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Extra-Gradient Method with Flexible Anchoring: Strong Convergence and Fast Residual Decay

Radu Ioan Boţ, Enis Chenchene

Abstract

In this paper, we introduce a novel Extra-Gradient method with anchor term governed by general parameters. Our method is derived from an explicit discretization of a Tikhonov-regularized monotone flow in Hilbert space, which provides a theoretical foundation for analyzing its convergence properties. We establish strong convergence to specific points within the solution set, as well as convergence rates expressed in terms of the regularization parameters. Notably, our approach recovers the fast residual decay rate $O(k^{-1})$ for standard parameter choices. Numerical experiments highlight the competitiveness of the method and demonstrate how its flexible design enhances practical performance.

Extra-Gradient Method with Flexible Anchoring: Strong Convergence and Fast Residual Decay

Abstract

In this paper, we introduce a novel Extra-Gradient method with anchor term governed by general parameters. Our method is derived from an explicit discretization of a Tikhonov-regularized monotone flow in Hilbert space, which provides a theoretical foundation for analyzing its convergence properties. We establish strong convergence to specific points within the solution set, as well as convergence rates expressed in terms of the regularization parameters. Notably, our approach recovers the fast residual decay rate for standard parameter choices. Numerical experiments highlight the competitiveness of the method and demonstrate how its flexible design enhances practical performance.

Paper Structure

This paper contains 24 sections, 13 theorems, 81 equations, 4 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contribution
  3. Background and Preliminaries
  4. Continuous Time Analysis
  5. A General Framework for the Extra-Anchored-Gradient method
  6. Technical Lemmas
  7. Main Convergence Result
  8. Choices of epsilon
  9. Linear:
  10. Power:
  11. Miscellaneous:
  12. Numerical Experiments
  13. Preparation:
  14. Comparison
  15. Comments:
  16. ...and 9 more sections

Key Result

Lemma 1.3.1

Let $x\colon [t_0, +\infty)\to H$ be a solution trajectory to eq:tychonov_flow and $x^*\in \mathop{\mathrm{zer}}\limits M$. Then,

Figures (4)

  • Figure 2.1: Numerical study of the worst-case rate. First row: the parameter choice $(\varepsilon^k)_k$. Second row: The sequence $(r^k)_k$ in \ref{['eq:recursion_worst_case_rate']} as a function of $k$, the upper bound for $\psi^k = \frac{1}{2}\|M(x^k) + \varepsilon^k x^k \|^2$.
  • Figure 3.1: Results of the experiment in Section \ref{['sec:num_comparison']}. Comparing Algorithm \ref{['alg:gfeg']} with EAG-C, EAG-V, FEG and APV.
  • Figure 3.2: Result of experiment in Section \ref{['sec:num_comparison']}. Comparing Algorithm \ref{['alg:gfeg']} with EAG-C, EAG-V, FEG and APV.
  • Figure 3.3: Results of the Experiment in Section \ref{['sec:num_infinite_dimensional']}. Comparison of EAG-type methods to solve an infinite dimensional problem in $\ell^2$.

Theorems & Definitions (27)

  • Lemma 1.3.1: Proposition 6 in cps08
  • proof
  • Lemma 1.3.2: Theorem 2.5 in bk24
  • proof
  • Theorem 1.3.3
  • proof
  • Corollary 1.3.4
  • proof
  • Remark 1.1: The case $\alpha = 1$
  • Lemma 2.1.1: $\psi$-Analysis
  • ...and 17 more