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Journal of Global Optimization

, Volume 39, Issue 4, pp 529–542 | Cite as

Necessary optimality conditions for bilevel set optimization problems

  • S. Dempe
  • N. Gadhi
Original Paper

Abstract

Bilevel programming problems are hierarchical optimization problems where in the upper level problem a function is minimized subject to the graph of the solution set mapping of the lower level problem. In this paper necessary optimality conditions for such problems are derived using the notion of a convexificator by Luc and Jeyakumar. Convexificators are subsets of many other generalized derivatives. Hence, our optimality conditions are stronger than those using e.g., the generalized derivative due to Clarke or Michel-Penot. Using a certain regularity condition Karush-Kuhn-Tucker conditions are obtained.

Keywords

Bilevel optimization Convexificator Karush-Kuhn-Tucker multipliers Necessary Optimality conditions Regularity condition Set valued mappings Support function 

Mathematics Subject Classification (2000)

Primary 49J52 90C29 Secondary 49K99 

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References

  1. 1.
    Amahroq T. and Gadhi N. (2003). Second order optimality conditions for an extremal problem under inclusion constraints. J. Math. Anal. Appl. 285: 74–85 CrossRefGoogle Scholar
  2. 2.
    Amahroq, T., Gadhi, N., Riahi, H.: Epi-differentiability and optimality conditions for an extremal problem. J. Inequal. Pure Appl. Math., Victoria University, ISSN electronic 1443–5756 4(2) Article 41 (2003)Google Scholar
  3. 3.
    Babahadda, H., Gadhi, N.: Necessary optimality conditions for bilevel optimization problems using convexificators (To appear) in J. Global. Optim.Google Scholar
  4. 4.
    Bard J.F. (1991). Some properties of the bilevel programming problem. J. Optim. Theory Appl. 68: 371–378 CrossRefGoogle Scholar
  5. 5.
    Bank B., Guddat J., Klatte D., Kummer B. and Tammer K. (1982). Nonlinear Parametric Optimization. Akademie-Verlag, Berlin Google Scholar
  6. 6.
    Chanas S., Delgado M., Verdegay J.L. and Vila M.A. (1993). Interval and fuzzy extensions of classical transportation problems. Transport. Plan. Technol. 17: 203–218 CrossRefGoogle Scholar
  7. 7.
    Chen, Y., Florian, M.: On the geometry structure of linear bilevel programs: A dual approach, Technical Report CRT-867, Centre de Recherche sur les transports. Université de Montreal, Montreal, Quebec, Canada (1992)Google Scholar
  8. 8.
    Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley-Interscience (1983)Google Scholar
  9. 9.
    Clarke, F.H.: Necessary conditions for a general control problem in calculus of variations and control. In: Russel D. (ed.) Mathematics research center, Pub.36. University of Wisconsin, pp. 259–278, New York Academy (1976)Google Scholar
  10. 10.
    Craven B.D., Ralph D. and Glover B.M. (1995). Small convex-valued subdifferentials in mathematical programming. Optimization 32: 1–21 CrossRefGoogle Scholar
  11. 11.
    Dantzig G.B., Folkman J. and Shapiro N. (1967). On the continuity of the minimum set of a continuous function. J. Math. Anal. Appl. 17: 512–548 CrossRefGoogle Scholar
  12. 12.
    Demyanov V.F. and Jeyakumar V. (1997). Hunting for a smaller convex subdifferential. J. Global Optim. 10: 305–326 CrossRefGoogle Scholar
  13. 13.
    Dempe S. (1992). A necessary and a sufficient optimality condition for bilevel programming problems. Optimization 25: 341–354 CrossRefGoogle Scholar
  14. 14.
    Dempe S. (2002). Foundations of Bilevel Programming. Kluwer Academie Publishers, Dordrecht Google Scholar
  15. 15.
    Dempe S. (1997). First-order necessary optimality conditions for general bilevel programming problems. J. Optim. Theory Appl. 95: 735–739 CrossRefGoogle Scholar
  16. 16.
    Dempe S. and Schmidt H. (1996). On an algorithm solving two-level programming problems with nonunique lower level solutions. Comput. Optim. Appl. 6: 227–249 Google Scholar
  17. 17.
    Dien P.H. (1983). Locally Lipschitzian set-valued maps and general extremal problems with inclusion constraints. Acta. Math. Vietnam. 1: 109–122 Google Scholar
  18. 18.
    Dien P.H. (1985). On the regularity condition for the extremal problem under locally Lipschitz inclusion constraints. Appl. Math. Appl. 13: 151–161 Google Scholar
  19. 19.
    Ekeland I. (1974). On the variational principle. J. Math. Anal. Appl. 47: 324–353 CrossRefGoogle Scholar
  20. 20.
    Gadhi N. (2005). Necessary optimality conditions for Lipschitz multiobjective optimization problems. Georgian Math. J. 12: 65–74 Google Scholar
  21. 21.
    Huang H.X. and Pardalos P.M. (2002). A multivariate partition approach to optimization problems. Cybernet. Syst. Anal. 38: 265–275 CrossRefGoogle Scholar
  22. 22.
    Ioffe A.D. (1989). Approximate subdifferential and applications.. III : the metric theory. Mathematika 36: 1–38 Google Scholar
  23. 23.
    Jahn J. (2004). Vector optimization. Springer, Berlin Google Scholar
  24. 24.
    Jahn J. and Rauh R. (1997). Contingent epiderivatives and set-valued optimization. Math. Method. Oper. Res. 46: 193–211 CrossRefGoogle Scholar
  25. 25.
    Jeyakumar V. and Luc D.T. (1998). Approximate Jacobian matrices for continuous maps and C1-Optimization. SIAM J. Control Optim. 36: 1815–1832 CrossRefGoogle Scholar
  26. 26.
    Jeyakumar V., Luc D.T. and Schaible S. (1998). Characterizations of generalized monotone nonsmooth continuous maps using approximate Jacobians. J. Convex Anal. 5: 119–132 Google Scholar
  27. 27.
    Jeyakumar V. and Luc D.T. (1999). Nonsmooth calculus, minimality and monotonicity of convexificators. J. Optim. Theory Appl. 101: 599–621 CrossRefGoogle Scholar
  28. 28.
    Klose J. (1992). Sensitivity analysis using the tangent derivative. Numer. Funct. Anal. Optimiz. 13: 143–153 CrossRefGoogle Scholar
  29. 29.
    Kuroiwa D. (1998). Natural criteria of set-valued optimization, Manuscript. Shimane University, Japan Google Scholar
  30. 30.
    Li Z. (1999). A theorem of the alternative and its application to the optimization of set-valued maps. J. Optim. Theory Appl. 100: 365–375 CrossRefGoogle Scholar
  31. 31.
    Loewen P.D. (1992). Limits of Frechet normals in nonsmooth analysis. Optimization and Nonlinear Analysis. Pitman Research Notes Math, Ser. 244: 178–188 Google Scholar
  32. 32.
    Luc D.T. (1991). Contingent derivatives of set-valued maps and applications to vector optimization. Math. Program. 50: 99–111 CrossRefGoogle Scholar
  33. 33.
    Luc D.T. and Malivert C. (1992). Invex optimization problems. Bull. Austral. Math. Soc. 46: 47–66 CrossRefGoogle Scholar
  34. 34.
    Marti K. (2005). Stochastic Optimization Methods. Springer, Berlin Google Scholar
  35. 35.
    Michel, P., Penot, J-P.: Calcul sous-differentiel pour des fonctions Lipschitziennes et non Lipschitziennes. C.R. Acad. Sc. Paris 298 (1984)Google Scholar
  36. 36.
    Michel P. and Penot J-P. (1992). A generalized derivative for calm and stable functions. Diff. Integral Eq. 5(2): 433–454 Google Scholar
  37. 37.
    Migdalas, A., Pardalos, P.M., Värbrand, P.: Multilevel optimization : algorithms and applications. Kluwer Academic Publishers (1997)Google Scholar
  38. 38.
    Mordukhovich, B.S.: Variational analysis and generalized differentiation. Vol. I, II, Springer Verlag, Berlin et al. (2006)Google Scholar
  39. 39.
    Mordukhovich B.S. and Shao Y. (1995). On nonconvex subdifferential calculus in Banach spaces. J. Convex Anal. 2: 211–228 Google Scholar
  40. 40.
    Outrata J.V. (1993). On necessary optimality conditions for Stackelberg problems. J. Optim. Theory Appl. 76: 306–320 CrossRefGoogle Scholar
  41. 41.
    Outrata J.V. (1993). Optimality problems with variational inequality constraints. SIAM J. Optim. 4: 340–357 CrossRefGoogle Scholar
  42. 42.
    Penot J.P. (1988). On the mean-value theorem. Optimization 19: 147–156 CrossRefGoogle Scholar
  43. 43.
    Ralph D. and Dempe S. (1995). Directional derivatives of the solution of a parametric nonlinear program. Math. Program. 70: 159–172 Google Scholar
  44. 44.
    Thibault L. (1991). On subdifferentials of optimal value functions. SIAM J. Control Optim. 29: 1019–1036 CrossRefGoogle Scholar
  45. 45.
    Treiman J.S. (1995). The linear nonconvex generalized gradient and Lagrange multipliers. SIAM J. Optim. 5: 670–680 CrossRefGoogle Scholar
  46. 46.
    Wang S., Wang Q. and Romano-Rodriguez S. (1993). Optimality conditions and an algorithm for linear-quadratic bilevel programs. Optimization 4: 521–536 Google Scholar
  47. 47.
    Ye J.J. and Zhu D.L. (1995). Optimality conditions for bilevel programming problems. Optimization 33: 9–27 CrossRefGoogle Scholar
  48. 48.
    Ye, J.J.: Constraint qualifications and KKT conditions for bilevel programming problems. Math. Oper. Res. (2007, to appear)Google Scholar
  49. 49.
    Zhang R. (1993). Problems of hierarchical optimization in finite dimensions. SIAM J. Optim. 4: 521–536 CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  1. 1.Department of Mathematics and Computers SciencesTechnical University Bergakademie FreibergFreibergGermany
  2. 2.Department of MathematicsSidi Mohamed Ben Abdellah UniversityAtlasMarokko

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