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Multi-stage stochastic optimization: the distance between stochastic scenario processes

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Abstract

Approximation techniques are challenging, important and very often irreplaceable solution methods for multi-stage stochastic optimization programs. Applications for scenario process approximation include financial and investment planning, inventory control, energy production and trading, electricity generation planning, pension fund management, supply chain management and similar fields. In multi-stage stochastic optimization problems the amount of stage-wise available information is crucial. While some authors deal with filtration distances, in this paper we consider the concepts of nested distributions and their distances which allows to keep the setup purely distributional but at the same time to introduce information and information constraints. Also we introduce the distance between stochastic process and a tree and we generalize the concept of nested distance for the case of infinite trees, i.e. for the case of two stochastic processes given by their continuous distributions. We are making a step towards to a new method for distribution quantization that is the most suitable for multi-stage stochastic optimization programs as it takes into account both the stochastic process and the stage-wise information.

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Notes

  1. Here we consider the conditional Gaussian distribution that depends on the historical values \(\xi ^{t-1}\) but not on the bounds in which these historical values are lying. To improve the approximation one may construct a distribution function conditional on the bounds \([b_{1,i_1-1},b_{1,i_1}],\,[b_{n_2,i_2-1},b_{n_2,i_1}]\),...,\([b_{n_{t-1},i_{t-1}-1},b_{n_{t-1},i_{t-1}}]\) of the stochastic process.

  2. For the numerical calculation of the upper bound it is necessary to compute the multivariate probabilities \(P(\xi ^t\in \mathbb{B }^t)\). One can read more about the methods for it in Genz (1992), Szántai (2000).

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Acknowledgments

The author is grateful to o.Univ.Prof. Dr. Georg Ch. Pflug for his continuous supervision, inspirational advices and guidance at ISOR. She would also like to thank IIASA’s RPV group for their support and valuable feedbacks during the development of applications in the field of catastrophic risk-management.

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  1. Department of Statistics and Operations Research, University of Vienna, Universitätsstrasse, 5, 1010 , Vienna, Austria

    Anna V. Timonina

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  1. Anna V. Timonina
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Correspondence to Anna V. Timonina.

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Timonina, A.V. Multi-stage stochastic optimization: the distance between stochastic scenario processes. Comput Manag Sci 12, 171–195 (2015). https://doi.org/10.1007/s10287-013-0185-3

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