Smoothing Randomized Block-coordinate Proximal Gradient Algorithms for Nonsmooth Nonconvex Composite Optimization

Abstract

In this paper, we propose a smoothing randomized block-coordinate proximal gradient (S-RBCPG) algorithm and a Bregman randomized block-coordinate proximal gradient (B-RBCPG) algorithm for minimizing the sum of two nonconvex nonsmooth functions and one of which is separable. The pivotal tool of our analysis is the connection of the proximal gradient mapping with V-proximal mapping and Bregman proximal mapping. S-RBCPG algorithm overcome the non-smoothness issue of the objective function by utilizing the smoothing technique and we establish its subsequential convergence. Further, B-RBCPG algorithm is designed for cases where the separable function is separately relative smooth (that is, each separation part is relative smooth). Then, we establish the global convergence and R-linear convergence rate of the B-RBCPG algorithm under expectation by assuming the Kurdyka-Łojasiewicz property on the objective function without any convexity assumption. Finally, we use numerical experiments to illustrate the effectiveness and convergence of the proposed algorithms. MSC Classification: 90C30 , 90C26 , 65K05