广义非线性最小二乘问题的两个新方法

1．引言讨论如下的广义非线性最小二乘问题其中为常数（i＝1~m），W由于此问题的特殊形式，将此问题转化为如下两个子问题进行求解比较有效[1]子问题1．对每一固定的X，解得子问题2。对子问题1的解，解对两个子问题的求解，[1]中给出了一种有效的方法。然而在两个子问题的已有求解方法中，关于方法收敛速度的讨论非常少见，本文给出了求解这两个子问题的两个算法，并证明了算法的超线性收敛性．为书写简单，以下约定：一个符号在（，L）处的值略去（，L），如V‘F＝＊‘列X，L）等·一个具有上标k和＊的符号分别表示其在（x‘，t‘）和…     

边界约束二次规划问题的分解方法

１．引言本文考虑如下边界约束的二次规划问题：其中ＱＥ＊"""是对称的,Ｃ,人。Ｅ＊"是给定的常数向量,且Ｚ＜。这类问题经常出现在偏微分方程,离散化的连续时间最优控制问题、线性约束的最小二乘问题、工程设计、或作为非线性规划方法中的序列子问题．因此具有特殊的重要性．本文提出求解问题（１．１）的分解方法．它类似求解线性代数方程组的选代法,它是对Ｑ进行正则分裂【对即把Ｑ分裂为两个矩阵之和,Ｑ＝Ｎ十片而这两个矩阵之差（Ｎ一则是对称正定的．在每次迭代中用一个易于求解的矩阵Ｎ替代Ｑ进行计算一新的二次规划问题．在适…     

线性流形上的两类矩阵最佳逼近问题

1．引言设R”””表示所有实mxn阶矩阵的全体，OR”””表示所有n阶正交阵的全体，对于A＝（ail）eR”””，B＝（b；j）eRP”’，用A＠BeR’”“”’表示矩阵A与B的Kronecker积，用A二（all，ala，…，al。，aal，…aa。，…；a＿l，…，a＿＊“表m矩阵A拉直算子，l］·IF表示矩阵的Frobenius范数，11·11。表示向量的2一范数．设S＝｛X，X6R”””f（X）二llAIXBI—Dlll》＋IIAZXBZ—D。股一Zill｝其中AlER”X”，BIERPXq，DIER”XqAZERtX”BZERPXIDZERtXI考虑下列两类问题：问题I．给定CIERll“”．FIE…     

大型稀疏线性互补问题的行作用法

且引言考虑线性互补问题＊＊P（q，M）：求X二（X；，x。，…，x。厂E”使得x＞O，训x）E＊x＋g＞o，／U（X）一O（1）其中M一（m；。）为nXn矩阵（不必对称），q一切，q。，…，q。）rER“为给定常向量．通常情况下已有求解LCP（q，M）的若干著名算法［‘－’j．本文提出求解LCP（q，M）的一种新算法一行作用法，方法具有如下特点：（i）每次迭代只需n个简单的投影运算，每次投影只涉及矩阵M的一行；（n）生成新的迭代点x‘“‘时只利用前次迭代点／；（iii）对矩阵M不实施任何整体运算．因而适合于求解大型（巨型）稀疏问题，且…     

双障碍问题的逐次逼近阻尼牛顿法

1．引言及算法考虑Rn中的双障碍问题：求＊E年使其中f：Rn→Rn连续可微,c＝｛x-∈Rn|0≤x≤c｝,c∈Rn为常向量．若记c＝（c1,…,cn）T,则不准证明问题（1）等价于求解下面的非光滑方程组其中算子max,min是指分量的最大或最小．显然,由（2）式定义的函数H：Rn→Rn的第i个分量函数Hi：Rn→R为由（2）定义的函数H一般不是厂可微的,但我们可对H作如下分解：设住k｝是一单调递减且趋于O的正数序列,xk00,八k（）都是R”－＋R”的映射,其中bk（儿,W汕地分别由下式定义：其中圳的二v：二;＜人（x）一。k｝,B（x）二F：人（……     

对称不定问题的不精确Newton法

1.引 言 非线性方程组F（x）=0的数值求解,经典的算法是Newton迭代;xk 1=xk sk,k=0,1,2,…,（1.1）其中的sk满足F’（xk）sk=-F（xk）;k=0,1,2,….（1.2）这里x0为迭代的初始点,{xk}称为Newton迭代序列.当变量个数比较多时,每一步Newton迭代中计算Jacobi矩阵F’（xk）和求解线性方程组（1.2）的代价非常高;特别当xk远离方程组的解x＊时,高精度地求解线性方程组（1.2）     

一类非线性代数方程组的并行迭代算法

１．引言考虑非线性代数方程组这里,　　　　　　　　　　　　　　　　　　　　　　　　　　　　为连续的对角映射,二者的导函数均存在,但并不一定连续．这类非线性代数方程组具有丰富的实际背景．譬如,Ｓｔｅｆａｎ问题和许多弱非线性椭圆型偏微分方程,就可归结为（１．１）的数值求解问题．根据方程组（１．１）的特殊结构,并利用矩阵多重分裂思想,文ｔＺ］讨论了一类并行非线性Ｇａｕｓｓ－Ｓｅｉｄｅｌ型迭代算法．这类算法具有很好的数值性质和较高的并行效率·在此基础上,运用松弛加速技术,文［８］进一步研究了一类并行多分…     

最优化中的几个问题解的等价性

考虑如下非线性规划问题：众所周知，问题（NP）的解法主要有三类：1．直接处理约束，2．将约束最优化问题化为 无约束最优化问题来处理，3．将（NP）化为简单的约束最优化问题如线性规划或二次规划等来处理，而将约束最优化问题化为无约束最优化问题的主要手段是利用如下的Lagrange函数：L（X，X，X）一八X）＋（X，g（X》十（X，h（X》（1．I）定义1．1称点卜”，V”撤足互补性条件，如果对”（X）一ojE【I：c］（亚．2）根据Lagrange函数（1．1）定义如下问题：（SPP）：求点k”，u”，v」6H””，m二。；＋c，使b“，u“，v」…     

伴随矩阵的反问题

伴随矩阵的反问题王新哲（保定师范专科学校数学系０７１０５１）设Ａ＝（ａｉｊ）为一方阵，记Ａ＊＝（Ａｊｉ），其中Ａｊｉ为ａｊｉ在Ａ中的代数余子式，则称Ａ＊为Ａ的伴随矩阵．然而，当给定一方阵Ｂ时，却不一定有方阵人使得Ａ＊＝Ｂ．如：Ｂ＝因三阶方阵Ａ的伴随矩...     

由一道解析几何例题的解答所想到的

现行高中课本《平面解析几何》（必修）P38页中有这样一道例题：已知两条直线：l_1：x＋my＋b＝0，l_2：（m－2）x＋3y＋2m＝0．当m为何值时，l_1与l_2（i）相交；（ii）平行；（iii）重合．课本给出的解题过程是：解将两直线的方程组成方程组：解得m＝3．（i）当m≠3时，方程组有唯一解，l_1与l_2相交．（ii）当m＝－1时，方程组无解，l_1与l_2平行．（iii）当m＝3时，方程组有无穷多解，l_1与l_2重合．其实，当m＝2或m＝0时，这两条直线也相交，这正是及的分母为0的倩况．因此这类问题还应注意对分母为零的情况的讨论．下面，我们不妨再…     

Parallel multisplitting two-stage iterative methods for large sparse systems of weakly nonlinear equations

The finite difference or the finite element discretizations of many differential or integral equations often result in a class of systems of weakly nonlinear equations. In this paper, by reasonably applying both the multisplitting and the two-stage iteration techniques, and in accordance with the special properties of this system of weakly nonlinear equations, we first propose a general multisplitting two-stage iteration method through the two-stage multiple splittings of the system matrix. Then, by applying the accelerated overrelaxation (AOR) technique of the linear iterative methods, we present a multisplitting two-stage AOR method, which particularly uses the AOR-like iteration as inner iteration and is substantially a relaxed variant of the afore-presented method. These two methods have a forceful parallel computing function and are much more suitable to the high-speed multiprocessor systems. For these two classes of methods, we establish their local convergence theories, and precisely estimate their asymptotic convergence factors under some suitable assumptions when the involved nonlinear mapping is only directionally differentiable. When the system matrix is either an H-matrix or a monotone matrix, and the nonlinear mapping is a P-bounded mapping, we thoroughly set up the global convergence theories of these new methods. Moreover, under the assumptions that the system matrix is monotone and the nonlinear mapping is isotone, we discuss the monotone convergence properties of the new multisplitting two-stage iteration methods, and investigate the influence of the multiple splittings as well as the relaxation parameters upon the convergence behaviours of these methods. Numerical computations show that our new methods are feasible and efficient for parallel solving of the system of weakly nonlinear equations. This revised version was published online in August 2006 with corrections to the Cover Date.     

ON MONOTONE CONVERGENCE OF NONLINEARMULTISPLITTING RELAXATION METHODS

ＯＮＭＯＮＯＴＯＮＥＣＯＮＶＥＲＧＥＮＣＥＯＦＮＯＮＬＩＮＥＡＲＭＵＬＴＩＳＰＬＩＴＴＩＮＧＲＥＬＡＸＡＴＩＯＮＭＥＴＨＯＤＳ￥ＷＡＮＧＤＥＲＥＮ；ＢＡＩＺＨＯＮＧＺＨＩ（ＤｅｐａｚａｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＳｈｓｌｌｇｈａｉＵｎｉｖｅ...     

A smoothing self-adaptive Levenberg-Marquardt algorithm for solving system of nonlinear inequalities

In this paper, we consider the smoothing self-adaptive Levenberg-Marquardt algorithm for the system of nonlinear inequalities. By constructing a new smoothing function, the problem is approximated via a family of parameterized smooth equations H(x) = 0. A smoothing self-adaptive Levenberg-Marquardt algorithm is proposed for solving the system of nonlinear inequalities based on the new smoothing function. The Levenberg-Marquardt parameter μ_k is chosen as the product of μ_k = ∥H^k∥^δ with δ ∈ (0, 2] being a positive constant. We will show that if ∥H^k∥^δ provides a local error bound, which is weaker than the non-singularity, the proposed method converges superlinearly to the solution for δ ∈ (0, 1), while quadratically for δ ∈ [1, 2]. Numerical results show that the new method performs very well for system of inequalities.     

A class of two‐stage iterative methods for systems of weakly nonlinear equations

The discretizations of many differential equations by the finite difference or the finite element methods can often result in a class of system of weakly nonlinear equations. In this paper, by applying the two-tage iteration technique and in accordance with the special properties of this weakly nonlinear system, we first propose a general two-tage iterative method through the two-tage splitting of the system matrix. Then, by applying the accelerated overrelaxation (AOR) technique of the linear iterative methods, we present a two-tage AOR method, which particularly uses the AOR iteration as the inner iteration and is substantially a relaxed variant of the afore-presented method. For these two classes of methods, we establish their local convergence theories, and precisely estimate their asymptotic convergence factors under some suitable assumptions when the involved nonlinear mapping is only B-differentiable. When the system matrix is either a monotone matrix or an H-matrix, and the nonlinear mapping is a P-bounded mapping, we thoroughly set up the global convergence theories of these new methods. Moreover, under the assumptions that the system matrix is monotone and the nonlinear mapping is isotone, we discuss the monotone convergence properties of the new two-tage iteration methods, and investigate the influence of the matrix splittings as well as the relaxation parameters on the convergence behaviours of these methods. Numerical computations show that our new methods are feasible and efficient for solving the system of weakly nonlinear equations. This revised version was published online in June 2006 with corrections to the Cover Date.     

A MODIFIED LEVENBERG-MARQUARDT ALGORITHM FOR SINGULAR SYSTEM OF NONLINEAR EQUATIONS

Based on the work of paper,we propose a modified Levenberg-Marquardt algoithm for solving singular system of nonlinear equations F(x)=0,where F(x):R^n→R^n is continuously differentiable and F‘(x)is Lipschitz continuous.The algorithm is equivalent to a trust region algorithm in some sense,and the global convergence result is given.The sequence generated by the algorithm converges to the solution quadratically,if ||F(x)||2 provides a local error bound for the system of nonlinear equations.Numerical results show that the algorithm performs well.     

解约束优化的分段线性有理NCP函数

本文给出新的NCP函数,这些函数是分段线性有理正则伪光滑的,且具有良好的性质.把这些NCP函数应用到解非线性优化问题的方法中.例如,把求解非线性约束优化问题的KKT点问题分别用QP-free方法,乘子法转化为解半光滑方程组或无约束优化问题.然后再考虑用非精确牛顿法或者拟牛顿法来解决该半光滑方程组或无约束优化问题.这个方法是可实现的,且具有全局收敛性.可以证明在一定假设条件下,该算法具有局部超线性收敛性.     

在一个新步长规则下梯度投影算法的全局收敛性

考虑约束最优化问题：minx∈Ωf(x)其中:f:R^n→R是连续可微函数，Ω是一闭凸集。本文研究了解决此问题的梯度投影方法，在步长的选取时采用了一种新的策略，在较弱的条件下，证明了梯度投影响方法的全局收敛性。     

Modulus‐based synchronous multisplitting iteration methods for linear complementarity problems

By an equivalent reformulation of the linear complementarity problem into a system of fixed‐point equations, we construct modulus‐based synchronous multisplitting iteration methods based on multiple splittings of the system matrix. These iteration methods are suitable to high‐speed parallel multiprocessor systems and include the multisplitting relaxation methods such as Jacobi, Gauss–Seidel, successive overrelaxation, and accelerated overrelaxation of the modulus type as special cases. We establish the convergence theory of these modulus‐based synchronous multisplitting iteration methods and their relaxed variants when the system matrix is an H ₊ ‐matrix. Numerical results show that these new iteration methods can achieve high parallel computational efficiency in actual implementations. Copyright © 2012 John Wiley & Sons, Ltd.     

MIMD机上解非线性方程组的同步化并行拟Newton法

本文给出了适于在MIMD机上解非线性方程组的同步化并行Broyden方法和换列修正拟Newton法的迭代格式,以及它们的局部收敛性定理.数值试验结果也验证了收敛性.     

A class of asynchronous multisplitting two-stage iterations for large sparse block systems of weakly nonlinear equations

For the block system of weakly nonlinear equations Ax=G(x), where is a large sparse block matrix and is a block nonlinear mapping having certain smoothness properties, we present a class of asynchronous parallel multisplitting block two-stage iteration methods in this paper. These methods are actually the block variants and generalizations of the asynchronous multisplitting two-stage iteration methods studied by Bai and Huang (Journal of Computational and Applied Mathematics 93(1) (1998) 13–33), and they can achieve high parallel efficiency of the multiprocessor system, especially, when there is load imbalance. Under quite general conditions that is a block H-matrix of different types and is a block P-bounded mapping, we establish convergence theories of these asynchronous multisplitting block two-stage iteration methods. Numerical computations show that these new methods are very efficient for solving the block system of weakly nonlinear equations in the asynchronous parallel computing environment.