The Study Of Numerical Methods For Solving Several Types Of Complementarity Problems

The complementarity problem is a mathematical model in operations research.Linear complementarity problem and implicit complementarity problem are special cases of complementarity theory.The linear complementarity problem does not only push forward an immense influence on the respects of the scientific computing and engineering applications,e.g.,the free boundary problems,the bimatrix games as well as the market equilibrium,but also is conceived as a unifying formulation for the linear and quadratic programming problems.The implicit complementarity problem is frequently applied to stochastic optimal control problems.As for these special complementarity problems,we mainly proposes different numerical methods to solve these problems and prove the corresponding convergences in this dissertation.We show these effectiveness by some numerical examples.This paper is organized as below:In the first chapter,we mainly introduce the research background of these complementarity problems,and give some basic concepts and related lemmas needed to prove the convergence of the proposed algorithms.In the second chapter,by means of constructing the linear complementarity problems into the corresponding absolute value equation,we raise an iteration method,called as the nonlinear lopsided HSS-like modulus-based matrix splitting iteration method,for solving the linear complementarity problems whose coefficient matrix in Rn×n is large sparse and positive definite.From the convergence analysis,it is appreciable to see that the proposed method will converge to its accurate solution under appropriate conditions.Numerical examples demonstrate that the presented method precedes to other methods in practical implementation.In the third chapter,a generalized two-step modulus-based matrix splitting iteration method for solving the implicit complementarity problems has been presented.The convergent analysis with the system matrix being H+-matrices are also discussed.Numerical experiments illustrate that our method is advantageous to the existing methods.The the last chapter,we summarize the content described in this dissertation and put forward some ideas that can be studied in the future.