The Smoothing Adaptive Trust Region Method For The Nonlinear Complementarity Problem

The nonlinear complementarity problem (NCP) has many important applications in engineering and economics, and there have developed many numerical methods to solve it at the same time, global and local convergence results have also been obtained. In the last years more attention has been devoted to reformulating the nonlinear complementarity problem as a system of nonsmooth equations by using some NCP function.In this paper, we use the Kanzow function to approximate the Fischer-Burmeister function so that smooth and nonlinear equatons can be obtained and then changed into a optimization problem. With the combination of the adaptive technique,the nonmonotone technique and the trust-region method which is more stable and reliable than linear search method, two new algorithms for solving these equations are proposed in Chapter 3 and Chapter 4.In Chapter 3, we combine the trust region subproblem with the adptive techique to propose a smoothing adptive trust region algorithm for the nonlinear complementarity problem. With the adaptive technique, the radius of trust regionΔk can be adjusted automatically to improve the efficiency of trust region methods. On the other hand, the smoothing coefficient of the Kanzow function is updated when the object function gets some satisfying reduction.In Chapter 4, based on the algorithm in Chapter 3, the new 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.In Chapter 5, a series of numerical examples are tested, which shows the algorithm is quite promising.