Convergence of Bregman Peaceman–Rachford Splitting Method for Nonconvex Nonseparable Optimization

This work is about a splitting method for solving a nonconvex nonseparable optimization problem with linear constraints, where the objective function consists of two separable functions and a coupled term. First, based on the ideas from Bregman distance and Peaceman–Rachford splitting method, the Bregman Peaceman–Rachford splitting method with different relaxation factors for the multiplier is proposed. Second, the global and strong convergence of the proposed algorithm are proved under general conditions including the region of the two relaxation factors as well as the crucial Kurdyka–Łojasiewicz property. Third, when the associated Kurdyka–Łojasiewicz property function has a special structure, the sublinear and linear convergence rates of the proposed algorithm are guaranteed. Furthermore, some preliminary numerical results are shown to indicate the effectiveness of the proposed algorithm.