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Projection and proximal point methods: Convergence results and counterexamples. (English) Zbl 1059.47060

The paper under review is a valuable and deep contribution to the convergence theory of certain sequences in Hilbert space. These sequences base on projection and proximal point methods. Herewith, this paper from functional analysis with its clear structure and thorough proofs is meaningful also for applied mathematics, especially, optimization theory in abstract spaces. This relation is reflected by the extension of projector classes and by the references.
After preparations in terms of projector and mapping classes, as well as the classical J. von Neumann’s and L. M. Bregman’s results, two main lines of work on which the paper bases itself and which it continues, are as follows:
(i) In the workshop “Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications”, Haifa, 2000, H. Hundal presented a hyperplane \(H\), a cone \(K\) and an initial point \(y_0\) in \(\ell_2\) such that the sequence of iterates of stepwise alternating projections on firstly \(H\) and secondly \(K\), weakly converges but not norm converges to a point in the intersection of \(H\) with \(K\). The authors extend this result to a counterexample on norm convergence of iterates given by averaged projections.
(ii) Herewith, a question posed by S. Reich becomes answered. In the paper, further counterexamples are presented in the line of research done by
(iii) A. Genel and J. Lindenstrauss: firmly nonexpansive maps,
(iv) O. Güler: proximal point algorithms, and
(v) Y. Censor et al.: string-averaging projection methods. Finally, extensions to the Hilbert ball and Banach spaces are discussed, too.
This paper with its rich and wide results may in the future serve for a deeper understanding of the numerical treatment of various problems from optimization, calculus of variations and optimal control.

MSC:

47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47J25 Iterative procedures involving nonlinear operators
90C25 Convex programming
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