Convergence rate analysis for averaged fixed point iterations in common fixed point problems

J.M. Borwein University of Newcastle Guoyin Li University of New South Wales M.K. Tam University of Gottingen

Numerical Analysis and Scientific Computing Optimization and Control mathscidoc:1904.25004

SIAM Journal on Optimization, 27, (1), 1–33, 2017
In this paper, we establish sublinear and linear convergence of fixed point iterations generated by averaged operators in a Hilbert space. Our results are achieved under a bounded Holder regularity assumption which generalizes the well-known notion of bounded linear regularity. As an application of our results, we provide a convergence rate analysis for many important iterative methods in solving broad mathematical problems such as convex feasibility problems and variational inequality problems. These include Krasnoselskii–Mann iterations, the cyclic projection algorithm, forward-backward splitting and the Douglas–Rachford feasibility algorithm along with some variants. In the important case in which the underlying sets are convex sets described by convex polynomials in a finite dimensional space, we show that the Holder regularity properties are automatically satisfied, from which sublinear convergence follows.
Averaged fixed point methods, bounded Holder regularity, convergence rate analysis
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@inproceedings{j.m.2017convergence,
  title={CONVERGENCE RATE ANALYSIS FOR AVERAGED FIXED POINT ITERATIONS IN COMMON FIXED POINT PROBLEMS},
  author={J.M. Borwein, Guoyin Li, and M.K. Tam},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20190417224846595780239},
  booktitle={SIAM Journal on Optimization},
  volume={27},
  number={1},
  pages={1–33},
  year={2017},
}
J.M. Borwein, Guoyin Li, and M.K. Tam. CONVERGENCE RATE ANALYSIS FOR AVERAGED FIXED POINT ITERATIONS IN COMMON FIXED POINT PROBLEMS. 2017. Vol. 27. In SIAM Journal on Optimization. pp.1–33. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20190417224846595780239.
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