【作者】 白晓新；

【导师】 陈国庆；

【作者基本信息】 内蒙古大学 ， 运筹学与控制论， 2010， 硕士

【摘要】 通过将非线性互补问题转化为光滑方程组,本文给出求解非线性互补问题NCP(F)的一种光滑牛顿法.在F为P0+R0函数时,证明了算法的全局收敛性.然而,由于相应光滑方程组的Jacobi矩阵在解上为零矩阵,算法理论上不保证局部超线性收敛率.借鉴A. O. Griewank[38]和任玉芳[53]的工作,本文提出一个扰动策略,在迭代点列靠近NCP(F)之解时,使扰动点进入能保证快速线性收敛到NCP(F)某个近似解的星形域,继续单位步长的光滑牛顿迭代,可保证算法能够快速线性收敛到]NCP(F)的某个近似解.数值算例表明,任意初始点,算法能较快迭代到解附近,再结合保证快速线性收敛的扰动策略,实际计算中获得很好的数值结果.相对现有非光滑牛顿法和光滑化牛顿法本文所提出的光滑牛顿法构造简单,便于实际应用.更多还原

【Abstract】 In this dissertation, a smooth damped Newton method for solving the nonlinear comple-mentarity problem NCP(F) is presented by reformulating NCP(F) to a system of smooth equa-tions. The algorithm is proved to be globally convergent under the assumption that F is P0+R0 function. However, the local superlinear convergence is not guaranteed since the Jacobian ma-trix of the system of smooth equations at the solution is a zero matrix. To overcome this defect, a perturbation strategy is given by which an iterative point xk can be perturbed to xk which belong to the star-like region presented in A. O. Griewank[38] and Ren Yufang[53],it can be proved that the sequence generated by the smooth Newton iteration with unit step size starting from xk is quickly linearly convergent to an approximate solution of NCP(F). Numerical re-sults show that, starting from any initial point x0∈Rn, the sequence {xk} generated by the smooth damped Newton method is quickly convergent to near the solution, and starting from the perturbation point xk the sequence generated by the smooth Newton iteration with unit step size is quickly convergent to an approximate solution. Compared with the existing non-smooth or smoothing Newton methods, the smooth Newton method presented in this dissertation is simple and easy to the practical applications.更多还原