Abstract:

My research on ADMM dates back to 1997 when I considered the problems from traffic network analysis. Over the last 10 years, the ADMM based on variational inequalities is widely used in optimization. This paper summarizes our research on ADMM over the last 20 years, particularly, the developments in splitting and contraction methods based on ADMM for convex optimization over the last 10 years. We list the main results as well as the motivations. Our analysis is based on the variational inequalities. All methods mentioned fall in a simple unified prediction-correction framework, in which the convergence analysis is quite simple. A through reading will acquaint you with the ADMM, while a more carefully reading may make you familiar with the tricks on constructing splitting methods according to the problem you met. We should notice that the ADMM originates from ALM and PPA, which are good at utilizing the splitting structure. However, it also inherits the intrinsic shortcomings of these first order methods.

Key words: convex optimization, monotone variational inequality, alternating directions method of multipliers, contractive properties, O(1/t) convergence rate, prediction-correction, unified framework