Wolfe线搜索下一个新的全局收敛共轭梯度法

共轭梯度法是求解大规模无约束优化问题的有效方法之一,其研究十分活跃.本文给出了一个新的共轭梯度法公式,新公式在精确线搜索下与DY公式等价.基于新公式,采用Wolfe非精确线搜索确定步长,本文设计了一个新的共轭梯度算法,并证明了新算法的下降性和全局收敛性.数值试验结果表明所设计新算法是有效的.     

一个新的谱共轭梯度法

谱共轭梯度法含有两个方向调控参数,是求解无约束优化问题的一类有效方法.本文给出一对参数公式以构建新的谱共轭梯度法,该方法在精确线搜索下与标准FR方法等价,在Wolfe线搜索下具有类似标准DY方法的内在性质.我们证明了采用Wolfe线搜索的新算法在每一次迭代中均产生下降方向,并且具有全局收敛性.数值实验结果表明,新算法数值稳定、有效,适合于求解大规模无约束优化问题.     

基于稀疏对角拟牛顿方向的非单调超记忆梯度算法

超记忆梯度算法由于其迭代简单和较小的存储需求,在求解大规模无约束优化问题中起着特殊的作用.本文基于稀疏对角拟牛顿技术,结合修正Gu和Mo非单调线搜索步长规则,建立了求解大规模无约束最优化问题的非单调超记忆梯度新算法,给出了算法的全局收敛性分析.新算法具有算法稳定、计算简单的特点可用于求解病态和大规模问题.数值例子表明算法有效稳定.     

一类新的基于信赖域技术的非单调共轭梯度算法

为有效求解大规模无约束优化问题,本文基于信赖域技术和修正拟牛顿方程,同时结合Zhang H.C.策略和Gu N.Z.策略,设计了一种新的非单调共轭梯度算法,应用信赖域技术保证了算法的稳健性和收敛性,并给出了算法的全局收敛性分析.在适当条件下,证明了该算法具有线性收敛性.数值实验表明新算法能够有效求解病态和大规模问题.与单独结合其中一种非单调策略的算法相比,新算法需要较少的迭代次数和运行时间,利用其得到的函数值与最优值更接近.     

Armijo型线搜索下一种共轭梯度法的收敛性

对无约束非线性规划问题,本文分别在两种不同的Armijo型线搜索下证明了Liu-Storey共轭梯度法的所有搜索方向都是充分下降的,并进一步证明了该算法是全局强收敛的。对另一种放松了函数值下降条件可以获得更大步长的Armijo型线搜索,本文还证明了该算法是全局强收敛的。     

线搜索下带误差项的Dai-Yuan共轭梯度算法

本文在共轭梯度不能精确计算的情况下,采用Wolfe或Armijo步长规则研究了带误差项的Dai-Yuan(abbr．DY)共轭梯度法,我们的方法的一个很重要的特征就是步长不一定趋于零。这种特征使得我们的分析对许多实际问题很有用。我们在很一般的假设条件下证明了算法的全局收敛性。最后给出了数值算例。     

一个充分下降的修正 PRP 型谱共轭梯度法

谱共轭梯度法是共轭梯度法的一种重要延拓,可以通过共轭参数和谱参数二维度调整,使得所设计算法的搜索方向满足某一预设条件,比如充分下降条件或共轭条件等。谱参数和共轭参数的设计是谱共轭梯度法的两大核心工作,决定方法的收敛性和数值效果。基于 PRP 方法,构造了一个修正的 PRP 型共轭参数,该共轭参数不仅保持了 PRP 公式的结构和性能,而且具有 FR 方法的收敛性质。利用充分下降条件取定一个谱参数,与修正的 PRP 型共轭参数结合,建立一个新的谱共轭梯度算法。该算法不依赖于任何线搜索就可以满足充分下降条件。常规假设条件下,采用强 Wolfe 线搜索准则产生步长,证明了新算法的全局敛性。通过 100 个算例对该算法进行数值测试并与其他五个算法进行比较,同时采用性能图对数值结果进行直观展示,结果表明该算法是有效的。     

结合广义Armijo步长搜索的一类记忆梯度算法及其收敛特征

本文对求解无约束规划的超记忆梯度算法中线搜索方向中的参数,给了一个假设条件,从而确定了它的一个新的取值范围,保证了搜索方向是目标函数的充分下降方向,由此提出了一类新的记忆梯度算法.并在去掉迭代点列有界和广义Armijo步长搜索下,讨论了算法的全局收敛性,且给出了结合形如共轭梯度法FR,PR,HS的记忆梯度法的修正形式,数值实验表明,新算法比Armijo线搜索下的FR,PR,HS共轭梯度法和超记忆梯度法更稳定、更有效.     

结合广义Armijo步长搜索的一类新的共轭度算法及其收敛特征

对求解无约束规划的共轭梯度算法中共轭梯度方向中的参数给了一个假设条件。从而确定它的一个取值范围，使其在此范围内取值均能保证共轭梯度方向是目标函数的充分下降方向。提出了一类新的共轭梯度算法，在去掉迭代点列有界和广义Armijo步长搜索下讨论了算法的全局收敛性。同时给出了具有好的收敛性质和较快收敛速度的FR，PR，HS共轭梯度法的修正形式。数值例子表明新算法比Armijo搜索下的FR，PR，HS共轭梯算法更稳定更有效。算法需要较小的存储。特别适于求解大规模无约束最优化问题。     

求解大规模优化的混合共轭梯度法

为了有效求解大规模无约束优化问题,在PRP方法和FR方法的基础上,给出了满足共轭条件的新的混合共轭梯度法.在强Wolfe线性搜索下,证明了此算法的全局收敛性.在特定条件下,新公式与HS公式一致,因此可看作是对HS方法的修正.对7个经典无约束优化问题的数值实验结果表明,所提出的新方法数值稳定.相比已有方法,随着问题规模的增大,所提方法在迭代次数,优化精度及梯度调用次数方面表现出明显优势.     

在Armijo型线搜索下共轭下降法的收敛性

我们提出了两种Armijo型线搜索,进而证明了这两种Armijo型线搜索可保证共轭下降法的下降搜索方向的充分下降性。并在这两种Armijo型线搜索下得到共轭下降法的收敛性结果。     

散射问题中复线性系统的扰动预条件技术

利用稀疏策略可以控制不完全分解因子的稀疏度,对角扰动技术则通过对原系数矩阵的对角元的轻微扰动,提高不完全分解预条件方法的效率.本文结合稀疏策略和对角扰动技术的修正的不完全LLT分解预条件技术,用来加速共轭垂直共轭梯度法(COCG)求解离散散射问题得到的大型、稀疏的复对称线性系统的求解速率,数值试验验证了基于扰动的不完全分解预条件方法,对迭代求解散射问题有着很好的提速效果.     

A robust incomplete factorization based on value and space constraints

This paper describes an incomplete factorization method for computing a preconditioning matrix for the conjugate gradient method. The incomplete factorization satisfies the stability requirement that the incomplete factor remains positive definite throughout the factorization. When selecting a preconditioner for the conjugate gradient method, the number of non-zero entries to be retained in the incomplete factor should be limited so that the amount of computations involving the preconditioning matrix is minimized. This paper introduces a method to generate an effective preconditioning matrix within a predefined space. Numerical results are presented to demonstrate the effectiveness of the incomplete factor as a preconditioner for the conjugate gradient method for solving large-scale structural engineering problems.     

求解大型稀疏线性方程组的不完全SAOR预条件共轭梯度法

预条件共轭梯度法是求解大型稀疏线性方程组的有效方法之一,SSOR预条件方法是基于矩阵分裂的较有效的预条件共轭梯度法。通过矩阵分裂,本文讨论不完全SAOR预条件方法,研究此方法的预条件因子及系数矩阵的预条件数,并证明了此方法的预条件数小于SSOR预条件方法的预条件数。最后通过求解离散化波松(Poisson)方程组表明了该方法的有效性。     

An optimal control problem in estimating the optimal control force for the force vibration system

An optimal control problem for the force vibration system based on the iterative regularization method, i.e. the conjugate gradient method (CGM), is examined to estimate the optimal control force in a damped system having time‐dependent system parameters such that the desire (or design) system displacements can be satisfied. It is assumed that no prior information is available on the functional form of the unknown control function in the present study, thus, it is classified as the function estimation. Numerical simulations are performed to test the validity of the present algorithm by using different types of the desire system displacements. Results show that an excellent estimation on the optimal control force can be obtained with arbitrary initial guesses within a couple of second's CPU time at Pentium III‐500 MHz PC. Copyright © 2001 John Wiley & Sons, Ltd.     

High performance computational kernels for selected segments of a p finite element code

A high performance implementation is presented for three kernel routines commonly found in element-byelement preconditioned conjugate gradient finite element codes. These routines include forming the element stiffness matrices and loading vectors, or in the case of a non-linear problem, element residual vectors; and routines for applying element matrix–vector products. The present study considers tensor product elements of arbitrary mapping in 2-D, although the generalization to triangular elements and serendipity elements is straightforward. The implementation presented is most appropriate for high p type finite element methods, where the element matrices are relatively large and dense. This results in a set of high performance kernels for superscalar architectures, which otherwise may be memory bandwidth limited. Performance studies are presented for a representative superscalar microprocessor, the Intel i860. As these types of microprocessors are at the heart of modern workstations as well as several parallel supercomputing systems, this work is relevant across a variety of platforms. The resulting kernels yield both high performance on a variety of sequential architectures as well as a high degree of code portability through the basic linear algebra subprograms mechanism.     

An iterative regularization method in estimating the unknown energy source by laser pulses with a dual-phase-lag model

In the present study an inverse problem for hyperbolic heat conduction with a dual-phase-lag model is solved by the conjugate gradient method (CGM) in estimating the unknown heat generation, due to the ultra-short duration laser heating, based on the interior temperature measurements. Results obtained in this inverse problem will be justified based on the numerical experiments where two different heat source distributions are to be estimated. Results show that the inverse solutions can always be obtained when choosing the initial guesses of the heat sources equal to zero. Finally, it is concluded that accurate heat sources can be estimated in this study. Copyright © 2008 John Wiley & Sons, Ltd.     

A gradient‐only line search method for the conjugate gradient method applied to constrained optimization problems with severe noise in the objective function

A new implementation of the conjugate gradient method is presented that economically overcomes the problem of severe numerical noise superimposed on an otherwise smooth underlying objective function of a constrained optimization problem. This is done by the use of a novel gradient‐only line search technique, which requires only two gradient vector evaluations per search direction and no explicit function evaluations. The use of this line search technique is not restricted to the conjugate gradient method but may be applied to any line search descent method. This method, in which the gradients may be computed by central finite differences with relatively large perturbations, allows for the effective smoothing out of any numerical noise present in the objective function. This new implementation of the conjugate gradient method, referred to as the ETOPC algorithm, is tested using a large number of well‐known test problems. For initial tests with no noise introduced in the objective functions, and with high accuracy requirements set, it is found that the proposed new conjugate gradient implementation is as robust and reliable as traditional first‐order penalty function methods. With the introduction of severe relative random noise in the objective function, the results are surprisingly good, with accuracies obtained that are more than sufficient compared to that required for engineering design optimization problems with similar noise. Copyright © 2004 John Wiley & Sons, Ltd.