非光滑方程的方向牛顿法

通过引入广义梯度,将求解含n个未知量方程的方向牛顿法推广到非光滑的情形.证明了该方法在半光滑条件下的收敛性定理,给出了解的存在性以及先验误差界.     

解奇异非光滑方程组的牛顿法和不精确牛顿法

本文主要解决奇异非光滑方程组的解法。应用一种新的次微分的外逆，我们提出了牛顿法和不精确牛顿法，它们的收敛性同时也得到了证明。这种方法能更容易在一引起实际应用中实现。这种方法可以看作是已存在的解非光滑方程组的方法的延伸。     

求解对称张量绝对值方程问题的非光滑牛顿法

提出一类对称张量绝对值方程问题,给出了求解此类问题的一类非光滑牛顿法,并且在一般的假设条件下,给出了算法的局部收敛性.最后给出相关的数值实验表明了算法的有效性.     

解非光滑方程组的Krylov子空间迭代法

给出了求解非光滑方程组的Newton-FOM算法和Newton-GMRES算法,证明了这些Krylov子空间方法的局部平方收敛性．数值结果表明了算法的有效性．     

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 .     

Approximate Newton Methods for Nonsmooth Equations

We develop general approximate Newton methods for solving Lipschitz continuous equations by replacing the iteration matrix with a consistently approximated Jacobian, thereby reducing the computation in the generalized Newton method. Locally superlinear convergence results are presented under moderate assumptions. To construct a consistently approximated Jacobian, we introduce two main methods: the classic difference approximation method and the -generalized Jacobian method. The former can be applied to problems with specific structures, while the latter is expected to work well for general problems. Numerical tests show that the two methods are efficient. Finally, a norm-reducing technique for the global convergence of the generalized Newton method is briefly discussed.     

求解半光滑方程组的近似Newton法

本文提出了求解半光滑方程组的近似Newton法，并证明了该算法的局部超线性收敛性。数值结果表明 该算法是有效的。     

Newton Linearized Methods for Semilinear Parabolic Equations

In this study, Newton linearized finite element methods are presented for solving semi-linear parabolic equations in two- and three-dimensions. The proposed scheme is a one-step, linearized and second-order method in temporal direction, while the usual linearized second-order schemes require at least two starting values. By using a temporal-spatial error splitting argument, the fully discrete scheme is proved to be convergent without time-step restrictions dependent on the spatial mesh size. Numerical examples are given to demonstrate the efficiency of the methods and to confirm the theoretical results.     

The Lagrangian Globalization Method for Nonsmooth Constrained Equations

The difficulty suffered in optimization-based algorithms for the solution of nonlinear equations lies in that the traditional methods for solving the optimization problem have been mainly concerned with finding a stationary point or a local minimizer of the underlying optimization problem, which is not necessarily a solution of the equations. One method to overcome this difficulty is the Lagrangian globalization (LG for simplicity) method. This paper extends the LG method to nonsmooth equations with bound constraints. The absolute system of equations is introduced. A so-called Projected Generalized-Gradient Direction (PGGD) is constructed and proved to be a descent direction of the reformulated nonsmooth optimization problem. This projected approach keeps the feasibility of the iterates. The convergence of the new algorithm is established by specializing the PGGD. Numerical tests are given. This author's work was done when she was visiting The Hong Kong Polytechnic University. His work is also supported by the Research Grant Council of Hong Kong.     

Semismooth Newton Methods for Solving Semi-Infinite Programming Problems

In this paper we present some semismooth Newton methods for solving the semi-infinite programming problem. We first reformulate the equations and nonlinear complementarity conditions derived from the problem into a system of semismooth equations by using NCP functions. Under some conditions a solution of the system of semismooth equations is a solution of the problem. Then some semismooth Newton methods are proposed for solving this system of semismooth equations. These methods are globally and superlinearly convergent. Numerical results are also given.     

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.     

New Version of the Newton Method for Nonsmooth Equations

In this paper, an inexact Newton scheme is presented which produces a sequence of iterates in which the problem functions are differentiable. It is shown that the use of the inexact Newton scheme does not reduce the convergence rate significantly. To improve the algorithm further, we use a classical finite-difference approximation technique in this context. Locally superlinear convergence results are obtained under reasonable assumptions. To globalize the algorithm, we incorporate features designed to improve convergence from an arbitrary starting point. Convergence results are presented under the condition that the generalized Jacobian of the problem function is nonsingular. Finally, implementations are discussed and numerical results are presented.     

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.     

Newton Methods for Quasidifferentiable Equations and Their Convergence

The Newton method and the inexact Newton method for solving quasidifferentiable equations via the quasidifferential are investigated. The notion of Q-semismoothness for a quasidifferentiable function is proposed. The superlinear convergence of the Newton method proposed by Zhang and Xia is proved under the Q-semismooth assumption. An inexact Newton method is developed and its linear convergence is shown.Project sponsored by Shanghai Education Committee Grant 04EA01 and by Shanghai Government Grant T0502.     

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.     

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.     

Global and superlinear convergence of the smoothing Newton method and its application to general box constrained variational inequalities

The smoothing Newton method for solving a system of nonsmooth equations , which may arise from the nonlinear complementarity problem, the variational inequality problem or other problems, can be regarded as a variant of the smoothing method. At the th step, the nonsmooth function is approximated by a smooth function , and the derivative of at is used as the Newton iterative matrix. The merits of smoothing methods and smoothing Newton methods are global convergence and convenience in handling. In this paper, we show that the smoothing Newton method is also superlinearly convergent if is semismooth at the solution and satisfies a Jacobian consistency property. We show that most common smooth functions, such as the Gabriel-Moré function, have this property. As an application, we show that for box constrained variational inequalities if the involved function is -uniform, the iteration sequence generated by the smoothing Newton method will converge to the unique solution of the problem globally and superlinearly (quadratically).

     

一类求解反应扩散方程的Newton波形松弛方法

对反应扩散方程提出一种新型的Newton波形松弛方法,并给出此方法的误差估计式.通过与传统的波形松弛方法比较,这种Newton波形松弛方法有更快的收敛性,且收敛速度不随网格加密而减慢.这种方法可以保持传统波形松弛方法可并行的特点.最后通过数值算例验证这种方法的有效性.     

求解垂直互补问题的参数牛顿法

给出了求解垂直互补问题的一种参数牛顿法,在较为温和的条件下证明了该方法的局部超线性收敛结果,并且给出了具体数值计算.