| This dissertation is devoted to the study on differential equation approaches to nonlinear complementarity problems, including theories and numerical implementations of both first order and second order differential equation approaches.The class of complementarity problems is a important branch of mathematical programming. It provides a framework for both linear programming and quartic programming. It has an important relation with stationary point theories, variational inequality, linear and nonlinear analysis, and some applied mathematical problems such as economic and equilibrium problems. There are many effective algorithms, including stationary point methods, homotopy methods, project methods, Newton methods, smooth differential equation methods, differential unconstrainizing methods and interior point methods.In Chapter 3 of this paper, we present a first order differential equation system with barrier projection method for solving nonlinear complementarity problems. The main ideas lie in that nonlinear complementarity problems can be converted to optimization problems, and the resulting problems are solved by differential equation approaches. It possesses a simple structure for implementation in hardware and preserves feasibility. We prove that the solution of a nonlinear complementarity problem is exactly the equilibrium point of differential equation system, and prove the asymptotical stability and global convergence. In addition, numerical examples are reported to verify the validity of the differential equation system.In Chapter 4, we focus on second order differential equation system for solving nonlinear complementarity problems. We establish a differential equation system via Newton method, and prove that the solution of nonlinear complementarity problems is exact the equilibrium point of differential equation system. Finally, we present a numerical algorithm and demonstrate its local quadratic convergence rate.
|
PDF Full Text Request