Skip to main content
SpringerLink
Log in

On Stability of Multistage Stochastic Decision Problems

  • Conference paper
Recent Advances in Optimization

Part of the book series: Lecture Notes in Economics and Mathematical Systems ((LNE,volume 563))

  • 1369 Accesses

Summary

The paper considers a general multistage stochastic decision problem which contains Markovian decision processes and multistage stochastic programming problems as special cases. The objective functions, the constraint sets and the probability measures are approximated. Making use of the Bellman Principle, (semi) convergence statements for the optimal value functions and the optimal decisions at each stage are derived. The considerations rely on stability assertions for parametric programming problems which are extended and adapted to the multistage case. Furthermore, new sufficient conditions for the convergence of objective functions which are integrals with respect to decision-dependent probability measures are presented. The paper generalizes results by Langen(1981) with respect to the convergence notions, the integrability conditions and the continuity assumptions.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. B. Bank, J. Cuddat, D. Klatte, B. Kummer, and K. Tammer. Nonlinear Parametric Optimization. Akademie-Verlag Berlin, 1982.

    Google Scholar 

  2. D.P. Bertsekas and S.E. Shreve. Stochastic Optimal Control: The Discrete Time Case. Academic Press, New York, 1978.

    Google Scholar 

  3. J. Dupačová and K. Sladý. Comparison of Multistage Stochastic Programs with Recourse and Stochastic Dynamic Programs with Discrete Time. Z. Angew. Math. Mech. 82:11–12: 753–765, 2002.

    Article  MathSciNet  Google Scholar 

  4. O. Fiedler and W. Römisch. Stability of multistage stochastic programming. Ann. Oper. Res. 56: 79–93, 1995.

    Article  MathSciNet  Google Scholar 

  5. K. Hinderer. Foundations of Non-Stationary Dynamic Programming with Discrete Time Parameter. Lecture Notes in Oper. Res. and Math. Syst., 33, 1970.

    Google Scholar 

  6. V. Kaňková. A Remark on Analysis of Multistage Stochastic Programs: Markov Dependence. Z. Angew. Math. Mech. 82,11–12: 781–793, 2002.

    Article  MathSciNet  Google Scholar 

  7. L.A. Korf and R.J.-B. Wets. Random lsc functions: an ergodic theorem. Math. Oper. Res. 26: 421–445, 2001.

    Article  MathSciNet  Google Scholar 

  8. S. Vogel and P. Lachout. On continuous convergence and epi-convergence of random functions-Theory and relations. Kybernetika 39: 75–98, 2003.

    MathSciNet  Google Scholar 

  9. S. Vogel and P. Lachout. On continuous convergence and epi-convergence of random functions-Sufficient conditions and applications. Kybernetika 39: 99–118, 2003.

    MathSciNet  Google Scholar 

  10. H.-J. Langen. Convergence of Dynamic Programming Models. Math. Oper. Res. 6: 493–512, 1981.

    Article  MATH  MathSciNet  Google Scholar 

  11. A. Mänz. Stabilität mehrstufiger stochastischer Optimierungsprobleme. Diploma Thesis, TU Ilmenau, 2003.

    Google Scholar 

  12. P.H. Müller and V. Nollau (ed.). Steuerung stochastischer Prozesse. AkademieVerlag, Berlin, 1984.

    Google Scholar 

  13. S.M. Robinson. Local epi-continuity and local optimization. Math. Programming 37: 208–222, 1987.

    Article  MATH  MathSciNet  Google Scholar 

  14. R.T. Rockafeller and R. J.-B. Wets. Variational Analysis. Springer, 1998.

    Google Scholar 

  15. M. Schäl. Conditions for Optimality in Dynamic Programming and for the Limit of n-Stage Optimal Policies to Be Optimal. Z. Wahrscheinlichkeitstheorie verve. Gebiete 32: 179–196, 1975.

    Article  MATH  Google Scholar 

  16. S. Vogel. Stochastische Stabilitätskonzepte. Habilitation, TU Ilmenau, 1991.

    Google Scholar 

  17. S. Vogel. A stochastic approach to stability in stochastic programming. J. Comput. and Appl. Mathematics, Series Appl. Analysis and Stochastics 56: 65–96, 1994.

    MATH  Google Scholar 

  18. S. Vogel.On stability in stochastic programming-Sufficient conditions for continuous convergence and epi-convergence. Preprint TU Ilmenau, 1994.

    Google Scholar 

  19. J. Wang. Stability of multistage stochastic programming. Ann. Oper. Res. 56: 313–322, 1995.

    Article  MATH  MathSciNet  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. ASPECTA Lebensversicherung AG, Germany

    Alexander Mänz

  2. Technische Universitäet Ilmenau, Germany

    Silvia Voge1

Authors
  1. Alexander Mänz
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Silvia Voge1
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Avignon, 33, rue Louis Pasteur, 84000, Avignon, France

    Alberto Seeger

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Mänz, A., Voge1, S. (2006). On Stability of Multistage Stochastic Decision Problems. In: Seeger, A. (eds) Recent Advances in Optimization. Lecture Notes in Economics and Mathematical Systems, vol 563. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-28258-0_7

Download citation

Publish with us

Policies and ethics

Search

Navigation