Rapidly convergent Steffensen-based methods for unconstrained optimization
RAIRO - Operations Research - Recherche Opérationnelle, Tome 53 (2019) no. 2, pp. 657-666.

A problem with rapidly convergent methods for unconstrained optimization like the Newton’s method is the computational difficulties arising specially from the second derivative. In this paper, a class of methods for solving unconstrained optimization problems is proposed which implicitly applies approximations to derivatives. This class of methods is based on a modified Steffensen method for finding roots of a function and attempts to make a quadratic model for the function without using the second derivative. Two methods of this kind with non-expensive computations are proposed which just use first derivative of the function. Derivative-free versions of these methods are also suggested for the cases where the gradient formulas are not available or difficult to evaluate. The theory as well as numerical examinations confirm the rapid convergence of this class of methods.

Reçu le :
Accepté le :
DOI : 10.1051/ro/2017043
Classification : 65K10, 90C53
Mots-clés : Unconstrained optimization, derivative-free, newton’s method, Steffensen’s method
Afzalinejad, Mohammad 1

1
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     author = {Afzalinejad, Mohammad},
     title = {Rapidly convergent {Steffensen-based} methods for unconstrained optimization},
     journal = {RAIRO - Operations Research - Recherche Op\'erationnelle},
     pages = {657--666},
     publisher = {EDP-Sciences},
     volume = {53},
     number = {2},
     year = {2019},
     doi = {10.1051/ro/2017043},
     mrnumber = {3961735},
     zbl = {07127208},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ro/2017043/}
}
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Afzalinejad, Mohammad. Rapidly convergent Steffensen-based methods for unconstrained optimization. RAIRO - Operations Research - Recherche Opérationnelle, Tome 53 (2019) no. 2, pp. 657-666. doi : 10.1051/ro/2017043. http://www.numdam.org/articles/10.1051/ro/2017043/

[1] A. Burmen and T. Tuma, Unconstrained derivative-free optimization by successive approximation. J. Comput. Appl. Math. 223 (2009) 62–74. | MR | Zbl

[2] R. Fletcher, Practical Methods of Optimization, 2nd ed. Wiley (2000). | MR | Zbl

[3] E. Kahya, Modified Secant-type methods for unconstrained optimization. Appl. Math. Comput 181 (2006) 1349–1356. | MR | Zbl

[4] T.G. Kolda, R.M. Lewis and V. Torczon, Optimization by direct search: New perspectives on some classical and modern methods. SIAM Review 45 (2003) 385–482. | MR | Zbl

[5] J. Nocedal and S. Wright, Numerical Optimization. Springer Verlag, New York (1999). | MR | Zbl

[6] J.M. Ortega and W.C. Rheinboldt, Iterative Solutions of Nonlinear Equations in Several Variables, Academic Press (1970). | MR | Zbl

[7] H. Ren and Q. Wu, A class of modified Secant methods for unconstrained optimization. App. Math. Comput. 206 (2008) 716–720. | MR | Zbl

[8] C. Sainvitu, How much do approximate derivatives hurt filter methods? RAIRO: OR 43 (2009) 309–329. | MR | Zbl

[9] Z. Weia, G. Lia and L. Qib, New quasi-Newton methods for unconstrained optimization problems. Appl. Math. Comput. 175 (2006) 1156–1188. | MR | Zbl

[10] X. Wu and J. Xia, Some substantial modifications and improvements for derivative-free iterative methods and derivative-free transformation for multiple zeros. Appl. Math. Comput. 181 (2006) 1585–1599. | MR | Zbl

[11] Q. Zheng, P. Zhao, L. Zhang and W. Ma, Variants of Steffensen-secant method and applications. Appl. Math. Comput. 216 (2010) 3486–3496. | MR | Zbl

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