DMDICO

ISSN 1509-9407 (print version)

ISSN 2084-0365 (electronic version)

https://doi.org/10.7151/dmdico

Discussiones Mathematicae Differential  Inclusions, Control and  Optimization

Discussiones Mathematicae Differential  Inclusions, Control and  Optimization

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Differential Inclusions, Control and Optimization 20 (2000) 257-278
DOI: https://doi.org/10.7151/dmdico.1015

A PRIMAL-DUAL INTEGRAL METHOD IN GLOBAL OPTIMIZATION

Jens Hichert

Geomagic GmbH
Harbergstr. 73, D-99099 Erfurt, Germany

Armin Hoffmann

Institute of Mathematics
Technical University Ilmenau
PF 100565, D-98684 Ilmenau, Germany
e-mail: armin.hoffmann@mathematik.tu-ilmenau.de

Huan Xoang Phú

Institute of Mathematics Hanoi
P.O. Box 631 Bo Ho, 10 000 Hanoi, Vietnam

Rüdiger Reinhardt

Institute of Mathematics
Technical University Ilmenau
PF 100565, D-98684 Ilmenau, Germany

Abstract

Using the Fenchel conjugate Fc of Phú's Volume function F of a given essentially bounded measurable function f defined on the bounded box D ⊂ Rn, the integral method of Chew and Zheng for global optimization is modified to a superlinearly convergent method with respect to the level sequence. Numerical results are given for low dimensional functions with a strict global essential supremum.

Keywords: global optimization, integral method, Monte Carlo method, primal dual algorithm, level set method.

1991 Mathematics Subject Classification: 90C30, 65C05, 65K05, 49M29.

References

[1] Soo Hong Chew and Quan Zheng, Integral Global Optimization, Springer, Berlin 1988.
[2] C.A. Floudas and P.M. Pardalos, A Collection of Test Problems for Constrained Global Optimization Algorithms, Springer, Berlin 1990.
[3] J. Hichert, BARLO - Eine Software zur Wesentlichen Globalen Optimierung, Technical Report 17/98, Technische Universität Ilmenau 1998.
[4] J. Hichert, Methoden zur Bestimmung des wesentlichen Supremums mit Anwendung in der globalen Optimierung, PhD thesis, Technische Universität Ilmenau 1999.
[5] J. Hichert, A. Hoffmann, and H.X. Phu, Convergence speed of an integral method for computing the essential supremum, in: I.M. Bomze, T. Csendes, R. Horst and P.M. Pardalos (eds.): Developments in Global Optimization, pp. 153-170, Kluwer Academic Publishers, Dordrecht 1997.
[6] J.-B. Hiriart-Urruty and C. Lemaréchal, Convex Analysis and Minimization Algorithms, Vol. I (Fundamentals), Springer, Berlin 1993.
[7] J.-B. Hiriart-Urruty and C. Lemaréchal, Convex Analysis and Minimization Algorithms, Vol. II (Advanced Theory and Bundle Methods), Springer, Berlin 1993.
[8] http://solon.cma.univie.ac.at/˜neum/glopt.
[9] P.M. Pardalos, An open global optimization problem on the unit sphere, Journal of Global Optimization 6 (1995), 213.
[10] H.X. Phú and A. Hoffmann, Essential supremum and supremum of summable functions, Numerical Functional Analysis and Optimization 17 (1,2) (1996), 167-180.
[11] A. Törn and A. Zilinskas, Global Optimization, Springer, Berlin 1989.
[12] Z.B. Zabinsky and R.L. Smith, Pure adaptive search in global optimization, Mathematical Programming 53 (1992), 323-338.
[13] Quan Zheng, Integral Global Optimization of Robust Discontinuous Functions, PhD thesis, The Graduate School of Clemson University, Clemson 1992.
[14] Quan Zheng and Deming Zhuang, Integral global minimization: Algorithms, implementations and numerical tests, Journal of Global Optimization 7 (1995), 421-454.

Received 10 February 2000
Revised 19 May 2000

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