Abstract
We give an equivalence between the tasks of computing the essential supremum of a summable function and of finding a certain zero of a one-dimensional convex function. Interpreting the integral method as Newton-type method we show that in the case of objective functions with an essential supremum that is not spread the algorithm can work very slowly. For this reason we propose a method of accelerating the algorithm which is in some respect similar to the method of Aitken/Steffensen.
Key words
- essential supremum
- convergence speed
- integral global optimization
- Newton algorithm
This is a preview of subscription content, access via your institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Archetti, F. and Betrò, B. (1975), Recursive Stochastic Evaluation of the Level Set Measure in the Global Optimization Problems, Technical Report, University of Pisa, Pisa, Italy.
Chichinadze, V.K. (1967), Random Search to Determine the Extremum of the Function of Several Variables, Engeneering Cybernetics 1, 115–123.
Caselton, W.F, and Yassien, H. A. (1994), LSP4, public domain software, available via ftp://ftp.ruhr-uni-bochum.de/mirrors/simtel.coast.net/SimTel/msdos/statistic/lsp4.zip.
Chew S.H. and Zheng Q. (1988), Integral Global Optimization, Springer, Berlin, Heidelberg.
De Biase, L. and Frontini, F. (1978), A Stochastic Method for Global Optimization: Its Structure and Numerical Performance, in: Dixon, L.C.W. and Szegö, G.P. (eds.) (1978), Towards Global Optimization 2, North-Holland, Amsterdam, 85–102.
Dieudonné, J. (1975), Grundzüge der modernen Analysis, Band III, Deutscher Verlag der Wissenschaften, Berlin.
Hiriart-Urruty, J.-B. and Lemaréchal, C. (1993), Convex Analysis and Minimization Algorithms I, Springer, Berlin, Heidelberg.
Kosmol, P. (1993), Methoden zur numerischen Behandlung nichtlinearer Gleichungen und Optimierungsaufgaben, Teubner, Stuttgart.
Kostreva, M.M. and Zheng Q. (1994), Integral Global Optimization Method for Solution of Nonlinear Complementarity Problems, Journal of Global Optimization 5, 181–193.
Natanson, I.P. (1975), Theorie der Funktionen einer reellen Veränderlichen, Akademie-Verlag, Berlin.
Phú, H.X. and Hoffmann, A. (1996), Essential Supremum and Supremum of Summable Functions, Numerical Functional Analysis and Optimization 17 (to appear).
Stoer, J. (1994), Numerische Mathematik 1, Springer, Berlin, Heidelberg.
Zheng Q. (1992), Integral Global Optimization of Robust Discontinuous Functions, Dissertation, Graduate School of Clemson University, Clemson.
Zheng Q. and Zhuang D. (1995), Integral Global Minimization: Algorithms, Implementations and Numerical Tests, Journal of Global Optimization 7, 421–454.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1997 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Hichert, J., Hoffmann, A., Phú, H.X. (1997). Convergence Speed of an Integral Method for Computing the Essential Supremum. In: Bomze, I.M., Csendes, T., Horst, R., Pardalos, P.M. (eds) Developments in Global Optimization. Nonconvex Optimization and Its Applications, vol 18. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-2600-8_10
Download citation
DOI: https://doi.org/10.1007/978-1-4757-2600-8_10
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4419-4768-0
Online ISBN: 978-1-4757-2600-8
eBook Packages: Springer Book Archive