Alternating Direction Method Of Multipliers And Its Applications

With the rapid development of the Internet industry and the the advent of the big data age, there has been a growing usage of sparse representations in signal process-ing, physical inverse problems, data processing and so on. The alternating direction method of multipliers(ADMM) is an efficient method to solve large-scale problems. It divides high dimensional problems into low dimensional subproblems by augmented lagrangian function and solves them iteratively. Although there has been thirty years history for ADMM, it becomes popular recently because of the study of large-scale nonsmooth and sparse optimization problems. Its convergence results for nonconvex optimization problems and its direct generalization to more than three variables are still open problems, however the numerical results are well enough.This paper focusses on the above theoretical and application problems of ADM-M. An introduction of splitting algorithms and related subjects is given in chapter2. Chapter3shows an application in a famous spare representation nonconvex model, dictionary learning problem. A large quantity of experimental results show the prac-ticality of our algorithm. In chapter4, an usage for a special structure optimization problem, i.e., the Pair-wise linear constraint problem, is proposed, and the convergence results of ADMM for more than three variables are analysed. Numerical experiments are done on two special inverse problems. The last part gives a short summary.