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Applied Mathematics & Optimization

, Volume 77, Issue 2, pp 377–404 | Cite as

Two Approaches to Stochastic Optimal Control Problems with a Final-Time Expectation Constraint

  • Laurent PfeifferEmail author
Article

Abstract

In this article, we study and compare two approaches to solving stochastic optimal control problems with an expectation constraint on the final state. The case of a probability constraint is included in this framework. The first approach is based on a dynamic programming principle and the second one uses Lagrange relaxation. In this article, we focus on discrete-time problems, but the two discussed approaches can be applied to discretized continuous-time problems.

Keywords

Stochastic optimal control Expectation and probability constraints Dynamic programming Lagrange relaxation 

Mathematics Subject Classification

90C15 93E20 

Notes

Acknowledgments

The author thanks J. Frédéric Bonnans for his advice and the two anonymous referees for useful remarks. The research leading to these results has received funding from the Gaspard Monge Program for Optimization and operations research (PGMO). The author gratefully acknowledges the Austrian Science Fund (FWF) for financial support under SFB F32, “Mathematical Optimization and Applications in Biomedical Sciences”.

References

  1. 1.
    Alais, J.C., Carpentier, P., De Lara, M.: Multi-usage hydropower single dam management: chance-constrained optimization and stochastic viability. Energy Syst. 1–24 (2015)Google Scholar
  2. 2.
    Andrieu, L., Cohen, G., Vázquez-Abad, F.J.: Gradient-based simulation optimization under probability constraints. Eur. J. Oper. Res. 212(2), 345–351 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bouchard, B., Elie, R., Imbert, C.: Optimal control under stochastic target constraints. SIAM J. Control Optim. 48(5), 3501–3531 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bouchard, B., Elie, R., Touzi, N.: Stochastic target problems with controlled loss. SIAM J. Control Optim. 48(5), 3123–3150 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Camilli, F., Falcone, M.: An approximation scheme for the optimal control of diffusion processes. RAIRO Modél. Math. Anal. Numér. 29(1), 97–122 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Granato, G.: Optimisation de lois de gestion energétique des véhicules hybrides. PhD thesis, Ecole Polytechnique (2012)Google Scholar
  7. 7.
    Henrion, R.: On the connectedness of probabilistic constraint sets. J. Optim. Theory Appl. 112(3), 657–663 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Henrion, R., Strugarek, C.: Convexity of chance constraints with independent random variables. Comput. Optim. Appl. 41(2), 263–276 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Hiriart-Urruty, J.B., Lemarechal, C.: Convex Analysis and Minimization Algorithms: Part 2: Advanced Theory and Bundle Methods. Convex Analysis and Minimization Algorithms. Springer, Berlin (1993)zbMATHGoogle Scholar
  10. 10.
    Kushner, H.J.: Numerical methods for stochastic control problems in continuous time. SIAM J. Control Optim. 28(5), 999–1048 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Lemaréchal, C.: Lagrangian relaxation. In: Junger, M., Naddef, D. (eds.) Computational Combinatorial Optimization. Lecture Notes in Computer Science, vol. 2241, pp. 112–156. Springer, Berlin (2001)CrossRefGoogle Scholar
  12. 12.
    Øksendal, B.: Stochastic Differential Equations: An Introduction with Applications. Hochschultext/Universitext, Springer, New York (2003)CrossRefzbMATHGoogle Scholar
  13. 13.
    Pfeiffer, L.: Optimality conditions for mean-field type optimal control problems. SFB Report 2015–015 (2015)Google Scholar
  14. 14.
    Rockafellar, R.T.: Convex Analysis. Princeton Mathematical Series. Princeton University Press, Princeton (1970)CrossRefzbMATHGoogle Scholar
  15. 15.
    Shapiro, A., Dentcheva, D., Ruszczyński, A.P.: Lectures on Stochastic Programming: Modeling and Theory. MPS-SIAM Series on Optimization. Society for Industrial and Applied Mathematics, Philadelphia (2009)CrossRefzbMATHGoogle Scholar
  16. 16.
    Touzi, N.: Optimal Stochastic Control, Stochastic Target Problems, and Backward SDE. Fields Institute Monographs, Springer, Berlin (2012)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Institute for Mathematics and Scientific ComputingKarl-Franzens-UniversitätGrazAustria

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