Alternating Direction Method of Multipliers for Non-smooth and Non-convex Optimization: Convergence, Complexity and Applications

Abstract: 

In the era of big data, large scale structured non-convex and non-smooth optimization problem can be found in various contemporary applications such as engineering, machine learning and signal denoising. The objective functions are often expressed as the sum of a smooth function f and a non-smooth regularization function P. An efficient and popular first order numerical method for solving these problems is the so called Alternating Direction Method of Multipliers (ADMM). The convergence behaviour of the ADMM method in the general non-convex setting is poorly understood although it has been successfully applied with excellent numerical performance in practice. This talk will outline the convergence theory and implementation of the ADMM method and present some numerical experiments to highlight the effectiveness of the method.