Non-parametric approximation of non-anticipativity constraints in scenario-based multistage stochastic programming

Jean-Sébastien Roy; Arnaud Lenoir

Kybernetika (2008)

  • Volume: 44, Issue: 2, page 171-184
  • ISSN: 0023-5954

Abstract

top
We propose two methods to solve multistage stochastic programs when only a (large) finite set of scenarios is available. The usual scenario tree construction to represent non-anticipativity constraints is replaced by alternative discretization schemes coming from non-parametric estimation ideas. In the first method, a penalty term is added to the objective so as to enforce the closeness between decision variables and the Nadaraya–Watson estimation of their conditional expectation. A numerical application of this approach on an hydro-power plant management problem is developed. The second method exploits the interpretation of kernel estimators as a sum of basis functions.

How to cite

top

Roy, Jean-Sébastien, and Lenoir, Arnaud. "Non-parametric approximation of non-anticipativity constraints in scenario-based multistage stochastic programming." Kybernetika 44.2 (2008): 171-184. <http://eudml.org/doc/33920>.

@article{Roy2008,
abstract = {We propose two methods to solve multistage stochastic programs when only a (large) finite set of scenarios is available. The usual scenario tree construction to represent non-anticipativity constraints is replaced by alternative discretization schemes coming from non-parametric estimation ideas. In the first method, a penalty term is added to the objective so as to enforce the closeness between decision variables and the Nadaraya–Watson estimation of their conditional expectation. A numerical application of this approach on an hydro-power plant management problem is developed. The second method exploits the interpretation of kernel estimators as a sum of basis functions.},
author = {Roy, Jean-Sébastien, Lenoir, Arnaud},
journal = {Kybernetika},
keywords = {multistage stochastic programming; scenarios; discrete approximation; multistage stochastic programming; scenarios; discrete approximation},
language = {eng},
number = {2},
pages = {171-184},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Non-parametric approximation of non-anticipativity constraints in scenario-based multistage stochastic programming},
url = {http://eudml.org/doc/33920},
volume = {44},
year = {2008},
}

TY - JOUR
AU - Roy, Jean-Sébastien
AU - Lenoir, Arnaud
TI - Non-parametric approximation of non-anticipativity constraints in scenario-based multistage stochastic programming
JO - Kybernetika
PY - 2008
PB - Institute of Information Theory and Automation AS CR
VL - 44
IS - 2
SP - 171
EP - 184
AB - We propose two methods to solve multistage stochastic programs when only a (large) finite set of scenarios is available. The usual scenario tree construction to represent non-anticipativity constraints is replaced by alternative discretization schemes coming from non-parametric estimation ideas. In the first method, a penalty term is added to the objective so as to enforce the closeness between decision variables and the Nadaraya–Watson estimation of their conditional expectation. A numerical application of this approach on an hydro-power plant management problem is developed. The second method exploits the interpretation of kernel estimators as a sum of basis functions.
LA - eng
KW - multistage stochastic programming; scenarios; discrete approximation; multistage stochastic programming; scenarios; discrete approximation
UR - http://eudml.org/doc/33920
ER -

References

top
  1. Babuska I., Melenk J., The partition of unity method, International Journal for Numerical Methods in Engineering 40 (1998), 727–758 (1998) MR1429534
  2. Barty K., Contributions à la discrétisation des contraintes de mesurabilité pour les problčmes d’optimisation stochastiques, PhD. Thesis, Ecole nationale des ponts et chaussées, 2004 
  3. Cohen G., Optimal scenario tree topology and corresponding rate of convergence, In: Proc. 11th Conference on Stochastic Programming, 2007 
  4. Dallagi A., Méthodes particulaires en commande optimale stochastique, PhD. Thesis, Université Paris I, 2007 
  5. Devroye L. P., On the almost everywhere convergence of nonparametric regression function estimates, Ann. Statist. 9 (1981), 1310–1319 (1981) Zbl0477.62025MR0630113
  6. Devroye L. P., Wagner T., Distribution-free consistency results in nonparametric discrimination and regression function estimation, Ann. Statist. 8 (1980), 231–239 (1980) Zbl0431.62025MR0560725
  7. Gröwe-Kuska N., Heitsch, H., Römisch W., Scenario reduction and scenario tree construction for power management problem, In: Power Tech Conference Proceedings, IEEE Bologna, 2003 
  8. Heitsch H., Römisch W., Generation of multivariate scenario trees to model stochasticity in power management, IEEE St. Petersburg Power Tech 2005, 2005 
  9. Heitsch H., Römisch W., Scenario Tree Modeling for Multistage Stochastic Programs, Preprint Humboldt-University Berlin, Institute of Mathematics, 2005 Zbl1173.90007MR2470797
  10. Nadaraya E., On estimating regression, Theory Probab. Appl. 9 (1964), 141–142 (1964) Zbl0136.40902
  11. Pennanen T., Epi-convergent discretizations of multistage stochastic programs, Math. Oper. Res. 30 (2005), 245–256 Zbl1165.90014MR2125146
  12. Spiegelman C., Sacks J., Consistent window estimation in nonparametric regression, Ann. Statist. 8 (1980), 240–246 (1980) Zbl0432.62066MR0560726
  13. Watson G., Smooth regression analysis, Sankhya Ser. A 26 (1964), 359–372 (1964) Zbl0137.13002MR0185765

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.