基于亏基的摄动原始单纯形Ⅰ阶段算法

通过摄动技术来使问题强制获得对偶可行性,执行亏基对偶单纯形算法得到一个原始可行基,并采用修正的主元规则,以充分发挥这两种算法的优势,从而为亏基原始单纯形算法提供一个新的I阶段算法,以使其进一步克服退化所带来的困扰.初步的数值试验表明,亏基和摄动两种算法优势的结合,能有效地克服退化的影响,能有效地减少总迭代次数和运行时间,其效率远远优于传统两阶段单纯形算法.     

基于LU分解的亏基摄动对偶Ⅰ阶段算法

将摄动算法和亏基单纯形算法相结合,以充分发挥这两种算法的优势,从而为亏基对偶单纯形算法提供一个新的Ⅰ阶段算法,以使其进一步克服退化所带来的困扰.数值试验结果表明,新算法能够降低退化带来的不良影响,减少总迭代次数和运算时间,其效率不仅远远优于传统的单纯形算法,且优于原有的亏基单纯形算法,是一个非常吸引人且充满希望的新尝试.     

无比值检验的亏基原始Ⅰ阶段算法

首次将亏基和无比值检验列主元规则相结合,执行亏基对偶单纯形算法得到一个原始可行基,以充分发挥这两种算法的优势,从而为亏基原始单纯形算法提供一个新的I阶段算法,以使其进一步克服退化所带来的困扰.数值试验表明,亏基和无比值主元规则的结合,能有效地减少总迭代次数和运行时间,其效率远远优于传统两阶段单纯形算法.     

基于最钝角主元标规则的亏基算法

在最钝角原理基础上建立了新的主元标规则,它按最钝角原理赋予一组非基本变量较高优先权,先在其中选择进基变量,直到其相应的检验数均满足符号条件;如果此时剩下的检验数均已满足条件,则已达到最优.在亏基架构中引入新的主元规则,能有效地减少每次迭代可选的非基变量的个数.数值试验表明,新算法的效率优于亏基原始单纯形算法,表明了最钝角原理的可行性和有效性.     

基于最钝角主元标规则的亏基算法北大核心

在最钝角原理基础上建立了新的主元标规则,它按最钝角原理赋予一组非基本变量较高优先权,先在其中选择进基变量,直到其相应的检验数均满足符号条件;如果此时剩下的检验数均已满足条件,则已达到最优.在亏基架构中引入新的主元规则,能有效地减少每次迭代可选的非基变量的个数.数值试验表明,新算法的效率优于亏基原始单纯形算法,表明了最钝角原理的可行性和有效性.     

基于最陡边规则的亏基对偶Ⅰ阶段算法

将摄动算法和亏基原始单纯形算法相结合,采用最陡边的列主元规则,以充分发挥这两种算法的优势,从而为亏基对偶单纯形算法提供一个新的I阶段算法,以使其进一步克服了退化所带来的困扰.初步的数值试验表明,所提出的算法能有效地减少总迭代次数,其效率不仅远远优于传统的原始两阶段单纯形算法,且优于原有的亏基原始单纯形算法,是一个非常吸引人而充满希望的新尝试.     

基于最钝角规则的亏基对偶单纯形Ⅰ阶段算法

对偶单纯形算法或原始对偶单纯形算法都需要一个初始对偶可行基．就此目的而言，基于最钝角行主元规则的对偶Ⅰ阶段算法非常有效[15].本文将其思想应用于亏基情形，建立一个不含比值检验的新的亏基对偶Ⅰ价段算法．初步的数值实验表明，该算法可在总体上减少运行时间和迭代次数，极具竞争性．     

传统选主元规则的变形及其在亏基方法中的应用

从几何直观入手，对传统单纯形两阶段方法加以分析，得到了变形传统选主元规则的思想和动态选主元策略的思想，并将两种思想在亏基架构下加以实现。由此给出了三种具有动态选主元策略的变形的选主元规则及其相应的亏基算法。数值试验结果表明，两种思相具有可行性。     

基于亏基的最陡边主元标算法

在最陡边规则的基础上建立了新的主元标规则,并将其应用到亏基情形,在亏基的框架下建立了一个新的求对偶可行基的算法,数值结果表明,新算法能够减少迭代次数,算法效率较高,并且对于大规模问题的求解具有潜在优势,进一步表明了最陡边主元规则的可行性和有效性.     

有界上三角算子矩阵的亏谱

讨论了希尔伯特空间上有界上三角算子矩阵的亏谱扰动性质,当对角元算子给定时,得到上三角算子矩阵的亏谱恰等于对角元算子的亏谱之并集的充要条件,特别地,给出有界上三角Hamilton型算子矩阵相应问题成立的条件,并辅以实例佐证.     

Studies of lexicography in the generalized network simplex method

This paper introduces an analytical approach for studying lexicography in generalized network problems. The equations obtained can help us to understand and to extend the existing theory. First, it is verified that all nonzero elements have the same sign in each row vector of a basis inverse for a generalized network (GN) problem with positive multipliers. However, this property does not necessarily hold when there exist negative multipliers. Second, we developed a strategy to select the dropping arc in the GN simplex algorithm when addressing GN problems with positive andnegative multipliers. This strategy is also based on lexicography and requires performing some comparisons. However, the values to be compared are already known since they can be obtained as a by-product of the calculations necessary to compute the basis representation of the entering arc. Consequently, the computational effort per pivot step isO(n) in the worst case. This worst case effort is the same as that required by the strongly convergent rules for selecting the dropping arc in the method of strong convergence.     

The simplest examples where the simplex method cycles and conditions where expand fails to prevent cycling

This paper introduces a class of linear programming examples that cause the simplex method to cycle and that are the simplest possible examples showing this behaviour. The structure of examples from this class repeats after two iterations. Cycling is shown to occur for both the most negative reduced cost and steepest-edge column selection criteria. In addition it is shown that the expand anti-cycling procedure of Gill et al. is not guaranteed to prevent cycling.Work supported by EPSRC grant GR/J0842This paper is dedicated to Roger Fletcher, a friend and inspiration to us both. The discovery of Rogers book, Practical Methods of Optimization, whilst working in industry, set the first author on the road to Dundee and a career in optimization. Happy 65th birthday, Roger.     

Partial Pricing Rule Simplex Method with Deficient Basis

A new partial pricing column rule is proposed to the basis-deficiency-allowing simplex method developed by Pan.Computational results obtained with a set of small problems and a set of standard NETLIB problems show its promise of success.     

求解带界约束的非线性方程组的混合方法

基于非单调技术和L-M算法, 提出了一种新的求解带界约束的非线性方程组的混合方法. 在一定条件下, 该算法具有全局收敛性. 数值试验表明该算法是有效的.     

Meshless method with ridge basis functions

Meshless collocation methods for the numerical solutions of PDEs are increasingly adopted due to their advantages including efficiency and flexibility, and radial basis functions are popularly employed to represent the solutions of PDEs. Motivated by the advantages of ridge basis function representation of a given function, such as the connection to neural network, fast convergence as the number of terms is increased, better approximation effects and various applications in engineering problems, a meshless method is developed based on the collocation method and ridge basis function interpolation. This method is a truly meshless technique without mesh discretization: it neither needs the computation of integrals, nor requires a partition of the region and its boundary. Moreover, the method is applied to elliptic equations to examine its appropriateness, numerical results are compared to that obtained from other (meshless) methods, and influence factors of accuracy for numerical solutions are analyzed.     

边界约束非凸二次规划问题的分枝定界方法

本文是研究带有边界约束非凸二次规划问题,我们把球约束二次规划问题和线性约束凸二次规划问题作为子问题,分明引用了它们的一个求整体最优解的有效算法,我们提出几种定界的紧、松驰策略,给出了求解原问题整体最优解的分枝定界算法,并证明了该算法的收敛性,不同的定界组合就可以产生不同的分枝定界算法,最后我们简单讨论了一般有界凸域上非凸二次规划问题求整体最优解的分枝与定界思想。     

Dual simplex method for GUB problems

The dual simplex method for generalized upper bound (GUB) problems is presented. One of the major operations in the dual simplex method is to update the elements of therth row, wherer is the index for the leaving basic variable. Those updated elements are used for the ratio test to determine the entering basic variabble. A very simple formula for therth row update for the dual simplex method for a GUB problem is derived, which is similar to the formula for the standard linear program. This derivation is based on the change key operation, which is to exchange the key column and its counterpart in the nonkey section. The change key operation is possible because of a theorem that guarantees the existence of such a counterpart.     

A Derivative-Free Algorithm for Bound Constrained Optimization

In this work, we propose a new globally convergent derivative-free algorithm for the minimization of a continuously differentiable function in the case that some of (or all) the variables are bounded. This algorithm investigates the local behaviour of the objective function on the feasible set by sampling it along the coordinate directions. Whenever a suitable descent feasible coordinate direction is detected a new point is produced by performing a linesearch along this direction. The information progressively obtained during the iterates of the algorithm can be used to build an approximation model of the objective function. The minimum of such a model is accepted if it produces an improvement of the objective function value. We also derive a bound for the limit accuracy of the algorithm in the minimization of noisy functions. Finally, we report the results of a preliminary numerical experience.