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Preconditioned Halpern iteration with adaptive anchoring parameters and an acceleration to Chambolle--Pock algorithm

Fangbing Lv, Qiao-Li Dong

TL;DR

The paper tackles structured convex optimization in Hilbert spaces by recasting it as a monotone inclusion and a fixed-point problem amenable to splitting methods. It introduces a preconditioned Halpern iteration with adaptive anchoring parameters (PHA) and proves strong convergence with a guaranteed rate of $O(1/k)$, extending the results to Halpern-type preconditioned proximal point methods. It further develops an accelerated Chambolle--Pock (aCP) algorithm that achieves $O(1/\varphi_k)$ decay for both the residual and the primal-dual gap, and it analyzes these methods within the broader degenerate PPP framework. Numerical experiments on minimax matrix games and LASSO validate improved convergence speed and robustness, demonstrating practical benefits for large-scale structured problems.

Abstract

In this article, we propose a preconditioned Halpern iteration with adaptive anchoring parameters (PHA) by integrating a preconditioner and Halpern iteration with adaptive anchoring parameters. Then we establish the strong convergence and at least $\mathcal{O}(1/k)$ convergence rate of the PHA method, and extend these convergence results to Halpern-type preconditioned proximal point method with adaptive anchoring parameters. Moreover, we develop an accelerated Chambolle--Pock algorithm that is shown to have at least $\mathcal{O}(1/k)$ convergence rate concerning the residual mapping and the primal-dual gap. Finally, numerical experiments on the minimax matrix game and LASSO problem are provided to show the performance of our proposed algorithms.

Preconditioned Halpern iteration with adaptive anchoring parameters and an acceleration to Chambolle--Pock algorithm

TL;DR

The paper tackles structured convex optimization in Hilbert spaces by recasting it as a monotone inclusion and a fixed-point problem amenable to splitting methods. It introduces a preconditioned Halpern iteration with adaptive anchoring parameters (PHA) and proves strong convergence with a guaranteed rate of , extending the results to Halpern-type preconditioned proximal point methods. It further develops an accelerated Chambolle--Pock (aCP) algorithm that achieves decay for both the residual and the primal-dual gap, and it analyzes these methods within the broader degenerate PPP framework. Numerical experiments on minimax matrix games and LASSO validate improved convergence speed and robustness, demonstrating practical benefits for large-scale structured problems.

Abstract

In this article, we propose a preconditioned Halpern iteration with adaptive anchoring parameters (PHA) by integrating a preconditioner and Halpern iteration with adaptive anchoring parameters. Then we establish the strong convergence and at least convergence rate of the PHA method, and extend these convergence results to Halpern-type preconditioned proximal point method with adaptive anchoring parameters. Moreover, we develop an accelerated Chambolle--Pock algorithm that is shown to have at least convergence rate concerning the residual mapping and the primal-dual gap. Finally, numerical experiments on the minimax matrix game and LASSO problem are provided to show the performance of our proposed algorithms.

Paper Structure

This paper contains 6 sections, 14 theorems, 90 equations, 9 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A preconditioned Halpern iteration with adaptive anchoring parameters
  4. An acceleration of Chambolle--Pock algorithm
  5. Numerical experiments
  6. Conclusion

Key Result

Lemma 2.1

Let $\mathcal{A} : \mathcal{H} \to 2^{\mathcal{H}}$ be a maximal monotone operator with $\mathop{\mathrm{zer}}\limits(\mathcal{A}) \neq \emptyset$. Let $\mathcal{M}$ be an admissible preconditioner for $\mathcal{A}$ with a decomposition $\mathcal{M} = \mathcal{C} \mathcal{C}^*$ such that $(\mathcal{

Figures (9)

  • Figure 1: Increase of $\varphi_k$ with $k$ for Problem 5.1
  • Figure 2: Comparison results of all algorithms on Problem 5.1: PD gap versus iteration numbers
  • Figure 3: Comparison results of all algorithms on Problem 5.1: PD gap versus CPU times
  • Figure 4: Comparison results of all algorithms on Problem 5.1: Residual versus iteration numbers
  • Figure 5: Comparison results of all algorithms on Problem 5.1: Residual versus CPU times
  • ...and 4 more figures

Theorems & Definitions (30)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 20 more