solving
  • Lemma 2.5: opial1967weak
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 6 more