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

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Abstract

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.

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Acknowledgements

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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Authors and Affiliations

  1. CARMA, University of Newcastle, Callaghan, NSW, 2308, Australia

    Minh N. Dao

  2. Department of Mathematical Sciences, Kennedy College of Sciences, University of Massachusetts Lowell, Lowell, MA, 01854, USA

    Hung M. Phan

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  1. Minh N. Dao
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  2. Hung M. Phan
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Correspondence to Hung M. Phan.

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Dao, M.N., Phan, H.M. Computing the resolvent of the sum of operators with application to best approximation problems. Optim Lett 14, 1193–1205 (2020). https://doi.org/10.1007/s11590-019-01432-x

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  • DOI: https://doi.org/10.1007/s11590-019-01432-x

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