A Dynamical Splitting Method for Minimizing the Sum of Three Convex Functions. (25th January 2022) - Zhu, Ming; Hu, Rong; Fang, Ya-Ping
Abstract: In this paper, we develop a dynamical splitting method based on a nonlinear dynamical system with two proximal operators for solving the minimum value problem of the sum of three convex functions. The method endows some existing classical splitting methods such as the Forward-Backward splitting method and the Douglas-Rachford splitting method with continuous time behaviors. We demonstrate that the proposed dynamical system converges globally to a point whose image, imaged by each of the two proximal operators, is a solution to the minimum value problem. We also shows that both the residual of the two proximal operators and the objective error enjoy the convergence rates of o ( 1 t ) where t denotes time. This rate can be improved to O ( 1 t ) by using an ergodic average technique. Moreover, under certain smoothness and metric subregularity conditions, we further establish the global exponential convergence of the dynamical splitting method. Finally, we apply the dynamical splitting method to deal with the problem of compressive sensing.