A Proximal-Perturbed Bregman ADMM for Solving Nonsmooth and Nonconvex Composite Optimization

In this paper, we focus on a linearly constrained composite minimization problem involving a possibly nonsmooth and nonconvex objective function. Unlike the traditional construction of the augmented Lagrangian function, we design a proximal-perturbed augmented Lagrangian to develop a new Bregman-type alternating direction method of multipliers. Under mild assumptions, we prove that the augmented Lagrangian sequence converges to the limit of the objective function sequence, and the iterative sequence generated by our method converges to a stationary point of the problem. The sublinear convergence rate of the primal residuals is also analyzed. Comparative experiments on testing the linear equation problem, graph-guided fused lasso problem and robust principal component analysis problem demonstrate the efficiency and flexibility of the proposed method.