基于新光滑函数的P0映射非线性互补问题的光滑牛顿法(英文)

非线性互补问题(NCP)可以重新表述为一个非光滑方程组的解.通过引入一个新的光滑函数,将问题近似为参数化光滑方程组.基于这个光滑函数,我们提出了一个求解P₀映射和R₀映射非线性互补问题的光滑牛顿法.该算法每次迭代只求解一个线性方程和一次线搜索.在适当的条件下,证明了该方法是全局和局部二次收敛的.数值结果表明,该算法是有效的.     

求解无限维非线性互补问题的光滑化牛顿法

研究一类无限维非线性互补问题的光滑化牛顿法.借助于非线性互补函数,将无限维非线性互补问题转化为一个非光滑算子方程.构造光滑算子逼近非光滑算子,在光滑逼近算子满足方向可微相容性的条件下,证明了光滑化牛顿法具有超线性收敛性.     

基于一个新的NCP函数的光滑牛顿法求解非线性互补问题

本文研究了非线性互补的光滑化问题.利用一个新的光滑NCP函数将非线性互补问题转化为等价的光滑方程组,并在此基础上建立了求解P0-函数非线性互补问题的一个完全光滑化牛顿法,获得了算法的全局收敛性和局部二次收敛性的结果.并给出数值实验验证了理论分析的正确性.     

一个求解广义圆锥互补问题的无导数光滑算法

本文研究了一个求解广义圆锥互补问题的无导数光滑算法.利用光滑函数将广义圆锥互补问题等价转化成一个光滑方程组,然后再利用牛顿法求解此方程组.该算法采用了一种新的非单调无导数线搜索技术,并且在适当条件下具有全局和局部二次收敛性质.数值实验结果表明算法是非常有效的.     

一个新的求解加权线性互补问题的非单调光滑牛顿法

本文研究了求解加权线性互补问题的光滑牛顿法.利用一类光滑函数将加权线性互补问题等价转化成一个光滑方程组,然后提出一个新的光滑牛顿法去求解它.在适当条件下,证明了算法具有全局和局部二次收敛性质.与现有的光滑牛顿法不同,我们的算法采用一个非单调无导数线搜索技术去产生步长,从而具有更好的收敛性质和实际计算效果.     

求解广义非线性互补问题的光滑化拟牛顿法

利用光滑对称扰动Fischer-Burmeister函数将广义非线性互补问题转化为非线性方程组,提出新的光滑化拟牛顿法求解该方程组.然后证明该算法是全局收敛的,且在一定条件下证明该算法具有局部超线性(二次)收敛性.最后用数值实验验证了该算法的有效性.     

解非线性互补问题的非单调非精确Broyden-like算法

本文通过构造一个新的光滑互补函数,将非线性互补问题等价转换为光滑方程组问题.将非单调线搜索技术与非精确Broyden-like算法相结合,建立了解非线性互补问题的非单调非精确Broyden-like算法.在一定条件下证明了该算法的全局收敛性和局部二次收敛性.数值实验表明该算法对求解非线性互补问题是十分有效的.     

解非线性互补问题的光滑牛顿方法

由于退化解会导致再生方程的奇异性,非线性互补问题的求解通常采用基于半光滑技术的广义牛顿法.基于2-正则性的概念,提出了一类利用光滑互补函数求解互补问题的光滑牛顿算法.算法采用积极集技术,能在解的附近估计出退化指标,并把原问题降阶为一个非奇异方程组,从而保证了迭代效率.算法具有整体收敛性和局部超线性收敛性,数值实验显示算法是有效的.     

一类非线性二阶锥规划的非光滑牛顿法

本文研究了非线性二阶锥规划问题.利用投影映射将非线性二阶锥规划问题的KKT最优性条件转化成非光滑方程组,获得了一个修正的中心路径非光滑牛顿法.在适当的条件下保证方程组的B-次微分在任意点都可逆,并且证明算法具有全局收敛性.     

非光滑方程组牛顿法的全局收敛性分析

通过定义一种新的*-微分,本文给出了局部Lipschitz非光滑方程组的牛顿法,并对其全局收敛性进行了研究.该牛顿法结合了非光滑方程组的局部收敛性和全局收敛性.最后,我们把这种牛顿法应用到非光滑函数的光滑复合方程组问题上,得到了较好的收敛性.     

Convergence analysis of nonsmooth equations for the general nonlinear complementarity problem

This paper deals with a general nonlinear complementarity problem, where the underlying functions are assumed to be continuous. Based on a nonlinear complementarity function, it is transformed into a system of nonsmooth equations. Then, two kinds of approximate Newton methods for the nonsmooth equations are developed and their convergence are proved. Finally, numerical tests are also listed.     

求解非线性互补问题的逐次逼近阻尼牛顿法

针对非线性互补问题,提出了与其等价的非光滑方程的逐次逼近阻尼牛顿法,并 在一定条件下证明了该算法的全局收敛性.数值结果表明,这一算法是有效的.     

Generalized equations and the generalized Newton method

We give some convergence results on the generalized Newton method (referred to by some authors as Newton's method) and the chord method when applied to generalized equations. The main results of the paper extend the classical Kantorovich results on Newton's method to (nonsmooth) generalized equations. Our results also extend earlier results on nonsmooth equations due to Eaves, Robinson, Josephy, Pang and Chan. We also propose inner-iterative schemes for the computation of the generalized Newton iterates. These schemes generalize popular iterative methods (Richardson's method, Jacobi's method and the Gauss-Seidel method) for the solution of linear equations and linear complementarity problems and are shown to be convergent under natural generalizations of classical convergence criteria. Our results are applicable to equations involving single-valued functions and also to a class of generalized equations which includes variational inequalities, nonlinear complementarity problems and some nonsmooth convex minimization problems.     

求解摩擦接触问题的一个非内点光滑化算法

给出了一个求解三维弹性有摩擦接触问题的新算法,即基于NCP函数的非内点光滑化算法．首先通过参变量变分原理和参数二次规划法,将三维弹性有摩擦接触问题的分析归结为线性互补问题的求解;然后利用NCP函数,将互补问题的求解转换为非光滑方程组的求解;再用凝聚函数对其进行光滑化,最后用NEWTON法解所得到的光滑非线性方程组．方法具有易于理解及实现方便等特点．通过线性互补问题的数值算例及接触问题实例证实了该算法的可靠性与有效性．     

A parameterized Newton method and a quasi-Newton method for nonsmooth equations

This paper presents a parameterized Newton method using generalized Jacobians and a Broyden-like method for solving nonsmooth equations. The former ensures that the method is well-defined even when the generalized Jacobian is singular. The latter is constructed by using an approximation function which can be formed for nonsmooth equations arising from partial differential equations and nonlinear complementarity problems. The approximation function method generalizes the splitting function method for nonsmooth equations. Locally superlinear convergence results are proved for the two methods. Numerical examples are given to compare the two methods with some other methods.This work is supported by the Australian Research Council.     

Smoothing Newton method for nonsmooth second-order cone complementarity problems with application to electric power markets

This paper is dedicated to solving a nonsmooth second-order cone complementarity problem, in which the mapping is assumed to be locally Lipschitz continuous, but not necessarily to be continuously differentiable everywhere. With the help of the vector-valued Fischer-Burmeister function associated with second-order cones, the nonsmooth second-order cone complementarity problem can be equivalently transformed into a system of nonsmooth equations. To deal with this reformulated nonsmooth system, we present an approximation function by smoothing the inner mapping and the outer Fischer-Burmeister function simultaneously. Different from traditional smoothing methods, the smoothing parameter introduced is treated as an independent variable. We give some conditions under which the Jacobian of the smoothing approximation function is guaranteed to be nonsingular. Based on these results, we propose a smoothing Newton method for solving the nonsmooth second-order cone complementarity problem and show that the proposed method achieves globally superlinear or quadratic convergence under suitable assumptions. Finally, we apply the smoothing Newton method to a network Nash-Cournot game in oligopolistic electric power markets and report some numerical results to demonstrate its effectiveness.

     

非线性互补问题非精确逐次逼近法的全局收敛性

本文针对非线性互补问题,提出了与其等价的非光滑方程的非精确逐次逼近算法,并在一定条件下证明了该算法的全局收敛性。     

A B-differentiable equation-based,globally and locally quadratically convergent algorithm for nonlinear programs,complementarity and variational inequality problems

This paper presents a globally convergent, locally quadratically convergent algorithm for solving general nonlinear programs, nonlinear complementarity and variational inequality problems. The algorithm is based on a unified formulation of these three mathematical programming problems as a certain system of B-differentiable equations, and is a modification of the damped Newton method described in Pang (1990) for solving such systems of nonsmooth equations. The algorithm resembles several existing methods for solving these classes of mathematical programs, but has some special features of its own; in particular, it possesses the combined advantage of fast quadratic rate of convergence of a basic Newton method and the desirable global convergence induced by one-dimensional Armijo line searches. In the context of a nonlinear program, the algorithm is of the sequential quadratic programming type with two distinct characteristics: (i) it makes no use of a penalty function; and (ii) it circumvents the Maratos effect. In the context of the variational inequality/complementarity problem, the algorithm provides a Newton-type descent method that is guaranteed globally convergent without requiring the F-differentiability assumption of the defining B-differentiable equations.This work was based on research supported by the National Science Foundation under Grant No. ECS-8717968.     

A penalized Fischer-Burmeister NCP-function

We introduce a new NCP-function in order to reformulate the nonlinear complementarity problem as a nonsmooth system of equations. This new NCP-function turns out to have stronger theoretical properties than the widely used Fischer-Burmeister function and other NCP-functions suggested previously. Moreover, numerical experience indicates that a semismooth Newton method based on this new NCP-function performs considerably better than the corresponding method based on the Fischer-Burmeister function. Received: March 10, 1997 / Accepted: February 15, 2000?Published online May 12, 2000