Smoothing randomized block-coordinate proximal gradient algorithms for nonsmooth nonconvex composite optimization

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, one of which is block separable. The pivotal tool of our analysis is the connection of the proximal gradient mapping with V-proximal mapping and Bregman proximal mapping. The S-RBCPG algorithm overcomes the non-smoothness of the objective function by utilizing the smoothing technique and we establish its subsequential convergence. Further, the B-RBCPG algorithm is designed for the case where the separable function is relatively smooth (that is, each separation part is relatively smooth). Then, we establish the \(\varvec{\mathcal {R}}\) $\mathcal{R}$ -linear convergence rate of the B-RBCPG algorithm under expectation by assuming the Kurdyka-Łojasiewicz property on the objective function. Finally, we use some numerical experiments to illustrate the effectiveness and convergence of the proposed algorithms.