求解无约束优化问题的一类新的下降算法

【作者】 潘翠英；

【导师】 陈兰平；

【作者基本信息】 首都师范大学 ， 应用数学， 2006， 硕士

【摘要】 最优化是一门应用广泛，发展迅速的学科。尤其对于非线性优化问题来说，寻找快速有效的算法一直是优化专家们研究的热门方向之一。近年来，人们提出了不少有效的算法，如：Fletcher-Reeves算法和Polak-Ribiere算法等，并试图证明它们的全局收敛性。 本文对于求解无约束最优化问题提出了一类新的下降算法Pan-Chen方法，这种方法同Dai-Yuan方法一样有很好的收敛性。结合新方法的有效收敛性以及Hestenes-Stiefel方法和Polak-Ribiere方法好的数值表现，我们又提出了两类杂交算法，在Wolfe线搜索下不需给定下降条件即证明了算法的全局收敛性。为了说明新方法的有效性，我们做了一定的数值实验，将本文算法与Dai-Yuan方法以及Dai-Yuan方法和Polak-Ribiere方法[10]的混合算法进行了比较，取得了很好的数值结果。 在第一章我们首先简要的介绍了最优化问题的提出以及判断最优解常用的最优性条件，回顾了无约束最优化问题常用的几类导数下降类算法。 在第二章中我们提出了一类新的下降算法，并且在Wolfe线搜索下无需给定下降条件即证明了算法的全局收敛性。采用文献[20]中讨论Dai-Yuan方法的基本思想，我们总结了Pan-Chen方法的一些内在性质。 在第三章中我们将提出的新算法的好的收敛性与Hestenes-Stiefel方法和Polak-Ribi(?)re-Polyak方法好的数值表现结合起来，给出了求解无约束优化问题的两类杂交算法。同样采用Wolfe线搜索，在比较弱的条件下证明了算法的全局收敛性。对于几个有代表性的问题我们做了一定的数值实验，将我们提出的新方法以及杂交方法与Hestenes-Stiefel方法，Dai-Yuan方法以及二者的混合算法进行了比较，并以非常好的数值结果说明了我们新算法的有效性。更多还原

【Abstract】 With its rapid development, optimization is used on a large scale by us nowadays. Especially, finding high performance algorithms in nonlinear optimization is a very heated research topic for the optimization specialists. Many algorithms were developed in recent years , among these are Polak-Ribiere Algorithm, Fletcher-Reeves Algorithm, Hestenes-Stiefel, Dai-Yuan Algorithm, and so on. And they try to prove the global convergence of this algorithms.This paper presents us a class of efficient new descent methods for unconstrained optimization, which are developed from Dai-Yuan Method. And also, we prove their global convergence in Wolfe line search. As a attempt, we get to use both of the advantages of Hestenes-Stiefel Algorithm and our new methods synthetically. Simultaneously, we prove that the Wolfe line search condition can ensure their global convergence. Finally, we also get better numerical results with the new algorithm.In chapter 1 ,we introduce the development of optimization and some extensive optimality conditions to determine the optimum solution. Also, We review several extensive derivative descent methods for unconstrained programming.In chapter 2, we propose a class of efficient new descent methods. moreover, we show their global congenvence in Wolfe line search without the descent property. And, the conditions on the objective function are also weak, which is similar to those required by the Zoutendijk condition. Moreover, we give some property about our new methods, this property is the same with Dai-Yuan Method.In chapter 3, we propose two hybrid conjugate gradient algorithms based on Hestenes-Stiefel method and our new method. And, we proved their global congenvence in Wolfe line search without descent condition. Numerical experiments shows that our methods are very efficient, especially for large scale problems.更多还原