Isotonicity of Proximity Operators in General Quasi-Lattices and Optimization Problems
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Isotonicity of Proximity Operators in General Quasi-Lattices and Optimization Problems Dezhou Kong1
· Lishan Liu2,3 · Yonghong Wu3
Received: 20 October 2019 / Accepted: 3 September 2020 / Published online: 18 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Motivated by the recent works on proximity operators and isotone projection cones, in this paper, we discuss the isotonicity of the proximity operator in quasi-lattices, endowed with general cones. First, we show that Hilbert spaces, endowed with general cones, are quasi-lattices, in which the isotonicity of the proximity operator with respect to one order and two mutually dual orders is then, respectively, studied. Various sufficient conditions and examples are introduced. Moreover, we compare the proximity operator with the identity operator with respect to the orders. As applications, we study the solvability and approximation results for the nonconvex nonsmooth optimization problem by the order approaches. By establishing the increasing sequences, we, respectively, discuss the region of the solutions and the convergence rate, which vary with combinations of the mappings, and hence, one can choose the proper combination of the mappings under specific conditions. Compared to other approaches, the optimal solutions are obtained and inequality conditions hold only for comparable elements with respect to the orders. Keywords Quasi-lattice · Cone · Proximity operator · Isotonicity · Nonconvex nonsmooth optimization problem
Communicated by Sándor Zoltán Németh.
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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, Shandong, China
2
School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong, China
3
Department of Mathematics and Statistics, Curtin University, Perth, Australia
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Journal of Optimization Theory and Applications (2020) 187:88–104
89
Mathematics Subject Classification 41A65 · 47H07 · 06B30 · 47J20 · 47H10
1 Introduction The proximity operator was first introduced by Moreau [1–3]. Since then, proximity operators have become more and more prominent in many disciplinary areas such as convex analysis, optimization, data processing, image processing, signal processing and machine learning (see [4–11]). To solve optimization problems, one of the important methods is to establish an increasing or decreasing sequence to find the solution, by adopting the isotonicity of proximity operators with respect to orders, induced by cones. This method can be traced back to Isac and Németh [12] and has provided many new results on optimization problems [13–18]. Note that most of these isotonicity results are with respect to the metric projection operator and only one order under strong conditions; for instance, the cone is minihedral, self-dual and a special cone [19–24]. As far as we know, the isotonicity of the proximity operator is seldom considered, especially in
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