Research On Peaceman-rachford Splitting Method For Nonconvex Nonseparable Optimization Problems

Nonconvex optimization problems appear widely in various practical engineering applications,such as sparse optimization,machine learning,unit commitment and so on.Based on the research of the Peaceman-Rachford(PR)splitting method and its variant on convex optimization and nonconvex optimization,in this dissertation,we mainly explore the PR splitting method to solve nonconvex optimization with nonseparable structure.Firstly,we consider a class of nonconvex nonseparable optimization problems with linear constraints,which the objective function is the sum of two separable structures and one nonseparable structure.The idea of PR splitting method is applied to obtain the solution of these problems,and Bregman distance is added into the subproblem.Then,a Bregman-type PR splitting method is proposed.Under the general assumptions including the estimation region of relaxation factors in the multiplier updating formulas and the bounded assumption of iteration points,the global convergence of the proposed method is obtained.When the merit function satisfies the Kurdyka-?ojasiewicz(KL)property,the strong convergence of the proposed method is proved.Moreover,when the associated KL property function has a special structure,the corresponding convergence rate results are analyzed and obtained.Preliminary numerical experiments show the effectiveness of the two-multiplier correction technique.Furthermore,the idea of linear approximation is applied to the Bregman-type PR splitting method,and a linear approximation Bregman-type PR splitting method is obtained.The second method expands the threshold of relaxation factors in the first one.Under appropriate assumptions,the convergence and convergence rate results of the second method are also given.Finally,numerical experiments are performed to verify the effectiveness of the second proposed method.Secondly,in order to solve a wider class of nonconvex nonseparable optimization problems with linear constraints and separable closed convex set constraints,on the basis of equivalent processing for these optimization problems,we improve the Lagrange multiplier update technique in the traditional splitting algorithm,and propose an improved Bregman-type PR splitting method to solve them.The global convergence and strong convergence of the improved method are proved under appropriate assumptions.Preliminary experimental results show the numerical validity of the improved method.