Damped Techniques for the Limited Memory BFGS Method for Large-Scale Optimization

This paper is aimed to extend a certain damped technique, suitable for the Broyden–Fletcher–Goldfarb–Shanno (BFGS) method, to the limited memory BFGS method in the case of the large-scale unconstrained optimization. It is shown that the proposed technique maintains the global convergence property on uniformly convex functions for the limited memory BFGS method. Some numerical results are described to illustrate the important role of the damped technique. Since this technique enforces safely the positive definiteness property of the BFGS update for any value of the steplength, we also consider only the first Wolfe–Powell condition on the steplength. Then, as for the backtracking framework, only one gradient evaluation is performed on each iteration. It is reported that the proposed damped methods work much better than the limited memory BFGS method in several cases.     

大规模非线性互补问题的共轭梯度法

提出求解大规模非线性互补问题NCP(F)的PRP型共轭梯度法,算法自然满足充分下降条件.当F是可微P_0+R_0函数且F'(χ)在水平集上全局Lipschitz连续条件下,证明了算法的全局收敛性.数值结果表明算法的有效性.     

A Modified BFGS Algorithm for Unconstrained Optimization

In this paper we present a modified BFGS algorithm for unconstrainedoptimization. The BFGS algorithm updates an approximate Hessianwhich satisfies the most recent quasi-Newton equation. The quasi-Newtoncondition can be interpreted as the interpolation conditionthat the gradient value of the local quadratic model matchesthat of the objective function at the previous iterate. Ourmodified algorithm requires that the function value is matched,instead of the gradient value, at the previous iterate. Themodified algorithm preserves the global and local superlinearconvergence properties of the BFGS algorithm. Numerical resultsare presented, which suggest that a slight improvement has beenachieved.     

An Augmented Lagrangian Algorithm for Large Scale Multicommodity Routing

The linear multicommodity network flow (MCNF) problem has many applications in the areas of transportation and telecommunications. It has therefore received much attention, and many algorithms that exploit the problem structure have been suggested and implemented. The practical difficulty of solving MCNF models increases fast with respect to the problem size, and especially with respect to the number of commodities. Applications in telecommunications typically lead to instances with huge numbers of commodities, and tackling such instances computationally is challenging.In this paper, we describe and evaluate a fast and convergent lower-bounding procedure which is based on an augmented Lagrangian reformulation of MCNF, that is, a combined Lagrangian relaxation and penalty approach. The algorithm is specially designed for solving very large scale MCNF instances. Compared to a standard Lagrangian relaxation approach, it has more favorable convergence characteristics. To solve the nonlinear augmented Lagrangian subproblem, we apply a disaggregate simplicial decomposition scheme, which fully exploits the structure of the subproblem and has good reoptimization capabilities. Finally, the augmented Lagrangian algorithm can also be used to provide heuristic upper bounds.The efficiency of the augmented Lagrangian method is demonstrated through computational experiments on large scale instances. In particular, it provides near-optimal solutions to instances with over 3,600 nodes, 14,000 arcs and 80,000 commodities within reasonable computing time.     

Updating the Self-Scaling Symmetric Rank One Algorithm with Limited Memory for Large-Scale Unconstrained Optimization

A new limited memory quasi-Newton algorithm is developed, in which the self-scaling symmetric rank one update with Davidon's optimal condition is applied. Preliminary numerical tests show that the new algorithm is very efficient for large-scale problems as well as general nonlinear optimization.     

解决大规模信赖域子问题的一种新算法

信赖域方法是解决无约束优化问题的一类有效的方法,而求解信赖域子问题又是信赖域方法的一个重要的组成部分。在本文中,我们首先介绍Hager[4]的序列子空间方法,并分析了对于不同的子空间序列,该算法所具有的性质。随后我们在以上分析的启发下,给出SSM算法的一种改进算法,改进后的算法不仅是全局收敛的,而且进一步减少了矩阵运算量。最后我们给出一些初步的数值试验报告。     

非凸函数极小问题的BFGS算法

本对于非凸函数的无约束优化问题，给出一类修正的BFGS算法。算法的思想是对非凸函数的近似Hesse矩阵进行修正，得到下降方向，并且保证拟牛顿条件成立，当步长采用线性搜索一般模型时，证明了该算法的局部收敛性。     

BFGS算法对非凸函数优化问题的收敛性

ＢＦＧＳ算法是无约束最优化中最著名的数值算法之一,对非凸函数ＢＦＧＳ算法是否具有整体收敛性,这是一个ｏｐｅｎ问题,本文考虑Ｗｏｌｆｏ线搜索下目标函数非凸的ＢＦＧＳ算法,我们给出一个使该算法收敛的充分条件。     

大型水电工程物资调运建模及算法

根据小湾水电工程物资供应管理的实际情况,提出了物资调运的三级节点概念模型,论述了物资采购与调运的数学模型,并给出了具体算法。模型已应用于该工程物资调运决策支持系统中,运行实践表明,该模型能够辅助调度人员制定合理的物资采购与调运方案。     

Stopping Criterion in Order Reduction of Large Scale Systems Using Krylov Subspace Methods

We show that order reduction by a two‐sided Krylov subspace method is equivalent to an approximate TBR in the sense of Galerkin type conditions. The Hankel Singular Values of the original system are thereby approximated by the Hankel singular values of the Krylov reduced model which are used as a stopping criterion to find a suitable order of the reduced model. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

求解一般约束优化问题的修正BFGS信赖域算法

借鉴无约束优化问题的BFGS信赖域算法,建立了非线性一般约束优化问题的BFGS信赖域算法,并证明了算法的全局收敛性．数值实验表明,算法是有效的．     

Piecewise Partially Separable Functions and a Derivative-free Algorithm for Large Scale Nonsmooth Optimization

This paper introduces the notion of piecewise partially separable functions and studies their properties. We also consider some of many applications of these functions. Finally, we consider the problem of minimizing of piecewise partially separable functions and develop an algorithm for its solution. This algorithm exploits the structure of such functions. We present the results of preliminary numerical experiments.     

非线性系统运动稳定性的一种判据

对于自治的非线性系统来说,只要其线性部分系数矩阵的特征值不属于临界情形,其无扰运动在其足够小的邻域内的稳定性完全可以由其线性部分的特征值确定.关于线性系统的稳定性,已有不少简单易行的判别方法,而关于非线性系统的稳定性,很多数学家和力学家作了大量的研究工作;但大都是针对特殊类型的非线性系统解决了一些问题,直到现在为止,还没有普遍适用于任何的非线性系统的简单易行的判别方法.本文所给的是判别非线性系统稳定性的充要条件,常用的克拉索夫斯基方法只是这一方法的一个特例[1],[2].     

Criterion for the Stability of Difference Schemes for Nonlinear Differential Equations

Differential Equations - For abstract nonlinear difference schemes with operators acting in finite-dimensional Banach spaces, a stability criterion is stated and proved; namely, for a consistent...     

An Oscillation Criterion for Nonlinear Third-Order Differential Equations

Sufficient conditions for oscillation of a certain class of nonlinear third-order differential equations are found.     

解非线性对称方程组问题的具有下降方向的近似高斯-牛顿基础的BFGS方法

本本文给出了一个解非线性对称方程组问题的具有下降方向的近似高斯一牛顿基础BFGS方法。无论使用何种线性搜索此方法产生的方向总是下降的。在适当的条件下我们将证明此方法的全局收敛性和超线性收敛性。并给出数值检验结果。     

大规模系统的全局优化

本文讨论了可分非凸大规模系统的全局优化控制问题 .提出了一种 3级递阶优化算法 .该算法首先把原问题转化为可分的多目标优化问题 ,然后凸化非劣前沿 ,再从非劣解集中挑出原问题的全局最优解 .建立了算法的理论基础 ,证明了算法的收敛性 .仿真结果表明算法是有效的 .     

A Criterion for a Normed Algebra in the Scale of Sobolev Spaces

Mathematical Notes -     

混沌优化算法在不可分稳态大系统优化中的应用

讨论了整体目标函数关于各子系统不具有可加形式的大规模稳态系统的优化问题,将混沌优化算法应用于其最优值的求解,利用混沌运动的遍历性来得到优化问题的全局最优值.仿真结果表明,该算法简单易行,求解精度和可靠性较高,是解决不可分稳态大系统优化问题的一种有效方法.