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<p>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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathcal {O}(1/k)\)</EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathcal {O}(1/k)\)</EquationSource> </InlineEquation> 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.</p>

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

  • 期刊论文

  • 发表日期：

    2026年3月4日

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  • Fangbing Lv,
  • Qiao-Li Dong

摘要

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.