Two-step inertial ADMM for the solution of nonconvex nonsmooth optimization problems with nonseparable structure

In this paper, we propose an algorithmic framework called two-step inertial alternating direction methods of multipliers (TIADMM) to solve a class of nonconvex and nonsmooth optimization problems with nonseparable structure. The new algorithm adds two-step inertial effect to each subproblem and introduces three relaxed terms into the dual update step to improve the convergence. This work employs two assumptions; (1) the auxiliary function satisfies the Kurdyka-& Lstrok;ojasiewicz property, and (2) the parameters satisfy some conditions to prove the global convergence of the sequence generated by the algorithm. Furthermore, the algorithm is extended to a linearization vision for solving nonconvex optimization problems. Finally, tests are conducted on two numerical examples to show the effectiveness.