  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.
