Two Improved The Alternating Direction Method Of Multipliers For Nonconvex Problems

Nonconvex two blocks optimization problems are widely used in data mining,signal and image processing,sparse optimization,neural network,support vector machine and other practical problems.The alternating direc-tion method of multipliers is an effective method for solving the two blocks convex optimization problems.However,when there is a nonconvex func-tion in the objective function,the convergence of the classical of the alter-nating direction method of multipliers cannot be guaranteed.In this paper,two improved the alternating direction method of multipliers are proposed for nonconvex unconstrained optimization problems and nonconvex linear constrained optimization problems.The research content is as follows:Firstly,we propose a regularization of the alternating direction method of multipliers for solving the nonconvex unconstrained optimization problem.The global convergence of the algorithm is proved.And under the hypothe-sis that the augmented Lagrangian function satisfies the Kurdyka-Lojasiewicz property,the strong convergence of the algorithm are analyzed.The numer-ical results show the validity of the new algorithm of the numerical experi-ments.Secondly,for solving the nonconvex linear constrained optimization problem,an inertial proximal alternating direction method of multipliers is proposed.The algorithm combines the basic ideas of regularization technol-ogy and inertial technology,and analyzes the global convergence and strong convergence of the algorithm.The algorithm is applied to solve the noncon-vex economic dispatch problem.