A two-step inertial Bregman symmetric ADMM-type algorithm with KL-property for nonconvex nonsmooth nonseparable optimization problems with application

The Alternating Direction Method of Multipliers (ADMM) is a simple and effective approach for solving the separable optimization problems. However, research on the convergence of the ADMM algorithm which the objective function includes coupled term is still at an early stage. In this paper, we propose an algorithm that combines the two-step inertial technique, Bregman distance, and symmetric ADMM to address nonconvex and nonsmooth nonseparable optimization problems. Under certain assumptions, we proved the sequence generated by the algorithm is bounded and converges to the stability point of the generalized Lagrange function. Additionally, we establish the global convergence of the algorithm under the condition that the auxiliary function satisfies the Kurdyka–Łojasiewicz property. To evaluate the effectiveness of the proposed algorithm, we conduct experiments on the penalized regularization SCAD model and the Matrix decomposition. The results indicate that our algorithm performs effectively in practical applications.