Download Probability metrics and the stability of stochastic programs with

Probability metrics and the stability of stochastic
programs with recourse∗
Michal Houda
Department of Probability and Mathematical Statistics, Faculty of Mathematics and
Physics, Charles University Prague, Sokolovská 83, 186 75 Prague 8, Czech Republic.
e-mail: [email protected]
Ferda Mravenec
Department of Ant Research, ‘Formica’ University New Ant City, Anthill 15, 36 156 New
Ant City, The Ant Kingdom.
e-mail: [email protected]
Abstract. In stochastic optimization models the underlying probability measure must be
very often replaced by its approximations. This leads to the investigation of the stability
of such models with respect to changes in the probability measure. In this context, special
attention is paid to recourse problems and the Wasserstein and Kolmogorov metrics.
Keywords: Stochastic programs with recourse, stability, Wasserstein metrics,
Kolmogorov metrics.
JEL classification: C44
AMS classification: 90C15
1.
Introduction
Designing an optimization model in the stochastic programming, we usually assume
that the underlaying probability distribution is known. This is not the case in practical applications: we work with some approximation or estimate, e. g. discretizations,
scenaria approaches, or empirical estimates.
In such cases (and the others as well) we would query about the quality of such
approximation. It is obvious that a small change in probability distribution may
cause large changes in the optimal value and/or in the optimal solution set. We
speak then about stability analysis of stochastic programming problem with respect
to changes in the probability measure. Henceforth, we quantify these changes by
a suitably chosen distance, a (pseudo-)metrics in a selected space of probability
measures (we speak in short about probability metrics).
∗
This research was supported by the Grant Agency of the Czech Republic under Grants
402/01/0539, 402/02/1015 and 402/01/0034.
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Table 1: Caption for Table 1
first
second third
number number
number number
1.1.
number
number
Probability metrics: a survey
Choosing an appropriate metrics is still another important task with regard to the
stability of stochastic programs: there are examples where the inappropriate selection gives bad results. See e. g. [1].
This is Figure 1
Figure 1: Caption for Figure 1
The text continues here.
1.1.1. ‘Subsubsection’ heading
‘Subsubsection’ text.
Theorem 1. Theorem text comes here.
Acknowledgements
The author wish to thank Vlasta Kaňková for insightfull reading the manuscript
and many helpful comments.
References
[1] Dupačová, J., and Römisch, W.: Quantitative stability for scenario-based stochastic
programs. In: Proceedings of the Prague Stochastics ’98 (M. Hušková, P. Lachout,
and J. Á. Vı́šek, eds.), Union of Czech Mathematicians and Physicists, Prague, 1998,
119–124.
[2] Prékopa, A.: Stochastic Programming. Kluwer Academic Publishers, Dordrecht, and
Akadémiai Kiadó, Budapest, 1995.
[3] Ross, D. W.: Lysosomes and Storage Diseases. MA Thesis, Columbia University, New
York, 1977.
[4] Vallander, S. S.: Calculation of the Wasserstein distance between probability distributions on the line, Theory of Probability and its Applications 18 (1973), 784–786.
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