Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-05T14:46:43.812Z Has data issue: false hasContentIssue false
  • English
  • Français

Decomposition of large-scale stochastic optimal control problems

Published online by Cambridge University Press:  23 July 2010

Kengy Barty ,
Pierre Carpentier  and
Pierre Girardeau
Kengy Barty
Affiliation:
EDF R&D, 1, avenue du Général de Gaulle, 92141 Clamart Cedex, France
Pierre Carpentier
Affiliation:
ENSTA ParisTech, 32, boulevard Victor, 75739 Paris Cedex 15, France; [email protected]
Pierre Girardeau
Affiliation:
Université Paris-Est, CERMICS, École des Ponts, Champs sur Marne, 77455 Marne la Vallée Cedex 2, France, also with EDF R&D and ENSTA
Get access

Abstract

In this paper, we present an Uzawa-based heuristic that is adapted to certain type of stochastic optimal control problems. More precisely, we consider dynamical systems that can be divided into small-scale subsystems linked through a static almost sure coupling constraint at each time step. This type of problem is common in production/portfolio management where subsystems are, for instance, power units, and one has to supply a stochastic power demand at each time step. We outline the framework of our approach and present promising numerical results on a simplified power management problem.

Keywords

Stochastic optimal controldecomposition methodsdynamic programming
Type
Research Article
Information
RAIRO - Operations Research , Volume 44 , Issue 3 , July 2010 , pp. 167 - 183
Copyright
© EDP Sciences, ROADEF, SMAI, 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

K.J. Arrow, L. Hurwicz and H. Uzawa, Studies in linear and nonlinear programming. Stanford University Press (1958).
Artstein, Z., Sensitivity to σ-fields of information in stochastic allocation. Stoch. Stoch. Rep. 36 (1991) 41–63. CrossRef
Bellman, R. and Dreyfus, S.E., Functional approximations and dynamic programming. Math. Tables Other Aides comput. 13 (1959) 247–251. CrossRef
R. Bellman, Dynamic programming, Princeton University Press. New Jersey (1957).
D.P. Bertsekas, Dynamic programming and optimal control, 2nd edition, Vol. 1 & 2, Athena Scientific (2000).
K. Barty, J.-S. Roy and C. Strugarek, A stochastic gradient type algorithm for closed loop problems. Math. Program. (2007).
Blomvall, J. and Shapiro, A., Solving multistage asset investment problems by the sample average approximation method. Math. Program. 108 (2006) 571–595. CrossRef
D.P. Bertsekas and J.N. Tsitsiklis, Neuro-Dynamic Programming, Athena Scientific (1996).
Cohen, G. and Culioli, J.-C., Decomposition Coordination Algorithms for Stochastic Optimization. SIAM J. Control Optim. 28 (1990) 1372–1403. CrossRef
G. Cohen, Auxiliary Problem Principle and decomposition of optimization problems. J. Optim. Theory Appl. (1980) 277–305.
J.M. Danskin, The theory of max-min. Springer, Berlin (1967).
de Farias, D.P. and Van Roy, B., The Linear Programming Approach to Approximate Dynamic Programming. Oper. Res. 51 (2003) 850–856. CrossRef
I. Ekeland and R. Temam, Convex analysis and variational problems. SIAM, Philadelphia (1999).
P. Girardeau, A comparison of sample-based Stochastic Optimal Control methods. E-print available at: arXiv:1002.1812v1, 2010.
Heitsch, H., Römisch, W. and Strugarek, C., Stability of multistage stochastic programs. SIAM J. Optim. 17 (2006) 511–525. CrossRef
J.L. Higle and S. Sen, Stochastic decomposition. Kluwer, Dordrecht (1996).
Pennanen, T., Epi-convergent discretizations of multistage stochastic programs. Math. Oper. Res. 30 (2005) 245–256. CrossRef
A. Prékopa, Stochastic programming, Kluwer, Dordrecht (1995).
Shapiro, A., On complexity of multistage stochastic programs. Oper. Res. Lett. 34 (2006) 1–8. CrossRef
A. Shapiro and A. Ruszczynski (Eds.), Stochastic Programming, Elsevier, Amsterdam (2003).
C. Strugarek, Approches variationnelles et autres contributions en optimisation stochastique, Thèse de doctorat, École Nationale des Ponts et Chaussées, 5 (2006).
Turgeon, A., Optimal operation of multi-reservoir power systems with stochastic inflows. Water Resour. Res. 16 (1980) 275–283. CrossRef
Tstsiklis, J.N. and Van Roy, B., Feature-based methods for large scale dynamic programming. Mach. Lear. 22 (1996) 59–94.