Abstract
It is proved that the iterative sequence constructed by the Basic Trust-Region Algorithm (see Conn et al. in Trust-region methods, MPS-SIAM series on optimization, Philadelphia, 2000), which uses the Cauchy point method, is locally stable and linearly convergent in a neighborhood of a nonsingular local minimizer.
Similar content being viewed by others
References
Conn, A.R., Gould, N.I.M., Toint, P.L.: Trust-Region Methods. MPS-SIAM Series on Optimization, Mathematical Programming Society and Society for Industrial and Applied Mathematics, Philadelphia (2000)
Nocedal, J., Wright, S.J.: Numerical Optimization. Springer, New York (1999)
Polyak, B.T.: Introduction to Optimization. Optimization Software, Inc., New York (1987)
Ruszczynski, A.: Nonlinear Optimization. Princeton University Press, Princeton (2005)
Vasilev, F.P.: Numerical Methods for Solving Extremal Problems (in Russian), 2nd edn. Nauka, Moscow (1988)
Le Thi, H.A., Pham Dinh, T., Yen, N.D.: Properties of two DC algorithms in quadratic programming. J. Global Optim. 49, 481–495 (2011)
Leong, W.J., Goh, B.S.: Convergence and stability of line search methods for unconstrained optimization. Acta Appl. Math. 127, 155–167 (2013)
Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York (1970)
Acknowledgements
Research of B.N. Muoi was supported by Hanoi Pedagogical University No. 2 (Vietnam) under Grant Number C.2016.12. Research of N.D. Yen was supported by the project VAST.HTQT.Pháp.03/16-17 between VAST and CNRS. We thank the anonymous referee for helpful comments and suggestions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Suliman Saleh Al-Homidan.
Dedicated to Professor B.T. Polyak on the occasion of his 80th birthday.
Rights and permissions
About this article
Cite this article
Muoi, B.N., Yen, N.D. Local Stability and Local Convergence of the Basic Trust-Region Method. J Optim Theory Appl 172, 578–593 (2017). https://doi.org/10.1007/s10957-016-1038-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10957-016-1038-3
Keywords
- Unconstrained smooth optimization problem
- Basic trust-region method
- Iterative sequence
- Stability
- Linear convergence