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A globally convergent BFGS method for nonlinear monotone equations without any merit functions


Authors: Wei-Jun Zhou and Dong-Hui Li
Journal: Math. Comp. 77 (2008), 2231-2240
MSC (2000): Primary 90C53
Published electronically: May 19, 2008
MathSciNet review: 2429882
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Abstract: Since 1965, there has been significant progress in the theoretical study on quasi-Newton methods for solving nonlinear equations, especially in the local convergence analysis. However, the study on global convergence of quasi-Newton methods is relatively fewer, especially for the BFGS method. To ensure global convergence, some merit function such as the squared norm merit function is typically used. In this paper, we propose an algorithm for solving nonlinear monotone equations, which combines the BFGS method and the hyperplane projection method. We also prove that the proposed BFGS method converges globally if the equation is monotone and Lipschitz continuous without differentiability requirement on the equation, which makes it possible to solve some nonsmooth equations. An attractive property of the proposed method is that its global convergence is independent of any merit function.We also report some numerical results to show efficiency of the proposed method.


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Additional Information

Wei-Jun Zhou
Affiliation: College of Mathematics and Computational Science, Changsha University of Science and Technology, Changsha 410076, China
Email: weijunzhou@126.com

Dong-Hui Li
Affiliation: College of Mathematics and Econometrics, Hunan University, Changsha 410082, China
Email: dhli@hnu.cn

DOI: https://doi.org/10.1090/S0025-5718-08-02121-2
Keywords: BFGS method, monotone equation, hyperplane projection method, global convergence.
Received by editor(s): March 14, 2006
Received by editor(s) in revised form: March 30, 2007
Published electronically: May 19, 2008
Additional Notes: This work was supported by the NSF (10771057 and 10701018) of China.
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.