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

TL;DR

This paper introduces a preconditioned Halpern iteration with adaptive parameters and an accelerated Chambolle--Pock algorithm, both with proven convergence rates, validated by numerical experiments on matrix games and LASSO.

Contribution

It develops new preconditioned and accelerated algorithms with convergence guarantees for optimization problems, extending existing methods with adaptive and preconditioning techniques.

Findings

Proven strong convergence of the proposed algorithms.

Achieved at least O(1/k) convergence rate.

Numerical experiments demonstrate improved performance.

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.