求解非线性方程组的L-M方法

本文研究了求解奇异非线性方程组的Levenberg-Marquardt方法的收敛性.利用选取新的迭代参数求解非线性方程组的L-M方法,获得点列的超线性收敛性和二阶收敛性,并把试验结果与文献[19,20]的结果进行了比较.     

混合布谷鸟搜索算法求解非线性方程组

针对当前算法求解非线性方程组存在求解个数不完整、精度低等问题,提出一种混合布谷鸟搜索算法(HCS).首先分析原始布谷鸟搜索算法不足,再结合差分进化算法和二次插值优势,将其进行深度融合.通过12个非线性方程组的仿真实验,结果表明算法能有效搜索到非线性方程组的较多解,并与其他算法进行比较,算法在解的数量和质量上具有优越性.     

带凸约束的非线性方程组的无导数记忆法

基于无导数线搜索技术和投影方法,本文提出了一种新的求解带凸约束的非线性方程组的无导数记忆法.该方法在每步迭代时不需要计算和贮存任何矩阵,因而适合求解大规模非线性方程组问题.在较弱条件下,该算法具有全局收敛性.数值试验结果及其相关的比较表明该算法是比较有效的.     

用非线性方程组求解等式约束非线性规划问题的降维算法

本文研究线性和非线性等式约束非线性规划问题的降维算法.首先,利用一般等式约束问题的降维方法,将线性等式约束非线性规划问题转换成一个非线性方程组,解非线性方程组即得其解;然后,对线性和非线性等式约束非线性规划问题用Lagrange乘子法,将非线性约束部分和目标函数构成增广的Lagrange函数,并保留线性等式约束,这样便得到一个线性等式约束非线性规划序列,从而,又将问题转化为求解只含线性等式约束的非线性规划问题.     

解凸约束非线性单调方程组的无导数谱PRP投影算法

本文在著名PRP共轭梯度算法的基础上研究了一种无导数谱PRP投影算法,并证明了算法在求解带有凸约束条件的非线性单调方程组问题的全局收敛性.由于无导数和储存量小的特性,它更适应于求解大规模非光滑的非线性单调方程组问题.数值试验表明,新算法对给定的测试问题是有效的和稳定的.     

求解非线性方程组的自适应拟Newton L—修正方法

在求解非线性方程组中认真采用的拟Newton法应是Broyden于1965年提出的方法,其迭代格式为:     

基于非单调自适应信赖域法求解非线性方程组

本文提出了求解非线性方程组的非单调自适应信赖域法.在适当的条件下证明了非单调自适应信赖域法的局部及全局收敛性质.基本的数值实验表明该方法在处理某些非线性方程组是非常有效的.     

谱HS投影算法求解非线性单调方程组

借助谱梯度法和HS共轭梯度法的结构, 建立一种求解非线性单调方程组问题的谱HS投影算法. 该算法继承了谱梯度法和共轭梯度法储存量小和计算简单的特征, 且不需要任何导数信息, 因此它适应于求解大规模非光滑的非线性单调方程组问题. 在适当的条件下, 证明了该算法的收敛性, 并通过数值实验表明了该算法的有效性.     

简单约束非线性方程组的射影尺度牛顿方法

基于射影尺度牛顿方法,本文使用新的势函数以取代原有的势函数,得到一类求解非线性方程组的数值算法.在合适的假设下,证明了算法的全局强收敛性和局部二次收敛速度.数值试验的结果说明了算法的有效性.     

凸约束非线性方程组的一种无导数投影方法

推广LCG共轭梯度方法并建立一种求解凸约束非线性单调方程组问题的无导数投影方法.在适当的条件下,证明了方法的全局收敛性.方法不需要任何导数信息,而且继承了共轭梯度方法储存量小的特征,因此它特别适合求解大规模非光滑的非线性单调方程组问题.大量数值结果和比较表明方法是有效的和稳定的.     

A BFGS trust-region method with a new nonmonotone technique for nonlinear equations

In this paper, we consider a trust-region method for solving nonlinear equations which employs a new nonmonotone technique. A strong nonmonotone strategy and a weaker nonmonotone strategy can be obtained by choosing the parameter adaptively. Thus, the disadvantages of the traditional nonmonotone strategy can be avoided. It does not need to compute the Jacobian matrix at every iteration, so that the workload and time are decreased. Theoretical analysis indicates that the new algorithm preserves the global convergence under classical assumptions. Moreover, superlinear and quadratic convergence are established under suitable conditions. Numerical experiments show the efficiency and effectiveness of the proposed method for solving nonlinear equations.     

A computational method for minimization with nonlinear constraints

An iterative technique is developed to solve the problem of minimizing a functionf(y) subject to certain nonlinear constraintsg(y)=0. The variables are separated into the basic variablesx and the independent variablesu. Each iteration consists of a gradient phase and a restoration phase. The gradient phase involves a movement (on a surface that is linear in the basic variables and nonlinear in the independent variables) from a feasible point to a varied point in a direction based on the reduced gradient. The restoration phase involves a movement (in a hyperplane parallel tox-space) from the nonfeasible varied point to a new feasible point.The basic scheme is further modified to implement the method of conjugate gradients. The work required in the restoration phase is considerably reduced when compared with the existing methods.     

A projection method for a system of nonlinear monotone equations with convex constraints

In this paper, we propose a projection method for solving a system of nonlinear monotone equations with convex constraints. Under standard assumptions, we show the global convergence and the linear convergence rate of the proposed algorithm. Preliminary numerical experiments show that this method is efficient and promising. This work was supported by the Postdoctoral Fellowship of The Hong Kong Polytechnic University, the NSF of Shandong China (Y2003A02).     

Efficient hybrid method for solving special type of nonlinear partial differential equations

In this article, an efficient hybrid method has been developed for solving some special type of nonlinear partial differential equations. Hybrid method is based on tanh–coth method, quasilinearization technique and Haar wavelet method. Nonlinear partial differential equations have been converted into a nonlinear ordinary differential equation by choosing some suitable variable transformations. Quasilinearization technique is used to linearize the nonlinear ordinary differential equation and then the Haar wavelet method is applied to linearized ordinary differential equation. A tanh–coth method has been used to obtain the exact solutions of nonlinear ordinary differential equations. It is easier to handle nonlinear ordinary differential equations in comparison to nonlinear partial differential equations. A distinct feature of the proposed method is their simple applicability in a variety of two‐ and three‐dimensional nonlinear partial differential equations. Numerical examples show better accuracy of the proposed method as compared with the methods described in past. Error analysis and stability of the proposed method have been discussed.     

Semilinear evolution equations with nonlinear constraints and applications

This paper deals with a general class of evolution problems for semilinear equations coupled with nonlinear constraints. Those constraints may contain compositions of nonlinear operators and unbounded linear operators, and hence the associated operators are not necessarily formulated in the form of continuous perturbations of linear operators. Accordingly, a family of equivalent norms is introduced to discuss 'quasidissipativity' in a local sense of the operators and a generation theory for nonlinear semigroups is employed to construct solution operators on bounded sets. It is a feature of our treatment that the resultant solution operators are obtained as nonlinear semigroups on the whole space which are not 'quasicontractive' but locally equi-Lipschitz continuous.     

Optimal control of a nonlinear singular system with state constraints

A control system described by a nonlinear equation of parabolic type is considered in the situation where there may be no global solution. A particular optimal control problem subject to state constraints is studied. A proof of the existence of an optimal control is presented. The penalty method is used to obtain necessary conditions for optimal control. A proof of the convergence of this method is given. The successive approximation method is used to obtain an approximate solution for the conditions derived. Translated fromMatematicheskie Zametki, Vol. 60, No. 4, pp. 511–518, October, 1996.     

Solution of a system of nonlinear equations by Adomian decomposition method

In this paper, the Adomian decomposition method is applied to solve a system of nonlinear equations. Convergence of the method is proved and some examples are presented to illustrate the method.     

Solving nonlinear simultaneous equations with a generalization of Brent's method

We describe an implementation of a generalization of Brent's method for solving systems of nonlinear equations. Some important features of the algorithm, like step control, discretization of derivatives and stopping criteria, are discussed. In particular we give numerical experiences which show that a stopping criterion proposed by D. Gay is efficient.     

Nonmonotone filter DQMM method for the system of nonlinear equations

In this paper, we propose a nonmonotone filter Diagonalized Quasi-Newton Multiplier (DQMM) method for solving system of nonlinear equations. The system of nonlinear equations is transformed into a constrained nonlinear programming problem which is then solved by nonmonotone filter DQMM method. A nonmonotone criterion is used to speed up the convergence progress in some ill-conditioned cases. Under reasonable conditions, we give the global convergence properties. The numerical experiments are reported to show the effectiveness of the proposed algorithm.