一种新线搜索下的FR共轭梯度法

基于传统的Wolfe线搜索,提出了一种新的非精确线搜索．在无需限制参数σ≤1/2的情况下(即盯的取值范围扩展至0＜σ＜1),证明了FR算法的全局收敛性．数值实验表明了这种线搜索下的FR算法的有效性．     

Wolfe线搜索下新的共轭梯度法的全局收敛性

共轭梯度法是求解无约束优化问题的一种重要的方法.本文提出一族新的共轭梯度法,证明了其在推广的Wolfe非精确线搜索条件下具有全局收敛性.最后对算法进行了数值实验,实验结果验证了该算法的有效性.     

Wolfe线搜索下一个全局收敛的混合共轭梯度法

对无约束优化问题, 本文给出了一个新的混合共轭梯度法公式. 在标准Wolfe非精确线搜索下,证明了由新公式所产生的算法具有下降性和全局收敛性, 并对算法进行了数值试验, 其结果表明该算法是有效的.     

一族新的共轭梯度法的全局收敛性

共轭梯度法是求解无约束优化问题的一种重要的方法,尤其适用于大规模优化问题的求解。本文提出一族新的共轭梯度法,证明了其在推广的Wolfe非精确线搜索条件下具有全局收敛性。最后对算法进行了数值试验,试验结果验证了该算法的有效性。     

推广的Wolfe搜索下一族共轭梯度法的全局收敛性

共轭梯度法是求解无约束优化问题的一种重要的方法,尤其适用于大规模优化问题的求解.本文提出一族包含FR方法和CD方法的新的共轭梯度法,证明了其在推广的Wolfe非精确线搜索条件下具有全局收敛性.最后对算法进行了数值试验,试验结果验证了该算法的有效性。     

求解无约束优化问题的新谱共轭梯度法及其收敛性

强Wolfe条件不能保证标准CD共轭梯度法全局收敛．本文通过建立新的共轭参数,提出无约束优化问题的一个新谱共轭梯度法,该方法在精确线搜索下与标准CD共轭梯度法等价,在标准wolfe线搜索下具有下降性和全局收敛性．初步的数值实验结果表明新方法是有效的,适合于求解非线性无约束优化问题．     

一个自调节Polak-Ribiere-Polyak型共轭梯度法

共轭梯度法是求解大规模无约束优化问题最有效的方法之一.基于Polak-RibièrePolyak(PRP)共轭梯度法具有较弱的收敛性和较好的数值表现,而Fletcher-Reeves(FR)共轭梯度法则反之,本文研究PRP共轭梯度法的一个自调节改进.在PRP公式引入调节因子,并据此提出了一个自调节PRP共轭梯度法.改进的方法具有PRP方法所特有的性质(*)及FR方法良好的收敛性·在强Wolfe非精确线搜索条件和常规假设下,证明了新方法不仅满足充分下降条件,而且全局收敛.最后,对新算法进行数值测试并与其他同类方法进行比较,结果表明所提方法是有效的.     

具有充分下降性的两个共轭梯度法

对无约束优化问题,本文给出了两个改进的共轭梯度法公式.在不依赖于任何线搜索条件下,由新公式所产生的算法方向均是充分下降的,且在标准Wolfe非精确线搜索条件下,算法都具有全局收敛性.最后,对新算法进行大量的比对试验,数值结果表明所提方法是有效的.     

具有精确线性搜索的改进共轭梯度法

其中g_k=f(x_k),β_k为参数.β_k的不同选法形成了各种共轭梯度法,其中Fletcher-Reeves法(简记为FR法)是理论较完整的一个方法,对水平集有界的二阶连续可微函数,Powell和Baali分别在精确和不精确线搜索下证明了其全局收敛性.Polak-Ribiere法     

利用Armijo型线性搜索HZ共轭梯度法的全局收敛性

由William W.Hager和张洪超提出的一种新的共轭梯度法(简称HZ方法),已被证明是一种有效的方法.本文证明了HZ共轭梯度法在Armijo型线性搜索下的全局收敛性.数值实验显示,在Armijo型线性搜索下的HZ共轭梯度法比在Wolfe线性搜索下更有效.     

共轭下降法的全局收敛性

共轭下降法最早由Ｆｌｅｔｃｈｅｒ提出，本文证明了一类非精确线搜索条件能保证共轭下的降法的收敛性，并且构造了反例表明，如果线搜索条件放松，则共轭下降法可能不收敛，此外，我们还得到了与Ｆｌｅｃｈｅｒ－Ｒｅｅｖｅｓ方法有关的一类方法的结论。     

在Armijo型线搜索下共轭梯度法簇的全局收敛性

本文我们讨论了一簇共轭梯度法，它可被看作是FR法和DY法的凸组合．我们提出了两种Armijo型线搜索，并在这两种线搜索下，讨论了共轭梯度法簇的全局收敛性．     

Conjugate Gradient Methods with Armijo-type Line Searches

Abstract Two Armijo-type line searches are proposed in this paper for nonlinear conjugate gradient methods.Under these line searches, global convergence results are established for several famous conjugate gradientmethods, including the Fletcher-Reeves method, the Polak-Ribiere-Polyak method, and the conjugate descentmethod.     

Further insight into the convergence of the Fletcher-Reeves method

The convergence properties of the Fletcher-Reeves method for unconstrained optimization are further studied with the technique of generalized line search. Two conditions are given which guarantee the global convergence of the Fletcher-Reeves method using generalized Wolfe line searches or generalized Arjimo line searches, whereas an example is constructed showing that the conditions cannot be relaxed in certain senses. Project supported by the National Natural Science Foundation of China (Grant No. 19801033).     

CG Global Convergence Properties with Goldstein Linesearch*

This paper explores the convergence of nonlinear conjugate gradient methods with Goldstein line search without regular restarts. Under this line search, global convergence for a subsequence is given for the famous conjugate gradient methods, Fletcher-Reeves method. The same result can be obtained for Polak-Ribiére-Polyak method and others. *This work was partially supported by National Hitech Program (863,2002AA104540) and National Natural Science Foundation of China (No.60373060).     

Convergence properties of the Fletcher-Reeves method

This paper investigates the global convergence properties ofthe Fletcher-Reeves (FR) method for unconstrained optimization.In a simple way, we prove that a kind of inexact line searchcondition can ensure the convergence of the FR method. Severalexamples are constructed to show that, if the search conditionsare relaxed, the FR method may produce an ascent search direction,which implies that our result cannot be improved.     

New spectral PRP conjugate gradient method for unconstrained optimization

In this paper, a new spectral PRP conjugate gradient algorithm has been developed for solving unconstrained optimization problems, where the search direction was a kind of combination of the gradient and the obtained direction, and the steplength was obtained by the Wolfe-type inexact line search. It was proved that the search direction at each iteration is a descent direction of objective function. Under mild conditions, we have established the global convergence theorem of the proposed method. Numerical results showed that the algorithm is promising, particularly, compared with the existing several main methods.     

A NOTE ON THE NONLINEAR CONJUGATE GRADIENT METHOD

The conjugate gradient method for unconstrained optimization problems varies with a scalar. In this note, a general condition concerning the scalar is given, which ensures the global convergence of the method in the case of strong Wolfe line searches. It is also discussed how to use the result to obtain the convergence of the famous Fletcher-Reeves, and Polak-Ribiere-Polyak conjugate gradient methods. That the condition cannot be relaxed in some sense is mentioned.     

由FR共轭梯度法控制的两类优化算法的全局收敛性

１引言 考虑无约束优化问题其中ｆ：Ｒｎ→Ｒ是一阶可微函数．求解（１）的非线性共轭梯度法具有如下形式：其中ｇｋ＝　ｆ（ｘｋ）,ａｋ是通过某种线搜索获得的步长,纯量βｋ的选取使得方法（２）—（３）在ｆ（ｘ）是严格凸二次函数且采用精确线搜索时化为线性共轭梯度法［１］．比较常见的βｋ的取法有Ｆｌｅｔｃｈｅｒ－Ｒｅｅｖｅｓ（ＦＲ）公式［２］和Ｐｏｌａｋ－Ｒｉｂｉｅｒｅ－Ｐｏｌｙａｋ（ＰＲＰ）公式［３－４］等．它们分别为其中　　　取欧几里得范数．对于一般非线性函数,ＦＲ方法具有较好的理论收敛性［５－６］,而…     

Applying the Powell's Symmetrical Technique to Conjugate Gradient Methods with the Generalized Conjugacy Condition

This article proposes new conjugate gradient method for unconstrained optimization by applying the Powell symmetrical technique in a defined sense. Using the Wolfe line search conditions, the global convergence property of the method is also obtained based on the spectral analysis of the conjugate gradient iteration matrix and the Zoutendijk condition for steepest descent methods. Preliminary numerical results for a set of 86 unconstrained optimization test problems verify the performance of the algorithm and show that the Generalized Descent Symmetrical Hestenes-Stiefel algorithm is competitive with the Fletcher-Reeves (FR) and Polak-Ribiére-Polyak (PRP⁺) algorithms.