Inertial Alternating Minimization Method With Bregman Distance For Nonconvex And Nonsmooth Optimization Problems

A great deal of application problems are actually nonconvex and nonsmooth,such as sparse signal recovery,image processing and matrix decomposition.Therefore,it is of great value to study the algorithm for solving nonconvex and nonsmooth optimization problems.In this paper,we consider a class of nonconvex nonsmooth optimization problems with structure that the objective function can be expressed as the sum of three functions.The alternating minimization method is the most intuitive way to solve such problems.Inertial technology can effectively improve the numercial results of algorithms.In our paper,an inertial alternating minimization algorithm with Bregman distance is proposed for solving nonconvex and nonsmooth optimization problems.Firstly,based on the proximal alternating method,combining with inertial technology and Bregman distance,we proposed an inertial alternating minimization algorithm with Bregman distance(BIAM)and proved the global convergence of this algorithm.Suppose that the merit function satisfies Kurdyka-?ojasiewicz property,we proved that each bounded sequence generated by BIAM strongly converges to a critical point.Furthermore,the linearized BIAM algorithm and its related theoretical results are presented.Finally,we apply the BIAM algorithm to signal recovery and image denoising problems,which shows the efficiency of the algorithm.Secondly,according to the characteristics of objective function,a Bregman inertial alternating linearized minimization with nonsmooth coupling term(IBALM)is proposed by combining the inertial technology.Under appropriate conditions,we analyzed the global convergence and strong convergence of IBALM algorithm.