Research On Theory And Algorithm Of Compementarity Problems

The Complementarity Theory and Algorithm, is a new domain in applied mathematics and its subject is the study of complementarity problems. Complementarity problems have the contact relations between fixed point theory and variational inequalities and represent a wide class of mathematical models related to optimization game theory, economics, engineering, mechanics, elasticity, fluid mechanics, stochatic optimal control etc. In this thesis, we study the existence of the solution, the boundness of the solution sets, the iterative algorithm and its convergence. This thesis is divided into five chapters.In chapter 1, we study the existence of the solutions for the Order Complementarity Problems. First, we introduce the concept of order relation and order spaces. And then, we define a new mixed monotone function and study the existence of solutions for order complementarity problems in the method of couple-mixed fixed point. Furthermore, we study the existence of solutions for order complementarity problems in the method of fixed point, respcetively. Finally, we give the new conditions of the solution for Implict Variational Inequalities using the relation bewteen the order complementarity problems and implict variational inequalities.In chapter 2, we introduce and study a class of generalized multi-valued implicit quasi-complementarity problem. First, we introduce the concept of generalized K-area boundness of a multi-valued map, and study the boundedness of the solution sets for the generalized multi-valued implicit quasi-complementarity problem. And then, we present the iterative algorithm for one kind of multi-valued implicit quasi-complementarity problem. Finally, we study the convergence of this algorithm by the properities of the projection mapping P_K and Nadler theorem. In chapter 3, we introduce and study a new system of generalized nonlinear co-complementarity problems and construct an iterative algorithm for approximating the solutions of the system of generalized nonlinear co-complementarity problems in Hilbert spaces. We prove the existence of the solutions for the system of generalized nonlinear co-complementarity problemsand using the relations between complementarity problems, variational ineqalities and fixed point, construct an iterative algorithm and prove the convergence of iterative sequences generated by the algorithm. We also study the convergence and stability of a new perturbed iterative algorithm for approximating the solution.In chapter 4, we study the iterative methods for linear complementarity problems with perturbation and interval data. We introduce the linear complementarity problems with perturbation and interval data which is the generalization of the linear complementarity problems with interval data. Using the properities of interval algorithm, we constructed (T)algorithm and (TI)algorithm for a class of linear complementarity problems with perturbation and interval data, and proved the convergence of these algorithms under some suitable conditions respcetively. In the end of this chapter, we give some numerical examples.In chapter 5, we introduce and study a class of generalized multi-valued nonlinear quasi-complementarity problem, which includes many complementarity problems as special cases. Using a new fixed point theorem of Ansari and Yao, we give a new existence result of solutions for generalized multi-valued nonlinear quasi-complementarity problem in Hausdorff topological linear spaces by the properities of upper semicontinuous multi-valued map with compact values.