Research On The Algorithms For Semi-definite Complementarity Problems

Compared with complementarity problem, nonnegative vector is replaced by block- diagonal symmetric positive semi-definite real matrice. Thus semi-definite com-plementarity problem is an extension of complementarity problem. The semi-definite complementarity problems have very important applications in the fields of economics,optimization and engineering, researchers in optimization area have paid great attention to it. The algorithm which is proposed based on merit functions is a class of important numerical method for the semi-definite complementarity problems. In this paper, semi-definite complementarity problems are effectively transformed into unconstrained optimization problems. Purpose of this paper is to study the method that is used to solve the semi-definite complementarity problems based on the merit function. For detail, we conclude them as follows:Firstly, the semi-definite complementarity problem is introduced. We know the merit functions have played an important role in the solution of complementarity prob-lems defined over the cone of nonnegative real vectors, and there we study the exten-sion of these merit functions to complementarity problems defined over the cone of block- diagonal symmetric positive semi-definite real matrices. And then based on these merit functions and the methods that are used to solve unconstrained optimization prob-lems, this paper proposed a descent algorithm for solving the semi-definite complemen-tarity problems. Under certain conditions, the convergence of the algorithm is proved, and some numerical tests are also presented. Being limited by merit functions, the de-creasing direction of the algorithm is determined. This paper discusses only the selec-tion of step-size factor, so the improved algorithm is given. Under the suitable condi-tions, the convergence of the improved algorithm above is known as well. At last, the conclusions are given.