Computing the resolvent of the sum of operators with application to best approximation problems
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We propose a flexible approach for computing the resolvent of the sum of weakly monotone operators in real Hilbert spaces. This relies on splitting methods where strong convergence is guaranteed. We also prove linear convergence under Lipschitz continuity assumption. The approach is then applied to computing the proximity operator of the sum of weakly convex functions, and particularly to finding the best approximation to the intersection of convex sets.
KeywordsBest approximation Douglas–Rachford algorithm Linear convergence Operator splitting Peaceman–Rachford algorithm Projector Proximity operator Resolvent Strong convergence
The authors thank two anonymous referees for their careful reading and helpful comments. MND was partially supported by Discovery Project 160101537 from the Australian Research Council. HMP was partially supported by Autodesk, Inc. via a gift made to the Department of Mathematical Sciences, University of Massachusetts Lowell.
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