Two Kinds Of Alternating Direction Methods Of Multipliers For Nonconvex Optimization Problems

Nonconvex optimization problems widely appear in many practical frontier problems such as sparse optimization,compressed sensing,data mining,image denoising,and machine learning.The alternating direction method of multipliers is an iterative algorithm that effectively solves convex optimization problems.When the objective function is nonconvex,in the case of,the convergence of the algorithm may not be guaranteed.This paper mainly studies two types of improved the alternating direction method of multipliers for solving two types of nonconvex separable optimization problems with linear equality constraints.The research content is as follows :In the first part,for a class of three-block separable nonconvex optimization problem.A type of the regularized alternating direction method of multipliers is proposed.First,this paper establishes the global convergence of the algorithm.Second,the augmented Lagrangian function satisfies under the condition of the KL property,the strong convergence of the algorithm is proved.Finally,the effectiveness of the algorithm is verified through numerical experiments.In the second part,for a class of two-block separable nonconvex optimization problem,an inertial symmetric regularized alternating direction method of multipliers is proposed,this method combines the basic ideas of regularization and inertia on the basis of the symmetric alternating direction method of multipliers.The current iteration information and the previous iteration information are used to generate new iteration points,thereby accelerating the convergence of the algorithm,and this method also has a certain range of relaxation parameters,which can be appropriately selected in practical applications.This paper establishes the global convergence of the algorithm,and when the potential function satisfies the KL property,the strong convergence of the algorithm is proved.Finally,the effectiveness of the algorithm is verified by numerical experiments.