The Conjugate Gradient Method For Solving Nonlinear Complementarity Problems

The main work of this dissertation includes two parts.First, based on quadratic model and Perry's conjugacy condition, two new formulas of the main parameter β_k of conjugate gradient method are proposed. They can be seen as a modification of HS and PRP method respectively. In comparison with classic conjugate gradient methods, the decrease of objective function is contained in the two new methods. The first method is modified by the method which satisfies the sufficient descent condition under Powell's restart rule and the standard Armijo line search. Imitating Powell's suggestion to PRP method, the second method is modified. In this dissertation, the global convergence of above—mentioned four methods are given respectively. Better numerical results are got which show that these conjugate gradient methods are efficient and promising.Second, the nonlinear complementarity problem is transformed into nonsmooth equation by using Fischer—Burmeister function. Meanwhile merit function is obtained. A PRP—Type conjugate gradient method for solving nonlinear complementarity problem is presented by analysing the properties of Fischer—Burmeister function and this merit function. The distinguishing feature of the new method is that it satisfies sufficient descent condition naturally. Its global convergence is proved when F is P₀ + R₀ and quadratic continuous differentiate function. Numerical results are given which indicate that the method is efficient.