The Nonmontonic Trust Region Methods For Nonlinear Complementarity Problems

In this paper,we study the numerical methods for nonlinear complementarity problems.Nonmontone strategy is introduced to accelerate the process of iteration since the traditional monotone algorithm will be of low efficiency when the point sequence enters some narrow areas.With the combination of the nonmonotone technique and the trust-region method which is more stable and reliable than linear search method,a new algorithm is proposed.By using the Fischer-Burmeister function,we reformulate complementarity problems to a system of nonlinear and nonsmooth equations.Furthermore,we use the Kanzow function to approximate the Fischer-Burmeister function so that smooth and nonlinear equations can be obtained.The trust-region algorithm using nonmonotone technique for solving these equations is proposed in Chapter three.The algorithm adopts a 'nonmontone ratio' to approximate the reduction of the object functions.An iteration is accepted when the ratio is not very small.On the other hand,the smoothing coefficient of the Kanzow function is updated when the object function gets some satisfying reduction.With the assumption that F is a P₀ function,we prove that the sequence generated by the algorithm remains in some level set.Furthermore,the method will generate at least one accumulation point if the level set is compact,which leads to the global convergence.In addition,the sequence will converge to one point and the superlinear convergence or even quadratic convergence under some conditions.A series of numerical examples are tested in Chapter 7,which shows the algorithm is quite promising.Especially in the case when the sequence enter some narrow areas,the efficience of the traditional monotone method canbe improved a lot.