几类具有充分下降性共轭梯度算法的研究

发布时间：2018-09-02 09:35

【摘要】：共轭梯度法由于算法简单,易于编程.存储需求少等特点,常作为解决大规模非线性无约束优化问题的一种重要的方法,在现实生活的众多领域频繁使用且行之有效.本文首先介绍了经典共轭梯度法,修正共轭梯度法,混合共轭梯度法,谱共轭梯度法的研究现状.在前面学者研究成果的基础之上,对共轭梯度法开展了进一步地分析和讨论.得到以下研究成果:1.基于共轭梯度法的下降性条件,提出了一类充分下降的混合型谱共轭梯度法.该方法结合了 FR法,WYL法,PRP法的优点;然后,在Wolfe线搜索下用反证法证明了新的混合型谱共轭梯度法的全局收敛性.最后,通过数值算例,将算法WS与WYL法,FR法进行比较,结果表明新算法在迭代次数与迭代总时间上均优于其他另外两种算法.算法的全局收敛性和数值效果的优越性表明新算法是值得研究的.2.在修正共轭梯度法的基础上,提出了一类WFR型谱共轭梯度法,该算法在任何线搜索下都具有充分下降性.在标准Wolfe线搜索下,证明了新算法具有全局收敛性.并对新算法进行数值实验,其实验结果表明新算法优于VFR法。
[Abstract]:The conjugate gradient method is easy to program because of its simple algorithm. As an important method to solve large-scale nonlinear unconstrained optimization problems, it is often used frequently and effectively in many fields of real life. In this paper, the classical conjugate gradient method, the modified conjugate gradient method, the mixed conjugate gradient method and the spectral conjugate gradient method are introduced. Based on the previous research results, the conjugate gradient method is further analyzed and discussed. Get the following research results: 1. Based on the descent condition of conjugate gradient method, a class of mixed spectral conjugate gradient method with sufficient descent is proposed. This method combines the advantages of FR method and WYL method, and proves the global convergence of the new hybrid spectral conjugate gradient method under Wolfe line search. Finally, a numerical example is given to compare the WS algorithm with the WYL method. The results show that the new algorithm is superior to the other two algorithms in the number of iterations and the total iteration time. The global convergence of the algorithm and the superiority of numerical effect show that the new algorithm is worth studying. Based on the modified conjugate gradient method, a class of WFR type spectral conjugate gradient method is proposed, which has sufficient descent under any line search. Under the standard Wolfe line search, the global convergence of the new algorithm is proved. The experimental results show that the new algorithm is superior to the VFR method.
【学位授予单位】：西南大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O224

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