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Baillon-Bruck-Reich revisited: divergent-series parameters and strong convergence in the linear case

Sedi Bartz, Heinz H. Bauschke, Yuan Gao

Abstract

The Krasnoselskii-Mann iteration is an important algorithm in optimization and variational analysis for finding fixed points of nonexpansive mappings. In the general case, it produces a sequence converging \emph{weakly} to a fixed point provided the parameter sequence satisfies a divergent-series condition. In this paper, we show that \emph{strong} convergence holds provided the underlying nonexpansive mapping is \emph{linear}. This improves on a celebrated result by Baillon, Bruck, and Reich from 1978, where the parameter sequence was assumed to be constant as well as on recent work where the parameters were bounded away from $0$ and $1$.

Baillon-Bruck-Reich revisited: divergent-series parameters and strong convergence in the linear case

Abstract

The Krasnoselskii-Mann iteration is an important algorithm in optimization and variational analysis for finding fixed points of nonexpansive mappings. In the general case, it produces a sequence converging \emph{weakly} to a fixed point provided the parameter sequence satisfies a divergent-series condition. In this paper, we show that \emph{strong} convergence holds provided the underlying nonexpansive mapping is \emph{linear}. This improves on a celebrated result by Baillon, Bruck, and Reich from 1978, where the parameter sequence was assumed to be constant as well as on recent work where the parameters were bounded away from and .
Paper Structure (3 sections, 2 theorems, 15 equations)

This paper contains 3 sections, 2 theorems, 15 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Conclusions

Key Result

Proposition 1

Recall that $T$ satisfies e:T. Assume in addition that $T$ is linear. Then

Theorems & Definitions (11)

  • proof
  • proof
  • Proposition 1
  • proof
  • Remark 1
  • Theorem 2.1: main result
  • proof
  • Remark 2: without linearity, strong convergence fails
  • Remark 3: without divergent-series parameters, convergence to a fixed point fails
  • Remark 4: extension to the affine case
  • ...and 1 more