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A Look at the Tool of BYRD and NOCEDAL

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DOI: 10.4236/ajcm.2011.14028    3,757 Downloads   7,011 Views  

ABSTRACT

A power tool for the analysis of quasi-Newton methods has been proposed by Byrd and Nocedal ([1], 1989). The purpose of this paper is to make a study to the basic property (BP) given in [1]. As a result of the BP, a sufficient condition of global convergence for a class of quasi-Newton methods for solving unconstrained minimization problems without convexity assumption is given. A modified BFGS formula is designed to match the requirements of the sufficient condition. The numerical results show that the proposed method is very encouraging.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

L. Huang, G. Li and G. Yuan, "A Look at the Tool of BYRD and NOCEDAL," American Journal of Computational Mathematics, Vol. 1 No. 4, 2011, pp. 240-246. doi: 10.4236/ajcm.2011.14028.

References

[1] R. Byrd and J. Nocedal, “A Tool for the Analysis of Quasi-Newton Methods with Application to Unconstrained Minimization,” SIAM Journal on Numerical Analysis, Vol. 26, No. 3, 1989, pp. 727-739. doi:10.1137/0726042
[2] M. J. D. Powell, “Some Global Convergence Properties of a variable Metric Algorithm for Minimization without Exact Line Searches,” In: R.W. Cottle and C. E. Lemke, Eds., Nonlinear Programming, SIAM-AMS Proceedings, Vol. 4, American Mathematical Society, Providence, 1976, pp.53-72.
[3] J. Werner, “über die Globale Knovergenz von Variable- Metric-Verfahre mit Nichtexakter Schrittweitenbestim- mung,” Numerische Mathematik, Vol. 31, No. 3, 1978, pp. 321-334. doi:10.1007/BF01397884
[4] R. Byrd, J. Nocedal and Y. Yuan, “Global Convergence of a Class of Quasi-Newton Methods on Convex Prob- lems,” SIAM Journal on Numerical Analysis, Vol. 24, No. 5, 1987, pp. 1171-1189. doi:10.1137/0724077
[5] D. Li and M. Fukushima, “A Global and Superlinear Con- vergent Gauss-Newton-Based BFGS Method for Sym- metric Nonlinear Equations,” SIAM Journal on Numeri- cal Analysis, Vol. 37, No. 1, 1999, pp. 152-172. doi:10.1137/S0036142998335704
[6] D. Li and M. Fukushima, “A Modified BFGS Method and Its Global Convergence in Nonconvex Minimiza- tion,” Journal of Computational and Applied Mathemat- ics, Vol. 129, No. 1-2, 2001, pp. 15-35. doi:10.1016/S0377-0427(00)00540-9
[7] D. Li and M. Fukushima, “On the Global Convergence of the BFGS Method for Nonconvex Unconstrained Optimi- zation Problems,” SIAM Journal on Optimization, Vol. 11, No. 4, 2001, pp. 1054-1064. doi:10.1137/S1052623499354242
[8] Z. Wei, G. Yu, G. Yuan and Z. Lian, “The Superlinear Convergence of a Modified BFGS-Type Method for Un- constrained Optimization,” Computational Optimization and Applications, Vol. 29, No. 3, 2004, pp. 315-332. doi:10.1023/B:COAP.0000044184.25410.39
[9] G. Yuan and Z. Wei, “Convergence Analysis of a Modi- fied BFGS Method on Convex Minimizations,” Compu- tational Optimization and Applications, Vol. 47, No. 2, 2010, pp. 237-255. doi:10.1007/s10589-008-9219-0
[10] E. G. Birgin and J. M. Martínez, “Structured Minimal- Memory Inexact Quasi-Newton Method and Secant Pre- conditioners for Augmented Lagrangian Optimization,” Computational Optimization and Applications, Vol. 39, No. 1, 2008, pp. 1-16. doi:10.1007/s10589-007-9050-z
[11] G. Yuan and Z. Wei, “The Superlinear Convergence Ana- lysis of a Nonmonotone BFGS Algorithm on Convex Ob- jective Functions,” Acta Mathematics Sinica, Vol. 24, No. 1, 2008, pp. 35-42. doi:10.1007/s10114-007-1012-y
[12] A. Griewank, “On Automatic Differentiation,” In: M. Iri and K. Tanabe, Eds., Mathematical Programming: Re- cent Developments and Applications, Academic Publish- ers, Cambridge, 1989, pp. 84-108.
[13] Y. Dai and Q. Ni, “Testing Different Conjugate Gradient Methods for Large-Scale Unconstrained Optimization,” Journal of Computational Mathematics, Vol. 21, No. 3, 2003, pp. 311-320.

  
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