Two Kinds Of Generalized Alternating Direction Method Of Multipliers For Nonconvex Optimization Problems

Nonconvex optimization plays an important role in nonlinear programming,and its application background is very wide,such as statistical learning,image processing,signal recovery,machine learning and so on.This thesis mainly focuses on the two-block nonconvex optimization with linear constraints.Based on the structure of the problem and its special properties,two new types of generalized alternating direction method of multipliers are designed,and under suitable conditions,the convergence of the our algorithms are proved.The main research content of the thesis and contribution are below:1.we propose an incremental aggregated proximal symmetric ADMM for minimizing the sum of smooth component functions and a nonsmooth function,both of which are allowed to be nonconvex.But for large-scale problems,the smooth subproblem is solved directly by the gradient descent method,which is computationally expensive.In order to reduce the calculation costs,combined with the idea of incremental gradient method,we only need to calculate part of the gradient to approximate the full gradient at each iteration.On the other hand,based on Kurdyka-?ojasiewicz property,the strongly convergence of our algorithm is analyzed.In order to verify the effectiveness of the algorithm,it is applied to sparse logistic regression problem.2.An inertial Bregman generalized alternating direction method of multiplies is proposed for solving the two-block nonconvex optimization.At each iteration,we choose appropriate the Bregman distance to simplify subproblem and improve numerical performance by combining inertial technology.Under some assumptions,the convergence of the algorithm is established.Finally,the efficiency of the proposed method is verified by numerical experiments of signal recovery and SCAD penalty problems.