一种保持对称性、稀疏性的拟Newton法

考虑非线性方程组 (1)F(x)=0其中,F(x)=(f_1(x),…,f_n(x))~T,x∈R~n.当F(x)为梯度算子时,F(x)的Jacobian是对称的.这类问题在实际计算工作中大量存在,比如近些年来研究很多的非线性泛函极小化问题就是如此,因此,人们自然地想到要把对于解非线性方程组很有效的Broyden方法发展到对称情形.1970年Powell提出PSB修正:     

换元Newton型方法的计算效能和最佳换元周期

考虑非线性方程组: F(x)=0, (1.1)其中F:R~n→R~n是二次连续可微函数.一般地说,解方程组(1.1)的拟Newton法较Newton法更为有效.我们可以将拟Newton法解释为逐次在R~n的子空间上构造F′(x)的近似(割线近似)得到的算法.按照这种思想,如果将子空间依次循环取成F′(x)的例     

阻尼牛顿法迭代中病态问题趋于稳定的渐变(瞬变)过程分析

<正>1引言设映像F:DR~n→R~n,考虑非线性方程组F(x)=0,x∈DR~n,其中F(x)=(f_1(x),f_2(x),…,f_n(x))T,分量f_i(x):R~n→R(i=1,2,…,n)是连续可微实值函数.目前,非线性方程组求解的数值方法有牛顿法、同伦型法、单纯形法与胞腔排除法等[1]~[3]牛顿法是一种非常实用的计算方法,迭代公式如下x=x+p,(2)其中x为前次迭代近似,x为紧接着x后的迭代近似,p=-[F'(x)]~(-1)F(x)为牛顿修正,F'(x)为x处的雅可比矩阵.     

关于正定可对称化线性代数方程组的预对称正则化算法

1引言考虑线性代数方程组A_x=b,A∈R~(n×n)非奇异,x,b∈R~n(1)的求解．当系数矩阵是大型稀疏的正定可对称化矩阵,文[1,2]讨论了一类预对称共轭梯度算法(LRSCG算法是其中之一),这类算法的实质是利用非对称的系数矩阵可对称化的性质,并结合共轭梯度法而构造的一种预处理的共轭梯度法[12,16,17]．但非对称的系数     

函数因子法——非线性方程组求解中处理奇异问题的一种新方法

§1.引言 非线性方程组 F(x)=0,F:D?R~n→R~n (1.1)嵌入参数t,构成同伦H:[0,T]×D?R~(n 1)→R~n,使得 H(0,x~0)=0,H(T,x)=F(x),(1.2)这里T可以是有限的或 ∞,当T为 ∞时以极限过程代替求值.若 H(t.x)=0(1.3)存在连续解x(t):[0,T]→D,则非线性方程组(1.1)的解x~*=x(T).若(1.3)的解     

求解非线性方程组的连续极小化方法

1.引言 求非线性方程组 F(x)=0 (1)(F:D?R~n→R~n)的各种方法中,牛顿法最为基本.但它只有局部收敛性和半局部收敛性,而且要求DF(x)~(-1)存在.为了扩大收敛范围及克服DF(x)奇异性带来的困难,用“连续化”的思想求方程(1)的解是一个有效的途径.这方面,已有许多工作,如[3—6].本文利用常微分方程几何理论,对连续化方法进行 些探讨,给出了沿积分曲线极小化求非线性方程组(1)的解的方法.考虑如下给定函数:     

再论Broyden方法的收敛性

本文用 Smale 提出的点估计理论,建立了在点估计条件下的求解非线性方程组 F(x)=0的著名的 Broyden 方法的收敛性及解的存在唯一性定理.从而在 R~n 空间的解析映射类上,解除了由于 F′的区域性 Lipschitz 条件带来的 Broyden 方法收敛判据之间的相互制约性.它为一大类修正算法点估计理论的建立,提供了新的途径.     

关于阻尼牛顿法收敛域的一个定理

设:D R~n→R~n是Frechet可导算子,以O(x,r)表示开球{x′|‖x′-x‖0. 为了求非线性方程F(x)-0的解x=x~*,常常使用牛顿迭代方法: x_(n+1)=x_n-F′(x_n)~(-1)F(x_n) (n∈N_0) (1)N_0={0,1,2,…}.但是在有些场合,为了取得更好的效果却需使用阻尼牛顿迭代法——一种修正的牛顿法:     

一类修正牛顿法的收敛域

设非线性方程 F(x)=0 (1) 其中F:DR~n→R~n是Fréchet可导算子。为求(1)的解x=x~*,通常用著名的牛顿迭代 x_(n+1)=x_n-(F′(x_n))~(-1)F(x_n),n=0,1,2,… (2) 有时为了取得更好效果,需要使用阻尼牛顿迭代 x_(n+1)=x_n-λ_n(F′(x_n))~(-1)F(x_n),n=0,1,2,… (3) 其中λ_n∈[0,1]称为阻尼因子。 迭代点列(2),(3)敛速虽高,缺点是要用到计算代价高昂的导算子,因此有导算子被近似替代所导出的种种修正牛顿迭代     

CONVERGENCE BALL OF ITERATIONS WITH ONE PARAMETER

§1Introduction OfthevariousiterationmethodsusedtosolvetheequationF(x)=0,thefamilyof iterativemethodswithoneparameter xn+1=xn-I+12PF(I-αPF)-1F′(xn)-1F(xn),PF(x)=F′(x)-1F″(x)F′(x)-1F(x),α∈[0,1],(1)isremarkablebecauseitincludesthesuper-Halleymethod,theChebyshevmethodand Halley'smethodasitsspecialcaseswhenαisequalto1,0and1/2,respectively.Thereareanumberofpapersconcerningtheabovethreeiterations.In[1-7],Halley's methodisstudied.Amongthesereferences,[1]givestheconvergenceofHalley'sme…     

A SUCCESSIVE LEAST SQUARES METHOD FOR STRUCTURED TOTAL LEAST SQUARES

A new method for Total Least Squares (TLS) problems is presented. It differs from previous approaches and is based on the solution of successive Least Squares problems.The method is quite suitable for Structured TLS (STLS) problems. We study mostly the case of Toeplitz matrices in this paper. The numerical tests illustrate that the method converges to the solution fast for Toeplitz STLS problems. Since the method is designed for general TLS problems, other structured problems can be treated similarly.     

无约束优化问题的对角稀疏拟牛顿法

对无约束优化问题提出了对角稀疏拟牛顿法,该算法采用了Armijo非精确线性搜索,并在每次迭代中利用对角矩阵近似拟牛顿法中的校正矩阵,使计算搜索方向的存贮量和工作量明显减少,为大型无约束优化问题的求解提供了新的思路．在通常的假设条件下,证明了算法的全局收敛性,线性收敛速度并分析了超线性收敛特征。数值实验表明算法比共轭梯度法有效,适于求解大型无约束优化问题．     

Generalized mixed quasi-equilibrium problems with trifunction

In this paper, we introduce a new class of equilibrium problems, which is called the generalized mixed quasi-equilibrium problems with trifunction. Using the auxiliary principle technique, we suggest and analyze a proximal point method for solving the generalized mixed quasi-equilibrium problems. It is shown that the convergence of the proposed method requires only pseudomonotonicity, which is a weaker condition than monotonicity. Our results represent an improvement and refinement of previously known results. Since the generalized mixed quasi-equilibrium problems include equilibrium problems and variational inequalities as special cases, results proved in this paper continue to hold for these problems.     

Using dual network bounds in algorithms for solving generalized set packing partitioning problems

This article deals with a method to compute bounds in algorithms for solving the generalized set packing/partitioning problems. The problems under investigation can be solved by the branch and bound method. Linear bounds computed by the simplex method are usually used. It is well known that this method breaks down on some occasions because the corresponding linear programming problems are degenerate. However, it is possible to use the dual (Lagrange) bounds instead of the linear bounds. A partial realization of this approach is described that uses a network relaxation of the initial problem. The possibilities for using the dual network bounds in the approximation techniques to solve the problems under investigation are described.     

Application of optimal homotopy asymptotic method for the analytic solution of singular Lane-Emden type equation

In this study, optimal homotopy asymptotic method is applied on singular initial value Lane-Emden type problems to check the effectiveness and performance of the method. It is observed that the method is easy to implement, quite valuable to handle singular phenomena and yield excellent results at minimum computational cost. Computational results of some of the test problems are presented to demonstrate the viability and practical usefulness of the method.     

求解大规模极大极小问题的光滑化三项共轭梯度算法

基于指数罚函数,对最近提出的一种求解无约束优化问题的三项共轭梯度法进行了修正,并用它求解更复杂的大规模极大极小值问题.证明了该方法生成的搜索方向对每一个光滑子问题是充分下降方向,而且与所用的线搜索规则无关.以此为基础,设计了求解大规模极大极小值问题的算法,并在合理的假设下,证明了算法的全局收敛性.数值实验表明,该算法优于文献中已有的类似算法.     

Sinc-Galerkin method for solving linear sixth-order boundary-value problems

There are few techniques available to numerically solve sixth-order boundary-value problems with two-point boundary conditions. In this paper we show that the Sinc-Galerkin method is a very effective tool in numerically solving such problems. The method is then tested on examples with homogeneous and nonhomogeneous boundary conditions and a comparison with the modified decomposition method is made. It is shown that the Sinc-Galerkin method yields better results.

     

带补偿的随机二次规划的近似Lagrange-Newton方法

In this paper, two-stage stochastic quadratic programming problems with equality constraints are considered. By Monte Carlo simulation-based approximations of the objective function and its first (second)derivative,an inexact Lagrange-Newton type method is proposed.It is showed that this method is globally convergent with probability one. In particular, the convergence is local superlinear under an integral approximation error bound condition.Moreover, this method can be easily extended to solve stochastic quadratic programming problems with inequality constraints.     

求弹性半平面问题基本解的一个直接方法

本文利用镜像法[2]和直接积分导出了弹性半平面问题的基本解.这个推导方法比Gladwell用镜像法和富氏变换要直观.这个基本解对于用边界元法分析半平面域具有任意外形孔洞的应力集中问题将起着十分重要的作用.     

Proximal Point Methods and Nonconvex Optimization

The goal of this paper is to discover some possibilities for applying the proximal point method to nonconvex problems. It can be proved that – for a wide class of problems – proximal regularization performed with appropriate regularization parameters ensures convexity of the auxiliary problems and each accumulation point of the method satisfies the necessary optimality conditions.