An Inexact Adaptive Operator Splitting Method For Solving Mixed Variational Inequality Problems

Variational inequality(VI)theory provides a unified model to study and investi-gate a wide class of problems arising in pure and applied sciences including optimiza-tion, elasticity, economics, transportation. In recent years, variational inequalities have been extended and generalized in different directions. An important and useful generalization of variational inequalities is a class of mixed variational inequalities involving a nonlinear convex, proper, and lower semicontinuous term. However, due to the presence of the nonlinear term, solving mixed variational inequalities are more generally difficult than variational inequality problems. Some simple iterative algorithms for solving variational inequality (such as projection) can not directly promote to solve the mixed variational inequalities.In the presence of this term, Glowinski[9] developed the auxiliary principle tech-nique to study the existence of a solution of mixed variational inequalities. Using the resolvent operator technique, Noor[19] proposed an implicit method for mixed vari-ational inequalities. But, the numerical experience shows that the method depends significantly on the initial penalty parameter. In this paper, in view of the implic-it method with variable parameters by [24] and the self-adaptive methods which permit step-sizes being selected self-adaptively for solving variational inequality, we suggest and analyze a new inexact operator splitting method with self-adaptive s-trategy for solving monotone and strongly monotone mixed variational inequalities. In this thesis, we establish the global convergence of the method under weaker con-dition that variable parameter is not limited-which. improve the algorithm efficiency relatively. Some numerical results are also presented to demonstrate the behavior of the new method.