Abstract
We extend the concept of the proximal map** of a real valued function on a convex set to the one of a multi-valued operator, and study its fixed point properties. We show that the extended proximal map** possesses certain contraction (resp., nonexpansiveness, approximate nonexpansiveness) properties when the multi-valued operator involved is strongly monotone (resp., cocoercive, monotone). Applications to multi-valued variational inequality and equilibrium problems are discussed.
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Acknowledgements
The authors are grateful to the editor and anonymous reviewers for insightful comments and suggestions that help improve the presentation of the paper. This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.01-2020.06. Part of this work was done during research stay of the second author (Xuan Thanh Le) at Vietnam Institute for Advanced Study in Mathematics.
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Muu, L.D., Le, X.T. & Hai, N.N. On the proximal map** for multi-valued monotone variational inequality problems. Optim Lett 17, 369–383 (2023). https://doi.org/10.1007/s11590-022-01879-5
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DOI: https://doi.org/10.1007/s11590-022-01879-5