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

• Published in: Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A, Matemáticas Físicas y Naturales. Serie A, Matemáticas, 2020-10, Vol.114 (4), Article 193
• Main Authors: Kong, Dezhou, Liu, Lishan, Wu, Yonghong
• Format: Article
• Language: English
• Subjects: Algorithms; Applications of Mathematics; Cones; Convergence; Hilbert space; Isotonicity; Iterative algorithms; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Optimization; Original Paper; Proximity; Theoretical
• ISSN: 1578-7303; 1579-1505
• DOI: 10.1007/s13398-020-00902-7

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.