A new proximal point iteration that converges weakly but not in norm

By Heinz H. Bauschke, James V. Burke, Frank Deutsch, Hein Hundal, J. Vanderwerff. 2005

54 citations recorded by the source corpus.

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

View Paper: https://www.semanticscholar.org/paper/528d87f23c96d62c2c3fe98df9cda12fcd94d625

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In Computational Theory and Mathematics

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Citations in the corpus

Primal-Dual Splitting Algorithm for Solving Inclusions with Mixtures of Composite, Lipschitzian, and Parallel-Sum Type Monotone Operators (2011, background): “50) which is known to converge weakly but not strongly [7, 25].”
SURVEY: SIXTY YEARS OF DOUGLAS–RACHFORD (2018, background): “An important result of Hundal shows that AP may not converge in norm for the convex case when H is infinite dimensional [102] (see also [20, 111]).”
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].”
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].”
Fitzpatrick functions, cyclic monotonicity and Rockafellar's antiderivative (2007, methodology): “A counterexample was provided by G¨uler [24] (see also [ 5 ] for a recent variant).”
Evolution equations for maximal monotone operators: asymptotic analysis in continuous and discrete time (2009, result): “A simpler example of this type can be found in [14] and can be retranslated to provide a new counterexample for strong convergence of the continuous trajectory, different from that of Baillon.”
Projection and proximal point methods: convergence results and counterexamples (2004, methodology): “In [ 10 ], Fact 4.1 has been interpreted as a simple counterexample to norm convergence of the proximal point algorithm with ∞n ! 0. We now realize that Corollary 7.1 provides another counterexample of this type, for the case when ∞n · 1.”