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Barycentric Bounds in Stochastic Programming: Theory and Application

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Stochastic Programming

Part of the book series: International Series in Operations Research & Management Science ((ISOR,volume 150))

Abstract

The design and analysis of efficient approximation schemes are of fundamental importance in stochastic programming research. Bounding approximations are particularly popular for providing strict error bounds that can be made small by using partitioning techniques. In this chapter we develop a powerful bounding method for linear multistage stochastic programs with a generalized nonconvex dependence on the random parameters. Thereby, we establish bounds on the recourse functions as well as compact bounding sets for the optimal decisions. We further demonstrate that our bounding methods facilitate the reliable solution of important real-life decision problems. To this end, we solve a stochastic optimization model for the management of nonmaturing accounts and compare the bounds on maximum profit obtained with different partitioning strategies.

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Notes

  1. 1.

    In the presence of concave recourse functions upper and lower bounds switch roles.

  2. 2.

    Sometimes, notation is simplified by further introducing a deterministic dummy random variable \(\boldsymbol \xi^0\) of the past.

  3. 3.

    Decision processes will also be referred to as strategies or policies.

  4. 4.

    Random variables appearing in both the objective function and the constraints of the stochastic program must be duplicated. The first copy of such a random variable is appended to η, while the second copy is appended to ξ. Note that the two copies will be discretized differently.

  5. 5.

    If the objective function coefficients and the rhs vectors are linear affine functions of the stochastic parameters, then the recourse functions exhibit a saddle shape themselves and no correction terms are needed. This situation is investigated in Frauendorfer (1994, 1996).

  6. 6.

    Analytical formulae for the masses and coordinates of the atoms of P l and P u are provided in Frauendorfer (1992); cf. also Frauendorfer (1994, 1996) and Kuhn (2004).

  7. 7.

    Continuity of A t follows inductively from the dominated convergence theorem.

  8. 8.

    Notice that \(F^l_t\) and \(F^u_t\) are concave polyhedral functions of x t for all fixed parameter values, since we are dealing with linear stochastic programs and since the barycentric measures have finite supports.

  9. 9.

    For a survey of the theory of set convergence see Rockafellar and Wets (1998, Chapter 4).

  10. 10.

    For instance, the well-known Cox, Ingersoll, and Ross model (Cox et al. 1985) involves a “square root process” with \(\gamma=0.5\).

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Acknowledgments

Daniel Kuhn thanks the Swiss National Science Foundation for financial support.

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Authors and Affiliations

  1. University of St. Gallen, St. Gallen, Switzerland

    Karl Frauendorfer & Michael Schürle

  2. Imperial College of Science, Technology and Medicine, London, UK

    Daniel Kuhn

Authors
  1. Karl Frauendorfer
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  2. Daniel Kuhn
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  3. Michael Schürle
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Correspondence to Karl Frauendorfer .

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Editors and Affiliations

  1. Dept. Management Science & Engineering, Stanford University, Stanford, 94305, California, USA

    Gerd Infanger

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Frauendorfer, K., Kuhn, D., Schürle, M. (2010). Barycentric Bounds in Stochastic Programming: Theory and Application. In: Infanger, G. (eds) Stochastic Programming. International Series in Operations Research & Management Science, vol 150. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-1642-6_5

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