A regularization method for ill-posed bilevel optimization problems

Maitine Bergounioux; Mounir Haddou

RAIRO - Operations Research (2006)

  • Volume: 40, Issue: 1, page 19-35
  • ISSN: 0399-0559

Abstract

top
We present a regularization method to approach a solution of the pessimistic formulation of ill-posed bilevel problems. This allows to overcome the difficulty arising from the non uniqueness of the lower level problems solutions and responses. We prove existence of approximated solutions, give convergence result using Hoffman-like assumptions. We end with objective value error estimates.

How to cite

top

Bergounioux, Maitine, and Haddou, Mounir. "A regularization method for ill-posed bilevel optimization problems." RAIRO - Operations Research 40.1 (2006): 19-35. <http://eudml.org/doc/249745>.

@article{Bergounioux2006,
abstract = { We present a regularization method to approach a solution of the pessimistic formulation of ill-posed bilevel problems. This allows to overcome the difficulty arising from the non uniqueness of the lower level problems solutions and responses. We prove existence of approximated solutions, give convergence result using Hoffman-like assumptions. We end with objective value error estimates. },
author = {Bergounioux, Maitine, Haddou, Mounir},
journal = {RAIRO - Operations Research},
keywords = {regularization method; ill-posed bilevel problems},
language = {eng},
month = {7},
number = {1},
pages = {19-35},
publisher = {EDP Sciences},
title = {A regularization method for ill-posed bilevel optimization problems},
url = {http://eudml.org/doc/249745},
volume = {40},
year = {2006},
}

TY - JOUR
AU - Bergounioux, Maitine
AU - Haddou, Mounir
TI - A regularization method for ill-posed bilevel optimization problems
JO - RAIRO - Operations Research
DA - 2006/7//
PB - EDP Sciences
VL - 40
IS - 1
SP - 19
EP - 35
AB - We present a regularization method to approach a solution of the pessimistic formulation of ill-posed bilevel problems. This allows to overcome the difficulty arising from the non uniqueness of the lower level problems solutions and responses. We prove existence of approximated solutions, give convergence result using Hoffman-like assumptions. We end with objective value error estimates.
LA - eng
KW - regularization method; ill-posed bilevel problems
UR - http://eudml.org/doc/249745
ER -

References

top
  1. A. Auslender and M. Teboulle, Asymptotic cones and functions in optimization and variational inequalities. Springer Monographs in Mathematics. Springer-Verlag, New York (2003).  Zbl1017.49001
  2. D. Azé and J.N. Corvellec, Characterizations of error bounds for lower semicontinuous functions on metric spaces. ESAIM: COCV10 (2004) 409–425.  Zbl1085.49019
  3. D. Azé and A. Rahmouni, On primal-dual stability in convex optimization. J. Convex Anal.3 (1996) 309–327.  Zbl0876.90085
  4. J.F. Bonnans and A. Shapiro, Perturbation analysis of optimization problems. Springer Series in Operations Research, Springer-Verlag, New York (2000).  Zbl0966.49001
  5. L. Brotcorne, M. Labbé, P. Marcotte and G. Savard, A bilevel model for toll optimization on a multicommodity transportation network. Transportation Sci.35 (2001) 1–14.  Zbl1069.90502
  6. S. Dempe, Foundations of bilevel programming, in Nonconvex optimization and its applications. Kluwer Acad. Publ., Dordrecht (2002).  Zbl1038.90097
  7. M. Labbé, P. Marcotte and G. Savard, On a class of bilevel programs, in Nonlinear optimization and related topics, (Erice, 1998). Appl. Optim. 36 (2000) 183–206.  Zbl1112.90365
  8. A.S. Lewis and J.-S. Pang, Error bounds for convex inequality systems, in Generalized convexity, generalized monotonicity: recent results (Luminy, 1996). Nonconvex Optim. Appl.27 (1998) 75–110.  
  9. W. Li, Abadie's constraint qualification, metric regularity, and error bounds for differentiable convex inequalities. SIAM J. Optim.7 (1997) 966–978.  Zbl0891.90132
  10. P. Loridan and J. Morgan, New results on approximate solutions in two-level optimization. Optimization20 (1989) 819–836.  Zbl0684.90089
  11. P. Loridan and J. Morgan, Regularizations for two-level optimization problems. Advances in optimization. Lect. Notes Econ. Math.382 (1992) 239–255.  Zbl0770.90102
  12. P. Marcotte and G. Savard, A bilevel programming approach to Price Setting in: Decision and Control in Management Science, in Essays in Honor of Alain Haurie, edited by G. Zaccour. Kluwer Academic Publishers (2002) 97–117.  
  13. K.F. Ng and X.Y. Zheng, Global error bounds with fractional exponents. Math. Program.Ser. B88 (2000) 357–370.  Zbl1016.90060
  14. T.Q. Nguyen, M. Bouhtou and J.-L. Lutton, D.C. approach to bilevel bilinear programming problem: application in telecommunication pricing, in Optimization and optimal control (Ulaanbaatar, 2002). Ser. Comput. Oper. Res.1 (2003) 211–231.  Zbl1069.90082
  15. Z. Wu and J. Ye, On error bounds for lower semicontinuous functions. Math. Program.Ser. A (2002).  Zbl1041.90053
  16. J. Zhao, The lower semicontinuity of optimal solution sets. J. Math. Anal. Appl.207 (1997) 240–254.  Zbl0872.90093

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.