从常步长梯度方法的视角看不可微凸优化增广Lagrange方法的收敛性

增广Lagrange方法是求解非线性规划的一种有效方法.从一新的角度证明不等式约束非线性非光滑凸优化问题的增广Lagrange方法的收敛性.用常步长梯度法的收敛性定理证明基于增广Lagrange函数的对偶问题的常步长梯度方法的收敛性,由此得到增广Lagrange方法乘子迭代的全局收敛性.     

A Steffensen-like method and its higher-order variants

For solving nonlinear equations, we suggest a second-order parametric Steffensen-like method, which is derivative free and only uses two evaluations of the function in one step. We also suggest a variant of the Steffensen-like method which is still derivative free and uses four evaluations of the function to achieve cubic convergence. Moreover, a fast Steffensen-like method with super quadratic convergence and a fast variant of the Steffensen-like method with super cubic convergence are proposed by using a parameter estimation. The error equations and asymptotic convergence constants are obtained for the discussed methods. The numerical results and the basins of attraction support the proposed methods.     

A study on the convergence of homotopy analysis method

In this study, a reliable approach for convergence of the homotopy analysis method when applied to nonlinear problems is discussed. First, we present an alternative framework of the method which can be used simply and effectively to handle nonlinear problems. Then, mainly, we address the sufficient condition for convergence of the method. The convergence analysis is reliable enough to estimate the maximum absolute truncated error of the homotopy series solution. The analysis is illustrated by investigating the convergence results for some nonlinear differential equations. The study highlights the power of the method.     

Block by block method for the systems of nonlinear Volterra integral equations

The approach given in this paper leads to numerical methods for solving system of Volterra integral equations which avoid the need for special starting procedures. The method has also the advantages of simplicity of application and at least four order of convergence which is easy to achieve. Also, at each step we get four unknowns simultaneously. A convergence theorem is proved for the described method. Finally numerical examples presented to certify convergence and accuracy of the method.     

关于Brown方法的半局部收敛性

在本文中,我们讨论解非线性方程组的Brown方法的半局部收敛性。通过对Brown方法的算法结构作深入的分析,我们将Brown方法变换成带有特殊误差项的近似Newton法,基于这种等价变形,我们建立了Brown方法的半局部收敛定理,从而完善了Brown方法的收敛理论。     

一个共轭梯度方法全局收敛性的判别准则

本文建立了一个共轭梯度方法全局收敛性的判别准则,基于这一准则证明了一类三参数共轭梯度法的全局收敛性及DY方法的一个变形的全局收敛性.     

一个四阶收敛的牛顿类方法

A fourth-order convergence method of solving roots for nonlinear equation,which is a variant of Newton's method given.Its convergence properties is proved.It is at least fourth-order convergence near simple roots and one order convergence near multiple roots. In the end,numerical tests are given and compared with other known Newton and Newtontype methods.The results show that the proposed method has some more advantages than others.It enriches the methods to find the roots of non-linear equations and it ...     

一维凸函数牛顿法的全局收敛性及其应用

牛顿法是求解非线性方程F（x)＝0的一种经典方法。在一般假设条件下，牛顿法只具有局部收敛性。本文证明了一维凸函数牛顿法的全局收敛性，并且给出了它在全局优化积分水平集方法中的应用。     

Point estimation and some applications to iterative methods

We consider one of the crucial problems in solving polynomial equations concerning the construction of such initial conditions which provide a safe convergence of simultaneous zero-finding methods. In the first part we deal with the localization of polynomial zeros using disks in the complex plane. These disks are used for the construction of initial inclusion disks which, under suitable conditions, provide the convergence of the Gargantini-Henrici interval method. They also play a key role in the convergence analysis of the fourth order Ehrlich-Aberth method with Newton's correction for the simultaneous approximation of all zeros of a polynomial. For this method we state the initial condition which enables the safe convergence. The initial condition is computationally verifiable since it depends only on initial approximations, which is of practical importance.     

无约束最优化锥模型拟牛顿信赖域方法的收敛性(英)

本文研究无约束最优化雄模型拟牛顿信赖域方法的全局收敛性．文章给出了确保这类方法全局收敛的条件．文章还证明了，当用拆线法来求这类算法中锥模型信赖域子问题的近似解时，确保全局收敛的条件得到满足     

Semismooth Newton and Newton iterative methods for HJB equation

In this paper, some semismooth methods are considered to solve a nonsmooth equation which can arise from a discrete version of the well-known Hamilton-Jacobi-Bellman equation. By using the slant differentiability introduced by Chen, Nashed and Qi in 2000, a semismooth Newton method is proposed. The method is proved to have monotone convergence by suitably choosing the initial iterative point and local superlinear convergence rate. Moreover, an inexact version of the proposed method is introduced, which reduces the cost of computations and still preserves nice convergence properties. Some numerical results are also reported.     

A globally convergent BFGS method for nonlinear monotone equations without any merit functions

Since 1965, there has been significant progress in the theoretical study on quasi-Newton methods for solving nonlinear equations, especially in the local convergence analysis. However, the study on global convergence of quasi-Newton methods is relatively fewer, especially for the BFGS method. To ensure global convergence, some merit function such as the squared norm merit function is typically used. In this paper, we propose an algorithm for solving nonlinear monotone equations, which combines the BFGS method and the hyperplane projection method. We also prove that the proposed BFGS method converges globally if the equation is monotone and Lipschitz continuous without differentiability requirement on the equation, which makes it possible to solve some nonsmooth equations. An attractive property of the proposed method is that its global convergence is independent of any merit function.We also report some numerical results to show efficiency of the proposed method.

     

Relationships between different types of initial conditions for simultaneous root finding methods

The construction of initial conditions of an iterative method is one of the most important problems in solving nonlinear equations. In this paper, we obtain relationships between different types of initial conditions that guarantee the convergence of iterative methods for simultaneously finding all zeros of a polynomial. In particular, we show that any local convergence theorem for a simultaneous method can be converted into a convergence theorem with computationally verifiable initial conditions which is of practical importance. Thus, we propose a new approach for obtaining semilocal convergence results for simultaneous methods via local convergence results.     

The Newton differential correction algorithm for rational Chebyshev approximation with constrained denominators

An algorithm for constrained rational Chebyshev approximation is introduced that combines the idea of an algorithm due to Hettich and Zencke, for which superlinear convergence is guaranteed, with the auxiliary problem used in the well-known original differential correction method. Superlinear convergence of the algorithm is proved. Numerical examples illustrate the fast convergence of the method and its advantages compared with the algorithm of Hettich and Zencke.     

Convergence and certain control conditions for hybrid viscosity approximation methods

Very recently, Yao, Chen and Yao [20] proposed a hybrid viscosity approximation method, which combines the viscosity approximation method and the Mann iteration method. Under the convergence of one parameter sequence to zero, they derived a strong convergence theorem in a uniformly smooth Banach space. In this paper, under the convergence of no parameter sequence to zero, we prove the strong convergence of the sequence generated by their method to a fixed point of a nonexpansive mapping, which solves a variational inequality. An appropriate example such that all conditions of this result are satisfied and their condition β_n→0 is not satisfied is provided. Furthermore, we also give a weak convergence theorem for their method involving a nonexpansive mapping in a Hilbert space.     

一类共轭梯度法的全局收敛性

共轭梯度法是求解大规模元约束优化同题的一种有效方法，本文提出一种新的共轭梯度法，证明了在推广的Wolfe线搜索条件下方法具有全局收敛性。最后对算法进行了数值试验，试验结果表明该算法具有良好的收敛性和有效性。     

一种新的迭代收敛阶数的证明与推广

迭代方法是求解非线性方程近似根的重要方法.本文基于隐函数存在定理,提出了一种新的迭代方法收敛性和收敛阶数的证明方法,并分别对牛顿(Newton)和柯西(Cauchy)迭代方法迭代收敛性和收敛阶数进行了证明.最后,利用本文提出的证明方法,证明了基于三次泰勒(Taylor)展式构成的迭代格式是收敛的,收敛阶数至少为4,并提出猜想,基于n次泰勒展式构成的迭代格式是收敛的,收敛阶数至少为(n+1).     

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.     

Waveform relaxation method for stochastic differential equations with constant delay

This paper extends the waveform relaxation method to stochastic differential equations with constant delay terms, gives sufficient conditions for the mean square convergence of the method. A lot of attention is paid to the rate of convergence of the method. The conditions of the superlinear convergence for a special case, which bases on the special splitting functions, are given. The theory is applied to a one-dimensional model problem and checked against results obtained by numerical experiments.     

The self-validated method for polynomial zeros of high efficiency

The improved iterative method of Newton’s type for the simultaneous inclusion of all simple complex zeros of a polynomial is proposed. The presented convergence analysis, which uses the concept of the R-order of convergence of mutually dependent sequences, shows that the convergence rate of the basic third order method is increased from 3 to 6 using Ostrowski’s corrections. The new inclusion method with Ostrowski’s corrections is more efficient compared to all existing methods belonging to the same class. To demonstrate the convergence properties of the proposed method, two numerical examples are given.