求解非线性互补问题的光滑化连续牛顿法

The smoothing technique-based continuation Newton method for nonlinear complementarity problems

本文主要研究了求解非线性互补问题的正则化连续方法。首先,研究通过引入Fischer-Burmeister函数和磨光滑化技术,将非线性互补问题转化为非线性方程组。然后,从传统的阻尼牛顿法出发,将隐式欧拉法应用到非线性方程组的连续牛顿流,从而得到一类连续牛顿法。最后,通过引入信赖域的思想,设计了一类基于信赖域更新策略的连续牛顿法求解非线性互补问题转化的非线性方程组,并将该互补问题算法与主流商业软件GAMS中的互补问题求解器做了数值实验比较。数值结果表明,本文所提出的非线性互补问题算法(简称CNMFN)比GAMS中的PATH和MILES求解器更健壮,且对于大部分的测试算例,CNMFN也比PATH和MILES求解器更高效。

This paper researches on the smoothing regularization methods for nonlinear complementarity problem.By introducing Fischer-Burmeister function and smoothing technqiues,nonlinear complementarity problems are converted into nonlinear equations problems.Then,the nonlinear equations are solved by a continuation Newton method with implicit Euler method applied to the continuous Newton flow of nonlinear equations under the thought of traditional damped Newton methods.Since the continuation method has the disadvantage of inefficiency when its time-stepping selection uses the line search,this paper further introduces the idea of the trust-region updating strategy to adjust the time step of the continuation Newton method adapatively.The proposed method(referred to as CNMFN)is compared with the state-of-the-art solvers such as PATH and MILES(the subroutines of the commercial software GAMS)for nonlinear complimentarity problems.Numerical results show that the prosposed algorithm is more robust than PATH and MILES solvers,and also faster than PATH and MILES in most of the test problems.