使用拟牛顿迭代公式的广义共轭梯度法

当容易计算目标函数f的梯度g(x)=■f(x),但存贮不允许使用完整的拟牛顿法(QN)时,广义共轭梯度法(CG)_H是一个很有前途的方法.它的基本迭代公式是     

Improved Hessian approximation with modified secant equations for symmetric rank-one method

Symmetric rank-one (SR1) is one of the competitive formulas among the quasi-Newton (QN) methods. In this paper, we propose some modified SR1 updates based on the modified secant equations, which use both gradient and function information. Furthermore, to avoid the loss of positive definiteness and zero denominators of the new SR1 updates, we apply a restart procedure to this update. Three new algorithms are given to improve the Hessian approximation with modified secant equations for the SR1 method. Numerical results show that the proposed algorithms are very encouraging and the advantage of the proposed algorithms over the standard SR1 and BFGS updates is clearly observed.     

Stair Laguerre pseudospectral method for differential equations on the half line

A stair Laguerre pseudospectral method is proposed for numerical solutions of differential equations on the half line. Some approximation results are established. A stair Laguerre pseudospcetral scheme is constructed for a model problem. The convergence is proved. The numerical results show that this new method provides much more accurate numerical results than the standard Laguerre spectral method. Dedicated to Charles A. Micchelli on the occasion of his 60th birthday Mathematics subject classifications (2000) 65N35, 41A10. Li-lian Wang: The work of this author is partially supported by The Shanghai Natural Science Foundation N. 00JC14057, The Shanghai Natural Science Foundation for Youth N. 01QN85 and The Special Funds for Major Specialities of Shanghai Education Committee. Ben-yu Guo: The work of this author is partially supported by The Special Funds for Major State Basic Research Projects of China G1999032804, The Shanghai Natural Science Foundation N. 00JC14057 and The Special Funds for Major Specialities of Shanghai Education Committee.     

降维梯度法

§1.引言 本文研究降维梯度法,它具有共轭梯度法的一切性质.对于正定二次函数,用不着精确的一维搜索,只要在每步加入两个校正项,即可将高阶问题转化为低阶问题,保证了二     

Practical Quasi-Newton algorithms for singular nonlinear systems

Quasi-Newton methods for solving singular systems of nonlinear equations are considered in this paper. Singular roots cause a number of problems in implementation of iterative methods and in general deteriorate the rate of convergence. We propose two modifications of QN methods based on Newton’s and Shamanski’s method for singular problems. The proposed algorithms belong to the class of two-step iterations. Influence of iterative rule for matrix updates and the choice of parameters that keep iterative sequence within convergence region are empirically analyzed and some conclusions are obtained.     

Approximate analysis of a gordon-newell like non-product-form queueing network

This paper is tutorial in nature. It demonstrates how a particular heuristic extension of the arrival theorem, which was introduced earlier for very special network topologies, can be effectively applied (in an essentially unchanged manner) to obtain all mean performance measures for a rich class of Gordon-Newell like non-product-form queueing networks (QNs). All nodes in the class of queueing networks discussed are either FIFO or IS (pure delay), there is a single closed chain with probabilistic routing and each FIFO node also processes customers from a dedicated open chain. The number of FIFO nodesK, the number of IS nodesL and the closed chain populationN are finite but arbitrary and closed chain customers route probabilistically according to an arbitrary routing matrixQ. The think time distribution at an IS node is general, the service time distribution for both closed chain and open chain customers at the FIFO nodes is exponential with distinct service times for each, and both IS think times and FIFO service times are node dependent.The approximation technique is enhanced by an analytic study which demonstrates that it mirrors the expected behavior of the QN in many essential respects: monotonicity, bottleneck and asymptotic behavior. Moreover, in the case of balanced QNs, the approximation yields simple and explicit expressions for all quantities of interest. The analytic study and the numerical experiments presented complement one another and suggest that this approximation technique captures the essential structure of the QN, insofar as mean performance quantities are concerned.Currently on leave of absence from Bell Laboratories, at The Chinese University of Hong Kong, Department of Information Engineering, New Territories, Hong Kong.     

The Existence of Silnikov＇s Orbit in Four-dimensional Duffing＇s Systems

The existence of Silnikov‘s orbits in a four-dimensional dynamical system is discussed. The existence of Silnikov‘s orbit resulting in chaotic dynamics is established by the fiber structure of invariant manifold and high-dimensional Melnikov method. Numerical simulations are given to demonstrate the theoretical analysis.     

Function-valued Padé-type approximant via the formal orthogonal polynomials and its applications in solving integral equations

A kind of function-valued Padé-type approximant via the formal orthogonal polynomials (FPTAVOP) is introduced on the polynomial space and an algorithm is sketched by means of the formal orthogonal polynomials. This method can be applied to approximate characteristic values and the corresponding characteristic function of Fredholm integral equation of the second kind. Moreover, theoretical analyses show that FPTAVOP method is the most effective one for accelerating the convergence of a sequence of functions. In addition, a typical numerical example is presented to illustrate when the estimates of characteristic value and characteristic function by using this new method are more accurate than other methods.     

Sparse quasi-Newton updates with positive definite matrix completion

Quasi-Newton methods are powerful techniques for solving unconstrained minimization problems. Variable metric methods, which include the BFGS and DFP methods, generate dense positive definite approximations and, therefore, are not applicable to large-scale problems. To overcome this difficulty, a sparse quasi-Newton update with positive definite matrix completion that exploits the sparsity pattern of the Hessian is proposed. The proposed method first calculates a partial approximate Hessian , where , using an existing quasi-Newton update formula such as the BFGS or DFP methods. Next, a full matrix H _k+1, which is a maximum-determinant positive definite matrix completion of , is obtained. If the sparsity pattern E (or its extension F) has a property related to a chordal graph, then the matrix H _k+1 can be expressed as products of some sparse matrices. The time and space requirements of the proposed method are lower than those of the BFGS or the DFP methods. In particular, when the Hessian matrix is tridiagonal, the complexities become O(n). The proposed method is shown to have superlinear convergence under the usual assumptions.      

Supply chain modelling and its analytical evaluation

A model-based analysis of supply chains (SCs) is proposed as a means of estimating performance measures like lead times and resource utilisations. SCs are regarded as discrete event dynamic systems (DEDS), described as process chains in an application-specific formalism, and analysed by state-of-the-art techniques developed for queueing networks (QNs) and Petri nets (PNs). The theoretical approach is accompanied by a toolset that provides automatic translations between a process-chain-specification front end and analysis engines for QN and PN models. In addition, the analysis technique also supports a hybrid approach, in which dedicated sub-models are mapped into a PN model, numerically analysed, and replaced with an aggregate. Finally, the resulting QN models may be analysed using analytical as well as simulative techniques. As an example application, we investigate the impact of an additional SC channel between a manufacturer and web-consumers on the overall performance of an SC. The example was inspired by the trend towards retail trade via the internet and illustrates the new challenges of electronic commerce for SC management.     

A description of the segment [1,T], where T is the meeting number of a set-lattice

To 80^th birthday of Paul Erds     

A systematization of the saddle point method. Application to the Airy and Hankel functions

The standard saddle point method of asymptotic expansions of integrals requires to show the existence of the steepest descent paths of the phase function and the computation of the coefficients of the expansion from a function implicitly defined by solving an inversion problem. This means that the method is not systematic because the steepest descent paths depend on the phase function on hand and there is not a general and explicit formula for the coefficients of the expansion (like in Watson's Lemma for example). We propose a more systematic variant of the method in which the computation of the steepest descent paths is trivial and almost universal: it only depends on the location and the order of the saddle points of the phase function. Moreover, this variant of the method generates an asymptotic expansion given in terms of a generalized (and universal) asymptotic sequence that avoids the computation of the standard coefficients, giving an explicit and systematic formula for the expansion that may be easily implemented on a symbolic manipulation program. As an illustrative example, the well-known asymptotic expansion of the Airy function is rederived almost trivially using this method. New asymptotic expansions of the Hankel function Hn(z) for large n and z are given as non-trivial examples.     

QN-like variable storage conjugate gradients

Both conjugate gradient and quasi-Newton methods are quite successful at minimizing smooth nonlinear functions of several variables, and each has its advantages. In particular, conjugate gradient methods require much less storage to implement than a quasi-Newton code and therefore find application when storage limitations occur. They are, however, slower, so there have recently been attempts to combine CG and QN algorithms so as to obtain an algorithm with good convergence properties and low storage requirements. One such method is the code CONMIN due to Shanno and Phua; it has proven quite successful but it has one limitation. It has no middle ground, in that it either operates as a quasi-Newton code using O(n ²) storage locations, or as a conjugate gradient code using 7n locations, but it cannot take advantage of the not unusual situation where more than 7n locations are available, but a quasi-Newton code requires an excessive amount of storage. In this paper we present a way of looking at conjugate gradient algorithms which was in fact given by Shanno and Phua but which we carry further, emphasize and clarify. This applies in particular to Beale's 3-term recurrence relation. Using this point of view, we develop a new combined CG-QN algorithm which can use whatever storage is available; CONMIN occurs as a special case. We present numerical results to demonstrate that the new algorithm is never worse than CONMIN and that it is almost always better if even a small amount of extra storage is provided.     

A new method for probabilistic assessments in power systems,combining monte carlo and stochastic‐algebraic methods

This article discusses a new methodology, which combines two efficient methods known as Monte Carlo (MC) and Stochastic‐algebraic (SA) methods for stochastic analyses and probabilistic assessments in electric power systems. The main idea is to use the advantages of each former method to cover the blind spots of the other. This new method is more efficient and more accurate than SA method and also faster than MC method while is less dependent of the sampling process. In this article, the proposed method and two other ones are used to obtain the probability density function of different variables in a power system. Different examples are studied to show the effectiveness of the hybrid method. The results of the proposed method are compared to the ones obtained using the MC and SA methods. © 2014 Wiley Periodicals, Inc. Complexity 21: 100–110, 2015     

A Trust Region Method for Optimization Problem with Singular Solutions

In this paper, we propose a trust region method for minimizing a function whose Hessian matrix at the solutions may be singular. The global convergence of the method is obtained under mild conditions. Moreover, we show that if the objective function is LC ² function, the method possesses local superlinear convergence under the local error bound condition without the requirement of isolated nonsingular solution. This is the first regularized Newton method with trust region technique which possesses local superlinear (quadratic) convergence without the assumption that the Hessian of the objective function at the solution is nonsingular. Preliminary numerical experiments show the efficiency of the method. This work is partly supported by the National Natural Science Foundation of China (Grant Nos. 70302003, 10571106, 60503004, 70671100) and Science Foundation of Beijing Jiaotong University (2007RC014).     

A globally convergent BFGS method with nonmonotone line search for non-convex minimization

In this paper, we propose a modified BFGS (Broyden–Fletcher–Goldfarb–Shanno) method with nonmonotone line search for unconstrained optimization. Under some mild conditions, we show that the method is globally convergent without a convexity assumption on the objective function. We also report some preliminary numerical results to show the efficiency of the proposed method.     

A characteristics-mixed finite element method for Burgers’ equation

In this paper, we propose a new mixed finite element method, called the characteristics-mixed method, for approximating the solution to Burgers’ equation. This method is based upon a space-time variational form of Burgers’ equation. The hyperbolic part of the equation is approximated along the characteristics in time and the diffusion part is approximated by a mixed finite element method of lowest order. The scheme is locally conservative since fluid is transported along the approximate characteristics on the discrete level and the test function can be piecewise constant. Our analysis show the new method approximate the scalar unknown and the vector flux optimally and simultaneously. We also show this scheme has much smaller time-truncation errors than those of standard methods. Numerical example is presented to show that the new scheme is easily implemented, shocks and boundary layers are handled with almost no oscillations. One of the contributions of the paper is to show how the optimal error estimates inL ²(Ω) are obtained which are much more difficult than in the standard finite element methods. These results seem to be new in the literature of finite element methods.     

A meshless method for Burgers’ equation using MQ-RBF and high-order temporal approximation

In this paper, a new numerical method is proposed to solve one-dimensional Burgers’ equation using multiquadric (MQ) radial basis function (RBF) for spatial approximation and a second-order compact finite difference scheme for temporal approximation. The numerical results obtained by this way for different Reynolds number have been compared with the existing numerical schemes to show the accuracy and efficiency of the approach. To show the superiority of this meshless method, numerical experiments with non-uniform MQ interpolation node distribution are also performed.     

基于非线性尺度不变性的CCG法

１．引言 ＣＧ法对于变量个数很多的问题,是很有用的．１９７０年后它有了许多改进和发展,ＣＣＧ法以正定圆锥函数为基础［１］,它的一般方法是：设圆锥函数为 ２］其中： Ｖ＝ Ｖ（ｘ）＝１＋ ａＴｘ ≠ ０;, ｒ ∈Ｒ１为常量; ａ,ｇ ∈ Ｒｎ为常向量;ｘ ∈ Ｒｎ为变向量;Ａ∈Ｒｎ×ｎ为对称正定矩阵．算法［１］：预先给出初始近似点ｘ０∈ Ｒｎ及初始搜索方向 ｐ０;满足：其中“Ｉ”是单位矩阵, Ｖ０＝ Ｖ（ｘ０）＝ １＋ ａｔｘ０及记号“”是函数的梯度．迭代格式为： ｘｋ＋１＝ ｘｋ ＋λｋｐｋ,ｋ＝ ０,１,２,…（３…     

Multicategory Classification by Support Vector Machines

We examine the problem of how to discriminate between objects of three or more classes. Specifically, we investigate how two-class discrimination methods can be extended to the multiclass case. We show how the linear programming (LP) approaches based on the work of Mangasarian and quadratic programming (QP) approaches based on Vapnik's Support Vector Machine (SVM) can be combined to yield two new approaches to the multiclass problem. In LP multiclass discrimination, a single linear program is used to construct a piecewise-linear classification function. In our proposed multiclass SVM method, a single quadratic program is used to construct a piecewise-nonlinear classification function. Each piece of this function can take the form of a polynomial, a radial basis function, or even a neural network. For the k > 2-class problems, the SVM method as originally proposed required the construction of a two-class SVM to separate each class from the remaining classes. Similarily, k two-class linear programs can be used for the multiclass problem. We performed an empirical study of the original LP method, the proposed k LP method, the proposed single QP method and the original k QP methods. We discuss the advantages and disadvantages of each approach.