Abstract

Using proof-theoretical techniques, we analyze a proof by Hong-Kun Xu regarding a result of strong convergence for the Halpern type proximal point algorithm. We obtain a rate of metastability (in the sense of Terence Tao) and also a rate of asymptotic regularity for the iteration. Furthermore, our final quantitative result bypasses the sequential weak compactness argument present in the original proof. This elimination is reflected in the extraction of primitive recursive quantitative information. This work follows from recent results in Proof Mining regarding the removal of sequential weak compactness arguments.

中文翻译：

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Halpern型近点算法的亚稳速率

摘要

使用证明理论技术，我们分析了徐洪坤关于Halpern型近端点算法的强收敛结果的证明。我们获得了亚稳态的速率（就Terence Tao而言）以及迭代的渐近正则率。此外，我们最终的定量结果绕过了原始证明中出现的顺序弱致密性的论点。这种消除反映在原始递归定量信息的提取中。这项工作是根据Proof Mining中有关删除连续弱紧凑性参数的最新结果得出的。