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.     

Banach空间中半光滑算子方程的不精确牛顿法(英文)

本文主要解决Banach空间中抽象的半光滑算子方程的解法.提出了两种不精确牛顿法,它们的收敛性同时得到了证明.这两种方法可以看作是有限维空间中已存在的解半光滑算子方程的方法的延伸.     

Ba空间中的多元加权光滑模与Bernstein Durrmeyer算子

该文引进Ba空间多元加权光滑模,推广L^p空间的DitzianTotik模, 证明该模与K泛函的等价性. 作为应用,讨论定义在单纯形上多元Bernstein-Durrmeyer算子与多元加权光滑模之间的关系. 即以多元加权光滑模为尺度, 建立Bernstein-Durrmeyer算子在Ba空间逼近阶的上界与下界估计.     

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.     

Orlicz空间中的多元光滑模及其应用

本文的目的是引进和应用Orlicz空间中一种新的多元光滑模，该光滑模是一元情形的一种自然推广．利用函数分解方法和归纳讨论证明它与K-泛函之间的等价关系．作为应用，给出定义在单纯形上Durrmeyer算子在Orlicz空间中的一个逼近逆定理．     

Baskakov算子加Jacobi权逼近及其导数的正逆定理

本文利用加权光滑模ω2ψλ(f,t)ω给出了Baskakov算子加Jacobi权逼近的正逆定理;另外,研究了加权下Baskakov算子导数与所逼近函数光滑性之间的关系.     

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.     

Baskakov算子加Jacobi权逼近及其导数的正逆定理

本文利用加权光滑模ω_~2λ(f,t)ω给出了Baskakov算子加Jacobi权逼近的正逆定理;另外,研究了加权下Baskakov算子导数与所逼近函数光滑性之间的关系.     

一类非光滑方程组的牛顿法及全局收敛性

借助于一种新的微分 - -微分 ,本文给出极大值函数及其光滑复合的非光滑方程组的牛顿法 .最后证明了该牛顿法具有全局收敛性 .     

非光滑非线性互补问题的牛顿法

研究了非光滑的非线性互补问题. 首先将非光滑的非线性互补问题转化为一个非光滑方程组,然后用牛顿法求解这个非光滑方程组. 在该牛顿法中,每次迭代只需一个原始函数B-微分中的一个元素. 最后证明了该牛顿法的超线性收敛性.     

Smoothing Newton method for operator equations in Banach spaces

In this paper, the global and superlinear convergence of smoothing Newton method for solving nonsmooth operator equations in Banach spaces are shown. The feature of smoothing Newton method is to use a smooth function to approximate the nonsmooth mapping. Under suitable assumptions, we prove that the smoothing Newton method is superlinearly convergent. As an application, we use the smoothing Newton method to solve a constrained optimal control problem.     

盒子型区域上变分不等式的一个光滑牛顿法(英文)

The box constrained variational inequality problem can be reformulated as a nonsmooth equation by using median operator.In this paper,we present a smoothing Newton method for solving the box constrained variational inequality problem based on a new smoothing approximation function.The proposed algorithm is proved to be well defined and convergent globally under weaker conditions.     

A globally and superlinearly convergent quasi-Newton method for general box constrained variational inequalities without smoothing approximation

A new quasi-Newton algorithm for the solution of general box constrained variational inequality problem (GVI(l, u, F, f)) is proposed in this paper. It is based on a reformulation of the variational inequality problem as a nonsmooth system of equations by using the median operator. Without smoothing approximation, the proposed quasi-Newton algorithm is directly applied to solve this class of nonsmooth equations. Under appropriate assumptions, it is proved that the algorithmic sequence globally and superlinearly converges to a solution of the equation reformulation and also of GVI(l, u, F, f). Numerical results show that our new algorithm works quite well.     

Newton and quasi-Newton methods for equations of smooth compositions of semismooth functions

The Newton method and the quasi-Newton method for solving equations of smooth compositions of semismooth functions are proposed. The Q-superlinear convergence of the Newton method and the Q-linear convergence of the quasi-Newton method are proved. The present methods can be more easily implemented than previous ones for this class of nonsmooth equations.     

Globally Convergent Broyden-Like Methods for Semismooth Equations and Applications to VIP, NCP and MCP

In this paper, we propose a general smoothing Broyden-like quasi-Newton method for solving a class of nonsmooth equations. Under appropriate conditions, the proposed method converges to a solution of the equation globally and superlinearly. In particular, the proposed method provides the possibility of developing a quasi-Newton method that enjoys superlinear convergence even if strict complementarity fails to hold. We pay particular attention to semismooth equations arising from nonlinear complementarity problems, mixed complementarity problems and variational inequality problems. We show that under certain conditions, the related methods based on the perturbed Fischer–Burmeister function, Chen–Harker–Kanzow–Smale smoothing function and the Gabriel–Moré class of smoothing functions converge globally and superlinearly.     

Smoothing Newton and Quasi-Newton Methods for Mixed Complementarity Problems

The mixed complementarity problem can be reformulated as a nonsmooth equation by using the median operator. In this paper, we first study some useful properties of this reformulation and then derive the Chen-Harker-Kanzow-Smale smoothing function for the mixed complementarity problem. On the basis of this smoothing function, we present a smoothing Newton method for solving the mixed complementarity problem. Under suitable conditions, the method exhibits global and quadratic convergence properties. We also present a smoothing Broyden-like method based on the same smoothing function. Under appropriate conditions, the method converges globally and superlinearly.     

数据挖掘中聚类中心问题的光滑化和填充函数方法

本文提出了数据挖掘中求解聚类中心问题的一种新方法.这类问题属于非凸非光滑全局最优化问题.我们首先利用光滑化方法将非光滑聚类函数用光滑函数逼近,然后对光滑化问题利用填充函数搜索其全局最优点.对不同数据库的数值试验表明,本文提出的算法是可行和有效的.     

An algorithm for composite nonsmooth optimization problems

Nonsmooth optimization problems are divided into two categories. The first is composite nonsmooth problems where the generalized gradient can be approximated by information available at the current point. The second is basic nonsmooth problems where the generalized gradient must be approximated using information calculated at previous iterates.Methods for minimizing composite nonsmooth problems where the nonsmooth function is made up from a finite number of smooth functions, and in particular max functions, are considered. A descent method which uses an active set strategy, a nonsmooth line search, and a quasi-Newton approximation to the reduced Hessian of a Lagrangian function is presented. The Theoretical properties of the method are discussed and favorable numerical experience on a wide range of test problems is reported.This work was carried out at the University of Dundee from 1976–1979 and at the University of Kentucky at Lexington from 1979–1980. The provision of facilities in both universities is gratefully acknowledged, as well as the support of NSF Grant No. ECS-79-23272 for the latter period. The first author also wishes to acknowledge financial support from a George Murray Scholarship from the University of Adelaide and a University of Dundee Research Scholarship for the former period.     

A new smoothing quasi-Newton method for nonlinear complementarity problems

A new smoothing quasi-Newton method for nonlinear complementarity problems is presented. The method is a generalization of Thomas’ method for smooth nonlinear systems and has similar properties as Broyden's method. Local convergence is analyzed for a strictly complementary solution as well as for a degenerate solution. Presented numerical results demonstrate quite similar behavior of Thomas’ and Broyden's methods.