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Study On Algorithms For Complementarity Problems And Nonlinear System

Posted on:2012-10-27Degree:DoctorType:Dissertation
Country:ChinaCandidate:J G ZhuFull Text:PDF
GTID:1110330338450234Subject:Applied Mathematics
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The design of an efficient algorithm is one of the most important research areas in numerical optimization community. This dissertation considers two class of wide appli-cation background problems, such as nonlinear complementarity problems and system of nonlinear inequalities, and mainly concentrates on the design of the algorithm, con-vergence analysis, numerical implementation. The main contributions are listed as follows:1. Smoothing functions play a key role in reformulation for the smoothing method. Firstly, a new smoothing function is proposed, which include many smoothing functions as special cases. The new smoothing functions possess a system of favorite properties, such as, the existence and continuity of a smooth path, boundedness of the iteration sequence and Jacobian consistency. Based on the new smoothing functions, we inves-tigate a smoothing Newton algorithm for the NCP, numerical results demonstrate that the new smoothing function introduced in this chapter is worth investigating.2. Based on the Fischer-Burmeister smoothing function, we propose a nonmonotone inexact regularized smoothing Newton method for nonlinear complementarity problems with a P0-function. Under the weaker condition, the boundedness of level set, global convergence and locally super linear convergence of the algorithm are established. Nu-merical results indicate that the newly proposed algorithm is effective, especially for the large scale nonlinear complementarity problems.3. By combining the trust region techniques and line searches, we propose a new semismooth Levenberg-Marquardt method for solving general (not necessarily P0) non-linear complementarity problems. Under suitable assumptions, global convergence and locally super linear convergence of the algorithm are established. Numerical results demonstrate that the proposed method is more efficient than some existing algorithms.4. Since the existing derivative-free descent algorithms for complementarity prob-lems are based on monotone line search, while their numerical implementations are based on nonmonotone line search, there lack theoretical analysis. Based on p-norm, we introduce a new generalized penalized Fischer-Burmeister merit function, and show that the function possesses a system of favorite properties. We propose a derivative-free algorithm for nonlinear comple-mentarity problems with a nonmonotone line search. The global convergence and locally convergence of the algorithm are established. Nu-merical results indicate that the newly proposed algorithm is effective and the new merit function is meaningful. 5. The system of nonlinear inequalities is reformulated as a nonsmooth equation by using a plus function. By using the Chen-Harker-Kanzow-Smale smoothing function of plus function, the nonsmooth equation is approximated by a family of parameterized smooth equations. A regularized smoothing Newton algorithm is proposed to solve the smooth equations and the solution of the system of nonlinear inequalities is founded. Numerical results indicate that the proposed algorithm is efficient.
Keywords/Search Tags:nonlinear complementarity problem, system of nonlinear inequalities, smoothing function, merit function, smoothing method, semis-mooth Levenberg-Marquardt method, nonmonotone derivative-free method, global convergence
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