Projection and proximal point methods: convergence results and counterexamples

By Heinz H. Bauschke, Eva Matouskova, Simeon Reich. 2004

268 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/8e314725f00eb56765bceb97a06bc5ce0f0fdd4b

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Citations in the corpus

The asymptotic behavior of the composition of two resolvents (2005, background): “On the other hand, it is known that strong convergence may fail to hold, even when both proximity operators are projectors (see [28] and [33]) or even when only a single proximity operator is iterated (see [13] and [27]).”
The proximal point algorithm in metric spaces (2012, background): “It is worth mentioning that counterexamples to strong convergence of the PPA are still very rare [2, 4].”
Strong asymptotic convergence of evolution equations governed by maximal monotone operators with Tikhonov regularization (2008, background): “More examples of this kind have appeared recently in [8,9], based on results of [23].”
A new proximal point iteration that converges weakly but not in norm (2005, background): “For another iteration of the proximal point algorithm that converges weakly but not in norm, also based on Hundal’s alternating projections example, we refer the reader to [5].”
The kernel average for two convex functions and its application to the extension and representation of monotone operators (2009, unknown): “We are now able to recover a known result about the proximal average; see [ 6, Theorem 6.1 ].”
Alternating projections in CAT(0) spaces (2011, background): “There is also a fruitful connection to other well-known algorithms (like the proximal point algorithm), see [4,8].”
Weak convergence of infinite products of operators in Hadamard spaces (2016, result): “In the Hilbert ball case this result was proved in [6].”
Two Strong Convergence Theorems for a Proximal Method in Reflexive Banach Spaces (2010, background): “1) converges weakly but not strongly; see [7] for a more recent and simpler example.”