# American Institute of Mathematical Sciences

July  2017, 13(3): 1587-1599. doi: 10.3934/jimo.2017008

## A fast continuous method for the extreme eigenvalue problem

 1 College of Applied Sciences, Beijing University of Technology, Beijing 100124, China 2 School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, China

* Corresponding author: zbli@lsec.cc.ac.cn

Received  July 2015 Revised  April 2016 Published  December 2016

Fund Project: This research was supported by the National Science Foundation of China (61179033), Collaborative Innovation Center on Beijing Society-building and Social Governance, and Shandong Provincial Natural Science Foundation of China (ZR2013FL032)

Matrix eigenvalue problems play a significant role in many areas of computational science and engineering. In this paper, we propose a fast continuous method for the extreme eigenvalue problem. We first convert such a nonconvex optimization problem into a minimization problem with concave objective function and convex constraints based on the continuous methods developed by Golub and Liao. Then, we propose a continuous method for solving such a minimization problem. To accelerate the convergence of this method, a self-adaptive step length strategy is adopted. Under mild conditions, we prove the global convergence of this method. Some preliminary numerical results are presented to verify the effectiveness of the proposed method eventually.

Citation: Huan Gao, Zhibao Li, Haibin Zhang. A fast continuous method for the extreme eigenvalue problem. Journal of Industrial & Management Optimization, 2017, 13 (3) : 1587-1599. doi: 10.3934/jimo.2017008
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##### References:
Sketch three trajectories of two different orders
Sketch three trajectories of two different orders
Results for Example 1
 n 100 500 1000 2000 3000 4000 5000 ODE45(Sec.) 0.328 5.297 17.2199 86.188 130.016 389.016 642.234 Iterations 31 55 43 51 39 55 60 Time (Sec.) 0.016 0.531 1.484 6.734 11.594 29.125 48.094
 n 100 500 1000 2000 3000 4000 5000 ODE45(Sec.) 0.328 5.297 17.2199 86.188 130.016 389.016 642.234 Iterations 31 55 43 51 39 55 60 Time (Sec.) 0.016 0.531 1.484 6.734 11.594 29.125 48.094
Results for Example 2
 n 100 500 1000 2000 3000 4000 5000 ODE45(Sec.) 0.234 5.359 44.343 80.469 149.234 314.516 452.859 Iterations 43 61 99 47 39 51 51 Time (Sec.) 0.031 0.578 3.438 6.328 11.594 26.781 41.234
 n 100 500 1000 2000 3000 4000 5000 ODE45(Sec.) 0.234 5.359 44.343 80.469 149.234 314.516 452.859 Iterations 43 61 99 47 39 51 51 Time (Sec.) 0.031 0.578 3.438 6.328 11.594 26.781 41.234

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