Journal of Industrial and Management Optimization (JIMO)

Barzilai-Borwein-like methods for the extreme eigenvalue problem

Pages: 999 - 1019, Volume 11, Issue 3, July 2015      doi:10.3934/jimo.2015.11.999

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Huan Gao - College of Applied Sciences, Beijing University of Technology, Beijing 100124, China (email)
Yu-Hong Dai - State Key Laboratory of Scientific and Engineering Computing, Institute of Computational Mathematics and Scientific/Engineering Computing, AMSS, Chinese Academy of Sciences, Beijing 100190, China (email)
Xiao-Jiao Tong - Hunan Province Key Laboratory of Smart Grids Operation and Control, Changsha University of Science and Technology, Changsha 410004, Hunan Province, China (email)

Abstract: We consider numerical methods for the extreme eigenvalue problem of large scale symmetric positive definite matrices. By the variational principle, the extreme eigenvalue can be obtained by minimizing some unconstrained optimization problem. Firstly, we propose two adaptive nonmonotone Barzilai-Borwein-like methods for the unconstrained optimization problem. Secondly, we prove the global convergence of the two algorithms under some conditions. Thirdly, we compare our methods with eigs and the power method for the standard test problems from the UF Sparse Matrix Collection. The primary numerical experiments indicate that the two algorithms are promising.

Keywords:  Extreme eigenvalue problems, Barzilai-Borwein-like methods, unconstrained optimization, global convergence.
Mathematics Subject Classification:  Primary: 90C25, 90C30; Secondary: 49M30.

Received: November 2012;      Revised: June 2014;      Available Online: October 2014.