Self-adaptive ADMM for semi-strongly convex problems

In this paper, we develop a self-adaptive ADMM that updates the penalty parameter adaptively. When one part of the objective function is strongly convex i.e., the problem is semi-strongly convex, our algorithm can update the penalty parameter adaptively with guaranteed convergence. We establish various types of convergence results including accelerated convergence rate of O(1/k^2), linear convergence and convergence of iteration points. This enhances various previous results because we allow the penalty parameter to change adaptively. We also develop a partial proximal point method with the subproblem solved by our adaptive ADMM. This enables us to solve problems without semi-strongly convex property. Numerical experiments are conducted to demonstrate the high efficiency and robustness of our method.