| This paper is devoted to the study on numerical algorithms for nonlinear complementary problems. It provides a new differential equation approach for nonlinear complementary problems and reports numerical implementations.Chapter 1 introduces the background of complementary problems. There are two aspects of the problems: theories and algorithms, In academic aspect, the research is primarily about how to introduce definitions and ideas to design the methods to resolve the complementary problems. It focuses on the solution's existence and uniqueness, In the last of 1980s, a lot of achievements in algorithms have been made. For example, we have stationary point methods, projection methods, and Newton methods. After 1990s, some effective algorithms have been constructed. For instance, smooth differential equation methods, non-smooth equation methods, differentiable unconstrained methods, and interior point methods. Recently, based on the polishing technique of Chen and Mangasarian, a non-interior point method has been developed.Chapter 3 is the main part of this dissertation, which studies differential equation approaches for solving nonlinear complementary problems. We convert a nonlinear complementary problem to an equivalent inequality constrained nonlinear programming (NLP) problem. After that, convert the inequality constrained NLP problem to an NLP problem with equality constraint by space transformation. A differential equation system is constructed to solve the NLP problems. It proves that solutions to the nonlinear complementary problem are equilibrium points to the differential equation system. A numerical algorithm, based on solving the differential system, is given and proved to be convergent.Chapter 4 provides numerical experiments by the algorithm for several examples of the nonlinear complementary problems. The numerical results show that the proposed algorithm is effective at least to solve operative problems with relative small scales.
|
PDF Full Text Request