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Isotonicity of the proximity operator and mixed variational inequalities in Hilbert spaces

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Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

In this paper, the isotonicity of the proximity operator and its applications are discussed. We first establish a few new conditions of the mappings such that their proximity operators are isotone with respect to orders induced different minihedral cones. Some properties and examples for these conditions are then introduced. We especially consider the isotonicity of the proximity operator with respect to one order induced by a subdual cone and two orders. To estimate the convergence rate of the iterative algorithms, some other inequality characterizations of the proximity operator with respect to the orders are then proved. As applications, some solvability and approximation theorems for the mixed variational inequality and optimization problems are established by order approaches, in which the mappings need not to be continuous and the solutions are optimal with respect to the orders. By using the isotonicity of the proximity operator with respect to two orders, we overcome the absence of the regularity of the order. The convergence rate of forward–backward algorithms is finally estimated by order approaches.

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Acknowledgements

The authors would like to thank the referees for their very important comments that improve the results and the quality of the paper. The authors were supported financially by the National Natural Science Foundation of China (11871302, 71773067), the Natural Science Foundation of Shandong Province of China (ZR2017MA034) and Key R&D Program of Shandong Province (2019GGX101024).

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Authors and Affiliations

  1. College of Information Science and Engineering, Shandong Agricultural University, Taian, 271018, Shandong, China

    Dezhou Kong

  2. School of Mathematical Sciences, Qufu Normal University, Qufu, 273165, Shandong, China

    Lishan Liu

  3. Department of Mathematics and Statistics, Curtin University of Technology, Perth, WA, 6845, Australia

    Lishan Liu & Yonghong Wu

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  1. Dezhou Kong
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  2. Lishan Liu
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  3. Yonghong Wu
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Correspondence to Dezhou Kong.

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Kong, D., Liu, L. & Wu, Y. Isotonicity of the proximity operator and mixed variational inequalities in Hilbert spaces. RACSAM 114, 193 (2020). https://doi.org/10.1007/s13398-020-00902-7

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  • DOI: https://doi.org/10.1007/s13398-020-00902-7

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