Isotonicity of Proximity Operators in General Quasi-Lattices and Optimization Problems

Dezhou Kong (), Lishan Liu () and Yonghong Wu ()
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Dezhou Kong: Shandong Agricultural University
Lishan Liu: Qufu Normal University
Yonghong Wu: Curtin University

Journal of Optimization Theory and Applications, 2020, vol. 187, issue 1, No 5, 88-104

Abstract: 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; 41A65; 47H07; 06B30; 47J20; 47H10 (search for similar items in EconPapers)
Date: 2020
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DOI: 10.1007/s10957-020-01746-2

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