An adaptive linearized alternating direction multiplier method with a relaxation step for convex programming

Alternating direction multiplication is a powerful technique for solving convex optimisation problems. When challenging subproblems are encountered in the real world, it is useful to solve them by introducing neighbourhood terms. When the neighbourhood matrix is positive definite, the algorithm converges but at the same time makes the iteration step small. Recent studies have revealed the potential non-positive definiteness of the neighbourhood matrix. In this paper, we present an adaptive linearized alternating direction multiplier method with a relaxation step, combining the relaxation step with an adaptive technique. The novelty of the method is to use the information of the current iteration point to dynamically select the neighbourhood matrix, increase the iteration step size, and speed up the convergence of the algorithm.We prove the global convergence of the algorithm theoretically and illustrate the effectiveness of the algorithm using numerical experiments.