Computing the resolvent of the sum of operators with application to best approximation problems

  • Minh N. Dao
  • Hung M. PhanEmail author
Original Paper


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.


Best 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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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.CARMAUniversity of NewcastleCallaghanAustralia
  2. 2.Department of Mathematical Sciences, Kennedy College of SciencesUniversity of Massachusetts LowellLowellUSA

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