A primal-dual integral method in global optimization

Jens Hichert; Armin Hoffmann; Huan Xoang Phú; Rüdiger Reinhardt

Discussiones Mathematicae, Differential Inclusions, Control and Optimization (2000)

  • Volume: 20, Issue: 2, page 257-278
  • ISSN: 1509-9407

Abstract

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Using the Fenchel conjugate F c 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.

How to cite

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Jens Hichert, et al. "A primal-dual integral method in global optimization." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 20.2 (2000): 257-278. <http://eudml.org/doc/271478>.

@article{JensHichert2000,
abstract = {Using the Fenchel conjugate $F^c$ 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.},
author = {Jens Hichert, Armin Hoffmann, Huan Xoang Phú, Rüdiger Reinhardt},
journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
keywords = {global optimization; integral method; Monte Carlo method; primal dual algorithm; level set method; integral global optimization; level set},
language = {eng},
number = {2},
pages = {257-278},
title = {A primal-dual integral method in global optimization},
url = {http://eudml.org/doc/271478},
volume = {20},
year = {2000},
}

TY - JOUR
AU - Jens Hichert
AU - Armin Hoffmann
AU - Huan Xoang Phú
AU - Rüdiger Reinhardt
TI - A primal-dual integral method in global optimization
JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization
PY - 2000
VL - 20
IS - 2
SP - 257
EP - 278
AB - Using the Fenchel conjugate $F^c$ 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.
LA - eng
KW - global optimization; integral method; Monte Carlo method; primal dual algorithm; level set method; integral global optimization; level set
UR - http://eudml.org/doc/271478
ER -

References

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  1. [1] Soo Hong Chew and Quan Zheng, Integral Global Optimization, Springer, Berlin 1988. 
  2. [2] C.A. Floudas and P.M. Pardalos, A Collection of Test Problems for Constrained Global Optimization Algorithms, Springer, Berlin 1990. 
  3. [3] J. Hichert, BARLO - Eine Software zur Wesentlichen Globalen Optimierung, Technical Report 17/98, Technische Universität Ilmenau 1998. 
  4. [4] J. Hichert, Methoden zur Bestimmung des wesentlichen Supremums mit Anwendung in der globalen Optimierung, PhD thesis, Technische Universität Ilmenau 1999. 
  5. [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. Zbl0878.65053
  6. [6] J.-B. Hiriart-Urruty and C. Lemaréchal, Convex Analysis and Minimization Algorithms, Vol. I (Fundamentals), Springer, Berlin 1993. 
  7. [7] J.-B. Hiriart-Urruty and C. Lemaréchal, Convex Analysis and Minimization Algorithms, Vol. II (Advanced Theory and Bundle Methods), Springer, Berlin 1993. 
  8. [8] http://solon.cma.univie.ac.at/~neum/glopt 
  9. [9] P.M. Pardalos, An open global optimization problem on the unit sphere, Journal of Global Optimization 6 (1995), 213. Zbl0822.90125
  10. [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. Zbl0877.49013
  11. [11] A. Törn and A. Zilinskas, Global Optimization, Springer, Berlin 1989. 
  12. [12] Z.B. Zabinsky and R.L. Smith, Pure adaptive search in global optimization, Mathematical Programming 53 (1992), 323-338. Zbl0756.90086
  13. [13] Quan Zheng, Integral Global Optimization of Robust Discontinuous Functions, PhD thesis, The Graduate School of Clemson University, Clemson 1992. 
  14. [14] Quan Zheng and Deming Zhuang, Integral global minimization: Algorithms, implementations and numerical tests, Journal of Global Optimization 7 (1995), 421-454. Zbl0846.90105

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