Differential Equation Methods For Complementarity Problems

This paper reviews the history of complementarity problems. It mainly provides two new differential equation methods for the linear and nonlinear complementarity problems. And to show the validity of these methods, we provide the corresponding numerical experiments.1. Complementarity problems was suggested in quartic programming in 1960s. It has very important values in mathematical physics and mathematical economy fields; At the same time, it also has widely useage in many practical problems. Such as, finance, transportation programming, electic system, regional development, ect. Thus the researches of complementarity problems attract more and more attaintion. And there are two aspacts of this problem: theories and algrithms. In academic aspect, the researche is primarily about how to introduce the correlative tecniques, definitions and ideas to design the methods to resolve the complementarity problems. It focuses on the solution's existance and uniqueness. In the last of 1980s, with the twenty-year hard work of many scholars, a lot of achievements of in this fields appear. For examples, stationary point method, homotopy method, projecton method, newton method and so on. After 1980s, scholars suggest some other effective algrithms. For instance, smooth differencial equation method, nonsmooth differencial equation method, differencial unconstrainizing method, inner point method. Recently, based on the burnishing tecnique of Chen and Mangasarian, someone has suggested non-inner point method.2. In Chapter 3 of this paper, we present a differential equation system for solving linear complementarity problem in real time. It possesses a very simple structure for implementation in hardware. In the theoretical aspect, this system is different from the existents which use the penalty functions or Lagrangians. We prove that the proposed differential equation system converges globally to the solution set of the problem starting from any initial point. In addition, the stability of it is analyzed and five numerical examples are given to verify the validity of the differential equation system.3. In Chapter 4, we introduce the definition of implicit complementarity problems and struct an equivalent projection differential equation system. And then we prove the equilibrium point of system is exact the solution of the implicit complementarity problem. After that, we choose an energic function to prove the stability of the differencial equation system.