Inertial Splitting Methods For Nonconvex And Nonsmooth Problems

Nonconvex and nonsmooth optimization problems widely appear in practical problems,such as sparse signal recovery,image noise reduction and so on.It is of great theoretical and practical value to study the optimization algorithm for nonconvex and nonsmooth optimization problems.In this paper,two kinds of nonconvex and nonsmooth optimization problems with block structure are considered.The splitting algorithm is commonly used to solve the optimization problem with block structure.In the construction of the algorithm,inertial technology can effectively reduce the number of iterations of the algorithm.In this paper,two kinds of inertial splitting algorithms for nonconvex and nonsmooth optimization problems are studied.Firstly,a class of nonconvex and nonsmooth optimization problems with linear constraints is considered.Peaceman-Rachford splitting method(PRSM)is an effective algorithm for solving two-block convex optimization problems.In this paper,an inertial proximal Peaceman-Rachford splitting method(IPPRSM)is proposed by combining the inertial technology and PRSM.In the case that the objective function is a nonconvex,nonsmooth function and a strongly convex continuously differentiable function,when the penalty parameters meet the appropriate conditions,it is proved that any cluster point of the sequence generated by the algorithm is the stationary point of the problem.The strongly convergence of IPPRSM is proved under the condition that the merit function satisfies the Kurdyka-?ojasiewicz property.Furthermore,this paper explores how to use the proposed algorithm to solve the nonconvex and nonsmooth optimization problems with linear constraints when the objective function is the sum of nonconvex and semiconvex function.The effectiveness and stability of the algorithm are verified by numerical experiments.Secondly,we consider a nonconvex and nonsmooth unconstrained optimization problem in which the objective function is the sum of three functions.The alternating minimization algorithm is an effective algorithm to solve this kind of structural optimization problem.In this paper,combined with the inertial technique and regularization technology,an efficient Bregman inertial alternating linearized minimization algorithm for solving the problem is proposed.The effectiveness of the algorithm are verified by numerical experiments.