Research On The Algorithms For Semidefinite Complementarity Problems

As a new mathematical model,"Complementarity Problem"was early called"Composite problem","Fundamental problem"or"Complementarity pivot problem"and so on, It is one of the basic topics in optimization. Its formation by promoting linear programming and nolinear programming. Complementarity Problems are closely associated with Mathematical programming, Variational inequalities, Fixed point problems, Generalized equations and so on. In the study of it, We use some theories in Nonlinear analysis and Topology. It is regarded as a crossover in Applied mathematics, Computational mathematics and the Basic mathematics. For example, Matrix game problems, Economic equilibrium problems, Traffic flow equilibrium problems, Contact problems, Free boundary problems and Goods issues in the supply chain are all translated into the model of Complementarity Problems.The Semidefinite Complementarity Problem is an intersecting research field for Semidefinite programming and complementarity theory. It plays important roles in many real sectors such as mechanics,engineering,economy,transportation and so on. So, The research on it changes a hotspot.The Non-Interior Continuation Method is a algorithm that is based on the Smoothing FB function (Abbreviation for FB function) theory and Centre path principle, By reducing the value of the smoothing parameter, We use the solution of the smoothing regenerated equations to approximate the solution of the nonsmoothing equations that we need to solve, gradually. Therefore, we can get the approximate solution of the problem. We can solve Complementarity Problems and Semidefinite Complementarity Problems effectively by this algorithm.In this paper, Firstly introduced are theoretical knowledge about Complementarity Problems and the nature of Smoothing FB function, and a Non-Interior Continuation Method for solving Complementarity Problems is introduced, The relevant analysis about this algorithm is followed. Then, Based on the smoothing FB function theory and centre path principle, We extend theoretical knowledge and the Non-Interior Continuation Method about Complementarity Problems to Semidefinite Complementarity Problem, and The algorithm is shown to be both globally linearly convergent and local quadratically convergent under some proper assumptions. Preliminary numerical experiments are reported to show the efficiency of the algorithm for solving Semidefinite Complementarity Problem. What is more, Numerical experiments result shows that the selections of parameterσhave certain effect on the efficiency of the algorithm (The greater the parameterσ(no more than 1), the moe efficient the algorithm).