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

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.