This paper deals with the
global optimization of several variables Holderian functions. An algorithm using a
sequence of overestimators of a single variable objective function was developed converging
to the maximum. Then by the use of α-dense curves, we show how to
implement this algorithm in a multidimensional optimization problem. Finally,
we validate
the algorithm by testing it on some test functions.
Cite this paper
Yahyaoui, A. and Ammar, H. (2017). Global Optimization of Multivariate Holderian Functions Using Overestimators. Open Access Library Journal, 4, e3511. doi: http://dx.doi.org/10.4236/oalib.1103511.
Miguel, R.L.
and Sahinidis, N.V. (2013) Derivative-Free
Optimization: A Review of Algorithms and Comparison of Software
Implementations. Journal of Global Optimization, 56,
1247-1293. https://doi.org/10.1007/s10898-012-9951-y
Lavigne, D. and Cherruault, Y. (1991) Alienor-Gabriel
Global Optimization of a Function of Several Variables. Mathematical and Computer Modelling, 15, 125-134. https://doi.org/10.1016/0895-7177(91)90097-Q
Mora, G., Cherruault, Y. and Ziadi, A. (2001) Global
Optimization. A New Variant of the Alienor Method. Computers and Mathematics with Applications, 41, 63-71. https://doi.org/10.1016/S0898-1221(01)85006-9
Shubert, B.O. (1972) A Sequential Method Seeking the
Global Maximum of a Function. SIAM Journal
on Numerical Analysis, 9, 379-388. https://doi.org/10.1137/0709036
Ammar, H. and Cherruault, Y. (1993) Approximation of
Several Variables Function by a One Variable Function and Application to Global
Optimization. Mathematical and Computer
Modelling, 18,
17-21. https://doi.org/10.1016/0895-7177(93)90003-H
Ammar, H. and Cherruault, Y. (1995) Implementation
of Alienor Technique in the Multidimensional Bissection Method. Application to
Global Optimization. A New Accelerated Algorithm. Kybernetes, 24, 31-40. https://doi.org/10.1108/03684929510147272
Evtushenko, Ya.G., Malkova, V.U.
and Stanevichyus, A.A.
(2009) Parallel
Global Optimization of Function of Several Variables. Computational Mathematics and Mathematical Physics, 49, 246-260. https://doi.org/10.1134/S0965542509020055
Gerge, V.P. and Sergeyev, Ya.D.
(1999) Sequential
and Parallel Algorithms for Global Minimizing Functions with Lipschitzian
Derivatives. Computers and Mathematics
with Applications, 37, 163-179. https://doi.org/10.1016/S0898-1221(99)00067-X
Sergeyev, Y.D. and Kvasov,
D.E. (2013) Lipschitz
Global Optimization Methods in Control Problems. Automation and Remote Control, 74, 1435-1448. https://doi.org/10.1134/S0005117913090014
Gourdin, E., Jaumard, B. and Ellaia, R.
(1996) Global
Optimization of Holder Functions. Journal
of Global Optimization, 8, 323-348. https://doi.org/10.1007/BF02403997
Lera, D. and Sergeyev, Ya.D. (2002) Global
Minimization Algorithms for Holder Functions. BIT Numerical Mathematics, 42, 119-133. https://doi.org/10.1023/A:1021926320198
Rahal, M. and Ziadi, A. (2008) A
New Extension of Piyavskii's Method to Holder Functions of Several Variables. Applied Mathematics and Computation, 197, 478- 488. https://doi.org/10.1016/j.amc.2007.07.067
Mishra, S.K. (2007) Some New Test Functions for
Global Optimization and Performance of Repulsive Particle Swarm Method.
University Library of Munich, Germany, MPRA Paper 2718.
Mora, G.
and Cherruault, Y.
(1997) Characterization
and Generation of α-Dense Curve. Computers & Mathematics with Applications, 33, 83-91. https://doi.org/10.1016/S0898-1221(97)00067-9
Mora, G., Cherruault, Y. and Ziadi, A.
(2000) Functional
Equations Generating Space-Densifying Curves. Computers and Mathematics with Applications, 39,
45- 55. https://doi.org/10.1016/S0898-1221(00)00085-7
Sergeyev, Y.D., Strongin, R.G.
and Lera, D.
(2013) Introduction
to Global Optimization Exploiting Space-Filling Curves. Springer Briefs
in Optimization. https://doi.org/10.1007/978-1-4614-8042-6
