Convergence of Peaceman-Rachford splitting method with Bregman distance for three-block nonconvex nonseparable optimization

Abstract It is of strong theoretical significance and application prospects to explore three-block nonconvex optimization with nonseparable structure, which are often modeled for many problems in machine learning, statistics, and image and signal processing. In this article, by combining the Bregman distance and Peaceman-Rachford splitting method, we propose a novel three-block Bregman Peaceman-Rachford splitting method (3-BPRSM). Under a general assumption, global convergence is presented via optimality conditions. Furthermore, we prove strong convergence when the augmented Lagrange function satisfies Kurdyka-Łojasiewicz property. In addition, if the association function possessing the Kurdyka-Łojasiewicz property exhibits a distinctive structure, then linear and sublinear convergence rate of 3-BPRSM can be guaranteed. Finally, a preliminary numerical experiment demonstrates the effectiveness.