The Study Of Filter Methods For Nonlinear Complementarity Problems

Anciently, Lagrangian function or penalty function was utilized to solve con-strained optimization problems. But there will occur any new questions. For example,to deal with the multiplier, it would induce Hessian matrix ill-condition while thepenalty parameter is too big. To avoid these questions, filter technique, a methodwithout multiplier and penalty parameter, is presented. Filter methods were proposedby Fletcher and Leyfer in 1997. Because of good numerical results, filter methods areattached importance to. The basic idea of filter methods lies in: to obtain a betterpoint than the current point, we try to reduce the objective function and constraintviolation degree. In every iteration, if the objective function and the constraint con-ditions become worse(the objective function and the constraint violation degree arebigger than all former points), the trial step is regarded as a unsuccessful step. Other-wise it is regarded as a successful step. There are some advantages of the filter methods.Firstly, it can balance the objective function and constraint conditions. Secondly, itdoes not require Lagrangian multipliers. Finally, it is a nonmonotonic method whichis beneficial to obtain the global optimization point. In this paper, we treat the non-linear complementarity problems(NCP) with view of multi-objective by using the filtertechnique to solve NCP. In this paper, we propose three methods for solving nonlinearcomplementarity problems(NCP), and give some numerical results to each algorithm.Our numerical results show that our algorithms are successful.In Chapter 2, we propose a new filter method for solving nonlinear complemen-tarity problems with two parametersεk andβk, where {εk} is monotone decreasing,as k→∞,εk→0. Under a proper condition, local Q-superlinear and Q-quadraticconvergence of the algorithm are established.In Chapter 3, we propose a filter interior-point algorithm for NCP. In the main al-gorithm, we get the step size by using Armijio-type linear search. And in the restorationalgorithm, the trust region method is used to ensure the convergence of the algorithm.Finally, we prove it's global convergence.In Chapter 4, a filter method for solving nonlinear complementarity problemsbased on derivative-free line search is presented. When a new point is not acceptedby the filter, by using restoration algorithm, we get a new point. In the convergentanalysis, the proposed algorithm is proved to be globally convergent under mild as-sumptions. Furthermore, we get superlinear convergence of the method under theproper conditions.