An extended proximal ADMM algorithm for three-block nonconvex optimization problems

We propose a new proximal alternating direction method of multipliers (ADMM) for solving a class of three-block nonconvex optimization problems with linear constraints. The proposed method updates the third primal variable twice per iteration and introduces semidefinite proximal terms to the subproblems with the first two blocks. The method can be regarded as an extension of the method proposed in Sun et al. (2015) which is specialized to the convex case with the third block of the objective function being quadratic. Based on the powerful Kurdyka–Łojasiewicz property, we prove that each bounded sequence generated by the proposed method converges to a critical point of the considered problem. Some numerical results are reported to indicate the effectiveness and superiority of the proposed method.