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大规模无约束优化的一族有限存储LBFGS类算法

 钱小燕, 施庆生, 刘浩, 石岿然   

  1. 南京工业大学 理学院 应用数学系
  • 收稿日期:2011-01-06 修回日期:2011-05-26 出版日期:2011-09-20 发布日期:2011-09-29
  • 通讯作者: 钱小燕 E-mail:xyqian122@163.com

A class of limited memory BFGS-type algorithms for large-scale unconstrainedoptimization

 QIAN  Xiao-Yan, SHI  Qing-Sheng, LIU  Hao, SHI  Kui-Ran   

  • Received:2011-01-06 Revised:2011-05-26 Online:2011-09-20 Published:2011-09-29
  • Contact: Yan XiaoQIAN E-mail:xyqian122@163.com

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2

摘要： 本文尝试在有限存储类算法中利用目标函数值所提供的信息. 我们首先利用插值条件构造了一个新的二次函数逼近目标函数,得到了一个新的弱割线方程，然后将此弱割线方程与袁\cite{yuan1991}的弱割线方程相结合，给出了一族包括标准LBFGS的有限存储BFGS类算法，证明了这族算法的收敛性. 从标准试验函数库CUTE中选择试验函数进行了数值试验, 试验结果表明这族算法的数值表现都与标准LBFGS类似.

关键词:  无约束优化, 弱割线方程, BFGS算法, 收敛性分析, 有限存储

Abstract: In this paper, objective function value information is exploited in limited memory BFGS-type algorithms. we first construct a new quadratic function satisfying some interpolation conditions to approximate the objective function, get a new weak secant equation. Combining the new weak secant equation and that obtained by Yuan\cite{yuan1991}, a class of limited memory BFGS--type algorithms including the classic LBFGS algorithm based on a new weak secant equation are proposed. The convergence of this class limited memory BFGS-type algorithms is proved. Numerical results for standard test problems from CUTE are reported, which indicate that all the algorithms in the proposed class perform quiet well.

Key words: unconstrained optimization, weak secant equation, LBFGS algorithm, convergence analysis, limited memory