一种WYL型谱共轭梯度法的全局收敛性

为解决大规模无约束优化问题,该文结合WYL共轭梯度法和谱共轭梯度法,给出了一种WYL型谱共轭梯度法.在不依赖于任何线搜索的条件下,该方法产生的搜索方向均满足充分下降性,且在强Wolfe线搜索下证明了该方法的全局收敛性.与WYL共轭梯度法的收敛性相比,WYL型谱共轭梯度法推广了线搜索中参数σ的取值范围.最后,相应的数值结果表明了该方法是有效的.     

求解无约束优化问题的两类修正的WYL共轭梯度方法

针对无约束优化问题,通过修正共轭梯度参数,构造新的搜索方向,提出两类修正的WYL共轭梯度法.在每次迭代过程中,两类算法产生的搜索方向均满足充分下降性.在适当条件下,证明了算法的全局收敛性.数值结果表明算法是可行的和有效的.     

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．     

一个具有充分下降性的混合共轭梯度法

混合共轭梯度法是一个改进的新共轭梯度法,有着比较好的数值表现.在Jia提出的混合共轭梯度法基础上,建立了一个新的具有充分下降性的混合共轭梯度算法;并证明了该算法在强Wolfe型线搜索下具有全局收敛性.数值实验结果表明该算法是有效的.     

Wolfe线搜索下一类混合共轭梯度法的全局收敛性

本文给出了一个新的共轭梯度公式,新公式在精确线搜索下与DY公式等价,并给出了新公式的相关性质.结合新公式和DY公式提出了一个新的混合共轭梯度法,新算法在Wolfe线搜索下产生一个下降方向,并证明了算法的全局收敛性,并给出了数值例子.     

一类新共轭梯度法的全局收敛性

共轭梯度法是求解无约束最优化问题的有效方法.本文在βkDY的基础上对βk引入参数,提出了一类新共轭梯度法,并证明其在强Wolfe线性搜索条件下具有充分下降性和全局收敛性.     

无约束优化的一个充分下降共轭梯度算法

在修正PRP共轭梯度法的基础上,提出了求解无约束优化问题的一个充分下降共轭梯度算法,证明了算法在Wolfe线搜索下全局收敛,并用数值实验表明该算法具有较好的数值结果.     

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.     

带参数共轭梯度法簇的全局收敛性

共轭梯度法是最优化中最常用的方法之一,广泛地应用于求解大规模优化问题,其中参数β_k的不同选取可以构成不同的共轭梯度法.给出了一类含有三个参数的共轭梯度算法,这种算法能够在给定的条件下证明选定的β_k在每一步都能产生一个下降方向,同时在强Wolfe线搜索下,这种算法具有全局收敛性.     

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．     

A new globalization technique for nonlinear conjugate gradient methods for nonconvex minimization

It is well-known that the HS method and the PRP method may not converge for nonconvex optimization even with exact line search. Some globalization techniques have been proposed, for instance, the PRP+ globalization technique and the Grippo-Lucidi globalization technique for the PRP method. In this paper, we propose a new efficient globalization technique for general nonlinear conjugate gradient methods for nonconvex minimization. This new technique utilizes the information of the previous search direction sufficiently. Under suitable conditions, we prove that the nonlinear conjugate gradient methods with this new technique are globally convergent for nonconvex minimization if the line search satisfies Wolfe conditions or Armijo condition. Extensive numerical experiments are reported to show the efficiency of the proposed technique.     

A three-parameter family of nonlinear conjugate gradient methods

In this paper, we propose a three-parameter family of conjugate gradient methods for unconstrained optimization. The three-parameter family of methods not only includes the already existing six practical nonlinear conjugate gradient methods, but subsumes some other families of nonlinear conjugate gradient methods as its subfamilies. With Powell's restart criterion, the three-parameter family of methods with the strong Wolfe line search is shown to ensure the descent property of each search direction. Some general convergence results are also established for the three-parameter family of methods. This paper can also be regarded as a brief review on nonlinear conjugate gradient methods.

     

Two descent hybrid conjugate gradient methods for optimization

In this paper, we propose two new hybrid nonlinear conjugate gradient methods, which produce sufficient descent search direction at every iteration. This property depends neither on the line search used nor on the convexity of the objective function. Under suitable conditions, we prove that the proposed methods converge globally for general nonconvex functions. The numerical results show that both hybrid methods are efficient for the given test problems from the CUTE library.     

A new two-parameter family of nonlinear conjugate gradient methods

Conjugate gradient methods are an important class of methods for unconstrained optimization, especially for large-scale problems. Recently, they have been much studied. In this paper, we propose a new two-parameter family of conjugate gradient methods for unconstrained optimization. The two-parameter family of methods not only includes the already existing three practical nonlinear conjugate gradient methods, but has other family of conjugate gradient methods as subfamily. The two-parameter family of methods with the Wolfe line search is shown to ensure the descent property of each search direction. Some general convergence results are also established for the two-parameter family of methods. The numerical results show that this method is efficient for the given test problems. In addition, the methods related to this family are uniformly discussed.     

A new general form of conjugate gradient methods with guaranteed descent and strong global convergence properties

Although the study of global convergence of the Polak–Ribière–Polyak (PRP), Hestenes–Stiefel (HS) and Liu–Storey (LS) conjugate gradient methods has made great progress, the convergence of these algorithms for general nonlinear functions is still erratic, not to mention under weak conditions on the objective function and weak line search rules. Besides, it is also interesting to investigate whether there exists a general method that converges under the standard Armijo line search for general nonconvex functions, since very few relevant results have been achieved. So in this paper, we present a new general form of conjugate gradient methods whose theoretical significance is attractive. With any formula β _k  ≥ 0 and under weak conditions, the proposed method satisfies the sufficient descent condition independently of the line search used and the function convexity, and its global convergence can be achieved under the standard Wolfe line search or even under the standard Armijo line search. Based on this new method, convergence results on the PRP, HS, LS, Dai–Yuan–type (DY) and Conjugate–Descent–type (CD) methods are established. Preliminary numerical results show the efficiency of the proposed methods.     

Two new conjugate gradient methods based on modified secant equations

Following the approach proposed by Dai and Liao, we introduce two nonlinear conjugate gradient methods for unconstrained optimization problems. One of our proposed methods is based on a modified version of the secant equation proposed by Zhang, Deng and Chen, and Zhang and Xu, and the other is based on the modified BFGS update proposed by Yuan. An interesting feature of our methods is their account of both the gradient and function values. Under proper conditions, we show that one of the proposed methods is globally convergent for general functions and that the other is globally convergent for uniformly convex functions. To enhance the performance of the line search procedure, we also propose a new approach for computing the initial steplength to be used for initiating the procedure. We provide a comparison of implementations of our methods with the efficient conjugate gradient methods proposed by Dai and Liao, and Hestenes and Stiefel. Numerical test results show the efficiency of our proposed methods.     

A descent hybrid conjugate gradient method based on the memoryless BFGS update

In this work, we present a new hybrid conjugate gradient method based on the approach of the convex hybridization of the conjugate gradient update parameters of DY and HS+, adapting a quasi-Newton philosophy. The computation of the hybrization parameter is obtained by minimizing the distance between the hybrid conjugate gradient direction and the self-scaling memoryless BFGS direction. Furthermore, a significant property of our proposed method is that it ensures sufficient descent independent of the accuracy of the line search. The global convergence of the proposed method is established provided that the line search satisfies the Wolfe conditions. Our numerical experiments on a set of unconstrained optimization test problems from the CUTEr collection indicate that our proposed method is preferable and in general superior to classic conjugate gradient methods in terms of efficiency and robustness.     

一种求解非线性无约束优化问题的充分下降的共轭梯度法

共轭梯度法是一类具有广泛应用的求解大规模无约束优化问题的方法. 提出了一种新的非线性共轭梯度(CG)法,理论分析显示新算法在多种线搜索条件下具有充分下降性. 进一步证明了新CG算法的全局收敛性定理. 最后,进行了大量数值实验,其结果表明与传统的几类CG方法相比,新算法具有更为高效的计算性能.     

The Dai–Liao nonlinear conjugate gradient method with optimal parameter choices

Minimizing two different upper bounds of the matrix which generates search directions of the nonlinear conjugate gradient method proposed by Dai and Liao, two modified conjugate gradient methods are proposed. Under proper conditions, it is briefly shown that the methods are globally convergent when the line search fulfills the strong Wolfe conditions. Numerical comparisons between the implementations of the proposed methods and the conjugate gradient methods proposed by Hager and Zhang, and Dai and Kou, are made on a set of unconstrained optimization test problems of the CUTEr collection. The results show the efficiency of the proposed methods in the sense of the performance profile introduced by Dolan and Moré.     

A family of three-term nonlinear conjugate gradient methods close to the memoryless BFGS method

Based on the memoryless BFGS quasi-Newton method, a family of three-term nonlinear conjugate gradient methods are proposed. For any line search, the directions generated by the new methods are sufficient descent. Using some efficient techniques, global convergence results are established when the line search fulfills the Wolfe or the Armijo conditions. Moreover, the r-linear convergence rate of the methods are analyzed as well. Numerical comparisons show that the proposed methods are efficient for the unconstrained optimization problems in the CUTEr library.