Study On Numerical Methods For Cone Constrained Variational Inequality Problems

It is well known that variational inequality problems have many important applications in operation research,computer science,system science,engineering technology,transportation, economics and management ect.In the last 20 years of the twentieth century,great attentions have been paid by many scholars in this direction.Meanwhile,conic optimization,especially semidefinite programming and second order cone programming,is an active field in optimization community.However,the study of variational inequality problems with cone constraints is not enough.Based on this observation,this dissertation focuses on the study of convergence analysis of numerical methods for variational inequality problems with cone constraints,including semismooth Newton methods and smoothing function methods for second-order cone constrained variational inequality problems and proximal point methods for variational inequality problems in Hilbert space.The main results of this dissertation can be summarized as follows:1.In the second chapter,the Karush-Kuhn-Tucker system of a second order cone constrained variational inequality problem is transformed into a semismooth system of equations with the help of Fischer-Burmeister operators over second order cones.The Clarke generalized differential of the semismooth mapping is presented.A modified Newton method with Armijo line search is proved to have global convergence with local superlinear rate of convergence under certain assumptions on the variational inequality problem.An illustrative example is given to show how the globally convergent method works.2.The third chapter mainly deals with the smooth method for second-order cone constrained variational inequality problems.The Karush-Kuhn-Tucker system of a second order cone constrained variational inequality problem is transformed into a smooth system of equations E=0 with the help of smoothing Fischer-Burmeister operators over second order cones.Under some conditions,we prove the Jacobin JE of E is nonsingular.A global convergent smooth Newton method is given for solving the smooth system of equations. 3.Chapter 4 discuss the method based on the resolvent operator for general variational inequality. A new monotonicity,M-monotonicity,is introduced.With the help of resolvent operator, an equivalence between the variational inequality VI(C,F+G) and the fixed point problem of a nonexpansive mapping is established.A proximal point algorithm is constructed to solve the fixed point problem,which is proved to have a global convergence under the condition that F in theⅥproblem is strongly monotone and Lipschitz continuous. Furthermore,a convergent path Newton method,which is based on the assumption that the projection mapping∏_C(·) is semismooth,is given for calculatingε-solutions to the sequence of fixed point problems,enabling the proximal point algorithm implementable.