Concrete convergence rates for common fixed point problems under Karamata regularity

• 成果类型: Article; Early Access
• 署名作者: Liu, Tianxiang; Lourenco, Bruno F.
• 署名单位: University of Tsukuba; Research Organization of Information & Systems (ROIS); Institute of Statistical Mathematics (ISM) - Japan
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-026-02332-6
• 发表日期: 2026-03-17
• 关键词: Common fixed point problem; Concrete rates; Karamata regularity; Quasi-cyclic algorithm; Regular Variation; Karamata theory; O-minimal structure; error-bounds; inequalities; projections

摘要

We introduce the notion of Karamata regular operators, which is a notion of regularity that is suitable for obtaining concrete convergence rates for common fixed point problems. This provides a broad framework that includes, but goes beyond, H & ouml;lderian error bounds and H & ouml;lder regular operators. By concrete, we mean that the rates we obtain are explicitly expressed in terms of a function of the iteration number k instead, of say, a function of the iterate xk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x<^>k$$\end{document}. While it is well-known that under H & ouml;lderian-like assumptions many algorithms converge linearly/sublinearly (depending on the exponent), little it is known when the underlying problem data does not satisfy H & ouml;lderian assumptions, which may happen if a problem involves exponentials and logarithms. Our main innovation is the usage of the theory of regularly varying functions which we showcase by obtaining concrete convergence rates for quasi-cylic algorithms in non-H & ouml;lderian settings. This includes certain rates that are neither sublinear nor linear but sit somewhere in-between, including a case where the rate is expressed via the Lambert W function. Finally, we connect our discussion to o-minimal geometry and show that, under mild assumptions, definable operators in any o-minimal structure are always Karamata regular.