Isotonicity of the proximity operator and mixed variational inequalities in Hilbert spaces
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Isotonicity of the proximity operator and mixed variational inequalities in Hilbert spaces Dezhou Kong1
· Lishan Liu2,3 · Yonghong Wu3
Received: 16 October 2019 / Accepted: 10 July 2020 © The Royal Academy of Sciences, Madrid 2020
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. Keywords Cone · Order · Proximity operator · Isotonicity · Mixed variational inequality Mathematics Subject Classification 41A65 · 47H07 · 06B30 · 47J20 · 47H10
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Dezhou Kong [email protected] Lishan Liu [email protected] Yonghong Wu [email protected]
1
College of Information Science and Engineering, Shandong Agricultural University, Taian 271018, Shandong, China
2
School of Mathematical Sciences, Qufu Normal University, Qufu 273165, Shandong, China
3
Department of Mathematics and Statistics, Curtin University of Technology, Perth, WA 6845, Australia 0123456789().: V,-vol
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1 Introduction In [1–3], motived by the problems in non-smooth mechanics, Moreau introduced the proximity operator. Since then, proximity operators have become prominent in convex optimization theory. For instance, they play a central theoretical role in convex analysis (see [4]). In term of application, their increasing presence is particularly manifested in the broad area of data processing, where they were introduced in [5]. Moreover, one has proven that proximity operators are very effective in the modeling and the numerical solution of a vast array of problems in disciplines such as image recovery, signal processing, computational statistics and machine learning (see [6–13]). Among methods for establishing the solvability of variational inequality problems and furnishing their solutions, an important one consists in deriving increasing and convergent iterative processes with respect to some order relations, induced by closed and convex cones. This approach was first introduced by Isac and Németh [14], where the isotonicity of the metric projection o
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