Projection and proximal point methods: convergence results and counterexamples

Bauschke, Heinz H. and Matouskova, Eva and Reich, Simeon (2003) Projection and proximal point methods: convergence results and counterexamples. [Preprint]

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      Recently, Hundal has constructed a hyperplane $H$, a cone $K$, and a starting point $y_0$ in $\ell_2$ such that the sequence of alternating projections $\big((P_KP_H)^ny_0\big)_{n \in \NN}$ converges weakly to some point in $H \cap K$, but not in norm. We show how this construction results in a counterexample to norm convergence for iterates of averaged projections; hence, we give an affirmative answer to a question raised by Reich two decades ago. Furthermore, new counterexamples to norm convergence for iterates of firmly nonexpansive maps (\`a la Genel and Lindenstrauss) and for the proximal point algorithm (\`a la G\"uler) are provided. We also present a counterexample, along with some weak and norm convergence results, for the new framework of string-averaging projection methods introduced by Censor, Elfving, and Herman. Extensions to Banach spaces and the situation for the Hilbert ball are discussed as well.

      Item Type: Preprint
      Additional Information: pubdom FALSE
      Uncontrolled Keywords: alternating projections, averaged projections, Hilbert space, nonexpansive, proximal point algorithm, weak convergence
      Subjects: 90-xx Economics, operations research, programming, games > 90Cxx Mathematical programming
      47-xx Operator theory > 47Jxx Equations and inequalities involving nonlinear operators
      47-xx Operator theory > 47Hxx Nonlinear operators and their properties
      Faculty: UNSPECIFIED
      Depositing User: Users 1 not found.
      Date Deposited: 27 Oct 2003
      Last Modified: 21 Apr 2010 11:13

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