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arXiv · 1510.06823

Convergence rate analysis for averaged fixed point iterations in the presence of Hölder regularity

Jonathan M. Borwein·Guoyin Li·Matthew K. Tam
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Abstract

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 Hölder 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 Krasnoselskii-Mann iterations, the cyclic projection algorithm, 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 Hölder regularity properties are automatically satisfied, from which sublinear convergence follows.

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Jonathan M. Borwein, Guoyin Li, Matthew K. Tam. 2016-08-18. Convergence rate analysis for averaged fixed point iterations in the presence of Hölder regularity. https://doi.org/10.1137/15m1045223

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