American Institute of Mathematical Sciences

Advanced
2005, 1(2): 181-192. doi: 10.3934/jimo.2005.1.181

Analysis of monotone gradient methods

Yuhong Dai 1, and  Ya-xiang Yuan 2,

1. 

State Key Laboratory of Scientific and Engineering Computing, Institute of Computational Mathematics and Scientific/Engineering Computing, P.O. Box 2719, Beijing 100080, China
2. 

State Key Laboratory of Scientific and Engineering Computing, Institute of Computational Mathematics and Scientific/Engineering computing, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, P. O. Box 2719, Beijing 100080, P. R., China

Received  August 2004 Revised  December 2004 Published  April 2005

The gradient method is one simple method in nonlinear optimization. In this paper, we give a brief review on monotone gradient methods and study their numerical properties by introducing a new technique of long-term observation. We find that, one monotone gradient algorithm which is proposed by Yuan recently shares with the Barzilai-Borwein (BB) method the property that the gradient components with respect to the eigenvectors of the function Hessian are decreasing together. This might partly explain why this algorithm by Yuan is comparable to the BB method in practice. Some examples are also provided showing that the alternate minimization algorithm and the other algorithm by Yuan may fall into cycles. Some more efficient gradient algorithms are provided. Particularly, one of them is monotone and performs better than the BB method in the quadratic case.
Keywords: monotone, cycle., nonmonotone, gradient method, strictly convex quadratics.
Mathematics Subject Classification: 65k05, 90c0.
Citation: Yuhong Dai, Ya-xiang Yuan. Analysis of monotone gradient methods. Journal of Industrial & Management Optimization, 2005, 1 (2) : 181-192. doi: 10.3934/jimo.2005.1.181
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