Non-convex Variational Inequality Problem And Its Algorithm Research

Bounkhel,Tauj and Hamdi[13]introduced and considered a new class of vari-ational inequalities in the context of nonconvex sets(which are uniformly r-prox-regular and includes the class of convex sets as special case)in 2003,so-called nonconvex variational inequalities.Subscquently,Noor[33,34]used the projection method to investigated the nonconvex variational inequalities,and prove the equiv-alence between the nonconvex variational inequalities and a fixed point problem which uses the concept of projection.Specially,Noor[33]established an explicit projection algorithm to solve the nonconvex variational inequality problem,and under the condition of T has strongly monotone operator proved that the itera-tive sequences which are obtained by the method of convergence to the solution of nonconvex variational inequality problem.There is no doubt that the condition is too strong and exclude many problems.In fact,in finite dimensional space,the condition can be weakened into a monotonicity.In 2010,Noor[36]using the equivalence between the nonconvex variational in-equalities and a fixed point problem studied the existence of nonconvex variational inequalities,and consider a new two-step iterative algorithm for finding the so-lution of the nonconvex variational inequalities.In addition,Ansari and Balooee[1,2],Balooee[8,9]also have considered different types of their generalizations in the context of nonconvex sets.In view of the important role in the real world for the non-convex variational inequality problem and its generalizations,it is worth further study.Inspired and motivated by the above works,in this paper,we mainly study the nonconvex variational inequalities from three aspects:the system of generalized regularized nonconvex variational inequalities,the system of extended regularized nonconvex variational inequalities,and the extended regularized nonconvex mixed bifunction variational inequalities,enrich the theory of nonconvex variational in-equalities.Among them,we establish the equivalence between the system of gen-eralized regularized nonconvex variational inclusions(the system of extended reg-ularized nonconvex variational inequalities)and a fixed point problem,and prove the existence and uniqueness of the solution of this system.In particular,we con-sider a class of new three-step projection iterative algorithms to find the common element of the set of fixed points of the nearly uniformly Lipschitzian mapping and the unique solution of the system of generalized regularized nonconvex variational inclusion.Moreover,we introduce a new perturbed projection iterative algorithm with mixed errors for finding the solution of the system of extended regularized nonconvex variational inequalities.Finally,with the help of the auxiliary principle technique,some predictor-corrector algorithms for solving the extended regularized nonconvex mixed bifunction variational inequalities are proposed and analyzed.