A Distributionally Robust Formulation for Stochastic Quadratic Bi-level Programming

Pablo Adasme, Abdel Lisser, Chen Wang

Abstract

In this paper, we propose a distributionally robust model for a (0-1) stochastic quadratic bi-level programming problem. To this purpose, we first transform the stochastic bi-level problem into an equivalent deterministic formulation. Then, we use this formulation to derive a bi-level distributionally robust model (Liao, 2011). The latter is accomplished while taking into account the set of all possible distributions for the input random parameters. Finally, we transform both, the deterministic and the distributionally robust models into single level optimization problems (Audet et al., 1997). This allows comparing the optimal solutions of the proposed models. Our preliminary numerical results indicate that slight conservative solutions can be obtained when the number of binary variables in the upper level problem is larger than the number of variables in the follower.

References

  1. Audestad, J., Gaivoronski, A., and Werner, A. (2006). Extending the stochastic programming framework for the modeling of several decision makers: Pricing and competition in the telecommunication sector. Annals of Operations Research, 142:19-39.
  2. Audet, C., Hansen, P., Jaumard, B., and Savard, G. (1997). Links between linear bilevel and mixed 01 programming problems. Journal of Optimization Theory and Applications, 93(2):273-300.
  3. Bertsimas, D., Brown, D., and Caramanis, C. (2010). Theory and applications of robust optimization. SIAM Review, 53(3):464501.
  4. Bertsimas, D. and Sim, M. (2004). The price of robustness. Operations Research, 52(1):35-53.
  5. Birge, J. and Louveaux, F. (1997). Introduction to stochastic programming. Springer-Verlag, New York.
  6. Carrion, M., Arroyo, J., and Conejo, A. (2009). A bilevel stochastic programming approach for retailer futures market trading. IEEE Transactions on Power Systems, 24(3):1446-1456.
  7. O zaltin, O., Prokopyev, O., and Schaefer, A. (2010). The bilevel knapsack problem with stochastic right-hand sides. Operations Research Letters, 38(4):328-333.
  8. Floudas, C. and Pardalos, P. (2001). Encyclopedia of Optimization. Kluwer Academic Publishers, Dordrecht. The Netherlands.
  9. Fortet, R. (1960). Applications de l'algebre de boole en recherche operationelle. Revue Francaise de Recherche Operationelle, 4:17-26.
  10. Gaivoronski, A., Lisser, A., and Lopez, R. (2011). Knapsack problem with probability constraints. Journal of Global Optimization, 49(3):397-413.
  11. Kalashnikov, V., Perez-Valdes, G., Tomasgard, A., and Kalashnykova, N. (2010). Natural gas cash-out problem: Bilevel stochastic optimization approach. European Journal of Operational Research, 206(1):18-33.
  12. Liao, S. (2011). Staffing a call center with uncertain nonstationary arrival rate and flexibility. To appear in OR Spectrum.
  13. Schultz, R., Leen, S., and Vlerk, M. V. D. (1996). Two-stage stochastic integer programming: a survey. Statistica Neerlandica, 50:404-416.
  14. Shapiro, A., Dentcheva, D., and RuszczyƁski, A. (2009). Lectures on Stochastic Programming: Modeling and Theory. SIAM Philadelphia, Series on Optimization, 9 of MPS/SIAM, Philadelphia, 436 edition.
  15. Wynter, L. (2009). Encyclopedia of Optimization, chapter Stochastic Bilevel Programs. Springer.
Download


Paper Citation


in Harvard Style

Adasme P., Lisser A. and Wang C. (2013). A Distributionally Robust Formulation for Stochastic Quadratic Bi-level Programming . In Proceedings of the 2nd International Conference on Operations Research and Enterprise Systems - Volume 1: ICORES, ISBN 978-989-8565-40-2, pages 24-31. DOI: 10.5220/0004207100240031


in Bibtex Style

@conference{icores13,
author={Pablo Adasme and Abdel Lisser and Chen Wang},
title={A Distributionally Robust Formulation for Stochastic Quadratic Bi-level Programming},
booktitle={Proceedings of the 2nd International Conference on Operations Research and Enterprise Systems - Volume 1: ICORES,},
year={2013},
pages={24-31},
publisher={SciTePress},
organization={INSTICC},
doi={10.5220/0004207100240031},
isbn={978-989-8565-40-2},
}


in EndNote Style

TY - CONF
JO - Proceedings of the 2nd International Conference on Operations Research and Enterprise Systems - Volume 1: ICORES,
TI - A Distributionally Robust Formulation for Stochastic Quadratic Bi-level Programming
SN - 978-989-8565-40-2
AU - Adasme P.
AU - Lisser A.
AU - Wang C.
PY - 2013
SP - 24
EP - 31
DO - 10.5220/0004207100240031