2022/06/14 by Walaa M. Moursi, Moursi, Walaa M.
Computer Science · Mathematics · #47H05 #47H09 #47H14 #49M27 #49M29 #49N15 #90C25 #90C46 #Advanced Optimization Algorithms Research #Contact Mechanics and Variational Inequalities #FOS: Mathematics #Functional Analysis (math.FA) #Optimization and Control (math.OC) #Optimization and Variational Analysis
paper · pdf · doi:10.48550/arxiv.2206.07204
openalex publication_date 2022/06/14 · openalex created_date 2022/06/18 · openalex updated_date 2026/07/28
The Douglas-Rachford algorithm is one of the most prominent splitting algorithms for solving convex optimization problems. Recently, the method has been successful in finding a generalized solution (provided that one exists) for optimization problems in the inconsistent case, i.e., when a solution does not exist. The convergence analysis of the inconsistent case hinges on the study of the range of the displacement operator associated with the Douglas-Rachford splitting operator and the corresponding minimal displacement vector. In this paper, we provide a formula for the range of the Douglas-Rachford splitting operator in (possibly) infinite-dimensional Hilbert space under mild assumptions on the underlying operators. Our new results complement known results in finite-dimensional Hilbert spaces. Several examples illustrate and tighten our conclusions.