一类约束非光滑优化的非单调信赖域算法

提出了求解一类带一般凸约束的复合非光滑优化的信赖域算法 .和通常的信赖域方法不同的是 :该方法在每一步迭代时不是迫使目标函数严格单调递减 ,而是采用非单调策略 .由于光滑函数、逐段光滑函数、凸函数以及它们的复合都是局部Lipschitz函数 ,故本文所提方法是已有的处理同类型问题 ,包括带界约束的非线性最优化问题的方法的一般化 ,从而使得信赖域方法的适用范围扩大了 .同时 ,在一定条件下 ,该算法还是整体收敛的 .数值实验结果表明 :从计算的角度来看 ,非单调策略对高度非线性优化问题的求解非常有效     

Solution of a laminar boundary layer flow via a numerical method

In this paper, the numerical solution of the Blasius problem is obtained using the collocation method based on rational Chebyshev functions. The Blasius equation is a nonlinear ordinary differential equation which arises in the boundary layer flow. The method reduces solving the equation to solving a system of nonlinear algebraic equations. The results presented here demonstrate reliability and efficiency of the method.     

非线性特征值问题的二次近似方法

本文研究求解非线性特征值问题的数值方法.基于矩阵值函数的二次近似,将非线性特征值问题转化为二次特征值问题,提出了求解非线性特征值问题的逐次二次近似方法,分析了该方法的收敛性.结合求解二次特征值问题的Arnoldi方法和Jacobi-Davidson方法,给出求解非线性特征值问题的一些二次近似方法.数值结果表明本文所给算法是有效的.     

The “Gauss-Seidelization” of iterative methods for solving nonlinear equations in the complex plane

In this paper we introduce a process we have called “Gauss-Seidelization” for solving nonlinear equations. We have used this name because the process is inspired by the well-known Gauss-Seidel method to numerically solve a system of linear equations. Together with some convergence results, we present several numerical experiments in order to emphasize how the Gauss-Seidelization process influences on the dynamical behavior of an iterative method for solving nonlinear equations.     

Inexact Newton methods for the nonlinear complementarity problem

An exact Newton method for solving a nonlinear complementarity problem consists of solving a sequence of linear complementarity subproblems. For problems of large size, solving the subproblems exactly can be very expensive. In this paper we study inexact Newton methods for solving the nonlinear, complementarity problem. In such an inexact method, the subproblems are solved only up to a certain degree of accuracy. The necessary accuracies that are needed to preserve the nice features of the exact Newton method are established and analyzed. We also discuss some extensions as well as an application. This research was based on work supported by the National Science Foundation under grant ECS-8407240.     

Multilevel augmentation methods for solving the sine-Gordon equation

In this paper we develop the multilevel augmentation method for solving nonlinear operator equations of the second kind and apply it to solving the one-dimensional sine-Gordon equation. We first give a general setting of the multilevel augmentation method for solving the second kind nonlinear operator equations and prove that the multilevel augmentation method preserves the optimal convergence order of the projection method while reducing computational cost significantly. Then we describe the semi-discrete scheme and the fully-discrete scheme based on multiscale methods for solving the sine-Gordon equation, and apply the multilevel augmentation method to solving the discrete equation. A complete analysis for convergence order is proposed. Finally numerical experiments are presented to confirm the theoretical results and illustrate the efficiency of the method.     

Local convergence analysis of inexact Gauss-Newton like methods under majorant condition

In this paper, we present a local convergence analysis of inexact Gauss-Newton like methods for solving nonlinear least squares problems. Under the hypothesis that the derivative of the function associated with the least squares problem satisfies a majorant condition, we obtain that the method is well-defined and converges. Our analysis provides a clear relationship between the majorant function and the function associated with the least squares problem. It also allows us to obtain an estimate of convergence ball for inexact Gauss-Newton like methods and some important, special cases.     

解可分离结构变分不等式的一种新的交替方向法

交替方向法是求解可分离结构变分不等式问题的经典方法之一, 它将一个大型的变分不等式问题分解成若干个小规模的变分不等式问题进行迭代求解. 但每步迭代过程中求解的子问题仍然摆脱不了求解变分不等式子问题的瓶颈. 从数值计算上来说, 求解一个变分不等式并不是一件容易的事情.因此, 本文提出一种新的交替方向法, 每步迭代只需要求解一个变分不等式子问题和一个强单调的非线性方程组子问题. 相对变分不等式问题而言, 我们更容易、且有更多的有效算法求解一个非线性方程组问题. 在与经典的交替方向法相同的假设条件下, 我们证明了新算法的全局收敛性. 进一步的数值试验也验证了新算法的有效性.     

Hybrid Newton-Type Method for a Class of Semismooth Equations

In this paper, we present a hybrid method for the solution of a class of composite semismooth equations encountered frequently in applications. The method is obtained by combining a generalized finite-difference Newton method to an inexpensive direct search method. We prove that, under standard assumptions, the method is globally convergent with a local rate of convergence which is superlinear or quadratic. We report also several numerical results obtained applying the method to suitable reformulations of well-known nonlinear complementarity problems.     

A new logarithmic-quadratic proximal method for nonlinear complementarity problems

In this paper, we propose a new modified logarithmic-quadratic proximal (LQP) method for solving nonlinear complementarity problems (NCP). We suggest using a prediction-correction method to solve NCP. The predictor is obtained via solving the LQP system approximately under significantly relaxed accuracy criterion and the new iterate is computed by using a new step size α_k. Under suitable conditions, we prove that the new method is globally convergent. We report preliminary computational results to illustrate the efficiency of the proposed method. This new method can be considered as a significant refinement of the previously known methods for solving nonlinear complementarity problems.     

Interior-point methods for nonconvex nonlinear programming: regularization and warmstarts

In this paper, we investigate the use of an exact primal-dual penalty approach within the framework of an interior-point method for nonconvex nonlinear programming. This approach provides regularization and relaxation, which can aid in solving ill-behaved problems and in warmstarting the algorithm. We present details of our implementation within the loqo algorithm and provide extensive numerical results on the CUTEr test set and on warmstarting in the context of quadratic, nonlinear, mixed integer nonlinear, and goal programming. Research of the first author is sponsored by ONR grant N00014-04-1-0145. Research of the second author is supported by NSF grant DMS-0107450.     

解非线性方程组的极大熵方法

１引言考虑非线性方程组．其中Ｆ（ｘ）＝（ｆ１（ｘ）ｆ２（ｘ）,ｆ２（ｘ）,…．ｆｎ（ｘ））Ｔ．ｆｉ：Ｒｎ（ｉ＝１,…,ｎ）是连续可微实值函数．求解非线性方程组的方法多种多样,例如．以Ｎｅｗｔｏｎ法为代表的迭代法及其一些变形．以及将问题（１．１）转换为ｆ（Ｆ（ｘ））的极小化问题,等等．Ｎｅｗｔｏｎ法在理论上有许多很好的结果,但在实际计算过程中,由于例如方法对初始点的严格要求以及计算Ｆ＇（ｘ）或其相应的近似估计的困难,使方法的使用受到一定的限制．用无约束优化方法求解（１．１）时,通常将其化成一个非线…     

On the convergence of Broyden-like methods for nonlinear equations with nondifferentiable terms

In this paper, the author studies a Broyden-like method for solving nonlinear equations with nondifferentiable terms, which uses as updating matrices, approximations for Jacobian matrices of differentiable terms. Local and semilocal convergence theorems are proved. The results generalize those of Broyden, Dennis and Moré.     

横向剪切对双模量复合材料叠层矩形板非线性弯曲的影响

本文探讨了动力松弛(DR)法在双模量复合材料叠层矩形板非线性弯曲问题中的应用。在分析中分别采用叠层板大挠度经典理论和计及大转动(在Von Karman意义上)的复合材料叠层板剪切变形理论。我们发现,对于考虑横向剪切变形的非线性弯曲问题,如何计算虚拟密度以控制数值计算的稳定性,仍然需要进一步研究。本文提出了一种虚拟密度的计算方法,从而保证了本课题数值计算的稳定性。文中介绍了用DR法求解双模量复合材料叠层板非线性弯曲的主要步骤,给出了由轻度双模量材料(Born-Epxy(B-E))和高度双模量材料(Aramid-Rubber(A-R)和Polyester-Rubber(P-R))的两层正交叠层简支矩形板在正弦分布载荷及均布载荷作用下的非线性弯曲特性的数值结果。将所得结果和小挠度分析结果及普通复合材料的结果作了比较,并分析了横向剪切变形对无量纲中心挠度的影响。     

解非线性有限元方程的逐层校正迭代法

本文提出一种求解非线性有限元方程的逐层校正迭代法.有关数值分析表明,当网格分划较细,网格分划参数h_j较小时,仅需一次简单的迭代和校正步骤就可满足数值计算的要求,使用该方法的计算复杂性是最佳阶的,即为O(N_j),其中N_j为最细网格层上离散结点变量的数目.     

Chaos in King’s iterative family

In this paper, the dynamics of King’s family of iterative schemes for solving nonlinear equations is studied. The parameter spaces are presented, showing the complexity of the family. The analysis of the parameter space allows us to find elements of the family that have bad convergence properties, and also other ones with stable behavior.     

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.     

On the Pironneau-Polak method of centers

The method of centers is a well-known method for solving nonlinear programming problems having inequality constraints. Pironneau and Polak have recently presented a new version of this method. In the new method, the direction of search is obtained, at each iteration, by solving a convex quadratic programming problem. This direction finding subprocedure is essentially insensitive to the dimension of the space on which the problem is defined. Moreover, the method of Pironneau and Polak is known to converge linearly for finite-dimensional convex programs for which the objective function has a positive-definite Hessian near the solution (and for which the functions involved are twice continuously differentiable). In the present paper, the method and a completely implementable version of it are shown to converge linearly for a very general class of finite-dimensional problems; the class is determined by a second-order sufficiency condition and includes both convex and nonconvex problems. The arguments employed here are based on the indirect sufficiency method of Hestenes. Furthermore, the arguments can be modified to prove linear convergence for a certain class of infinite-dimensional convex problems, thus providing an answer to a conjecture made by Pironneau and Polak.     

A SUPERLINEARLY CONVERGENT TRUST REGION ALGORITHM FOR LC^1 CONSTRAINED OPTIMIZATION PROBLEMS

In this paper, a new trust region algorithm for nonlinear equality constrained LC^1 optimization problems is given. It obtains a search direction at each iteration not by solving a quadratic programming subproblem with a trust region bound, but by solving a system of linear equations. Since the computational complexity of a QP-Problem is in general much larger than that of a system of linear equations, this method proposed in this paper may reduce the computational complexity and hence improve computational efficiency. Furthermore, it is proved under appropriate assumptions that this algorithm is globally and super-linearly convergent to a solution of the original problem. Some numerical examples are reported, showing the proposed algorithm can be beneficial from a computational point of view.     

Gauss-Newton-based BFGS method with filter for unconstrained minimization

One class of the lately developed methods for solving optimization problems are filter methods. In this paper we attached a multidimensional filter to the Gauss-Newton-based BFGS method given by Li and Fukushima [D. Li, M. Fukushima, A globally and superlinearly convergent Gauss-Newton-based BFGS method for symmetric nonlinear equations, SIAM Journal of Numerical Analysis 37(1) (1999) 152-172] in order to reduce the number of backtracking steps. The proposed filter method for unconstrained minimization problems converges globally under the standard assumptions. It can also be successfully used in solving systems of symmetric nonlinear equations. Numerical results show reasonably good performance of the proposed algorithm.