Bilevel Optimization: Reformulation and First Optimality Conditions

Chapter
Part of the Forum for Interdisciplinary Mathematics book series (FFIM)

Abstract

Bilevel optimization problems are nonsmooth, nonconvex optimization problems the feasible set of which is (in part) described using the graph of the solution set mapping of a second parametric optimization problem. To investigate them, their transformation into a one-level optimization problem is necessary. For that, different approaches can be used. Two of them are considered in this article: the transformation using the Karush–Kuhn–Tucker conditions of the (convex) lower level problem resulting in a mathematical program with equilibrium constraint (MPEC) and the use of the optimal value function of this problem which leads to a nonsmooth optimization problem. Besides the resulting necessary optimality conditions, first solution algorithms for the bilevel problem using these transformations are presented.

References

  1. 1.
    Bank, B., Guddat, J., Klatte, D., Kummer, B., Tammer, K.: Non-linear Parametric Optimization. Birkhäuser Verlag, Basel, Boston, Stuttgart (1983)MATHGoogle Scholar
  2. 2.
    Bard, J.F.: Practical bilevel optimization: algorithms and applications. Kluwer Academie Publishers, Dordrecht (1998)CrossRefMATHGoogle Scholar
  3. 3.
    Bazaraa, M.S., Sherali, H.D., Shetty, C.M.: Nonlinear Programming. Theory and Algorithms, vol. xv, 3rd edn. Wiley, Hoboken, NJ (2006) (English)Google Scholar
  4. 4.
    Bector, C.R., Chandra, S., Dutta, J.: Principles of Optimization Theory. Alpha Science, UK (2005)Google Scholar
  5. 5.
    Beer, K.: Lösung großer linearer Optimierungsaufgaben. Deutscher Verlag der Wissenschaften, Berlin (1977)MATHGoogle Scholar
  6. 6.
    Bracken, J., McGill, J.: Mathematical programs with optimization problems in the constraints. Oper. Res. 21, 37–44 (1973)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)MATHGoogle Scholar
  8. 8.
    Dempe, S.: A necessary and a sufficient optimality condition for bilevel programming problems. Optimization 25, 341–354 (1992)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Dempe, S.: A bundle algorithm applied to bilevel programming problems with non-unique lower level solutions. Comput. Optim. Appl. 15, 145–166 (2000)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Dempe, S.: Foundations of Bilevel Programming. Kluwer Academie Publishers, Dordrecht (2002)MATHGoogle Scholar
  11. 11.
    Dempe, S.: Annotated bibliography on bilevel programming and mathematical programs with equilibrium constraints. Optimization 52, 333–359 (2003)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Dempe, S., Dutta, J.: Is bilevel programming a special case of a mathematical program with complementarity constraints? Math. Program. 131, 37–48 (2012)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Dempe, S., Dutta, J., Mordukhovich, B.S.: New necessary optimality conditions in optimistic bilevel programming. Optimization 56, 577–604 (2007)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Dempe, S., Franke, S.: Solution algorithm for an optimistic linear stackelberg problem. Comput. Oper. Res. 41, 277–281 (2014)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Dempe, S., Franke, S.: On the solution of convex bilevel optimization problems. Comput. Optim. Appl. 63, 685–703 (2016)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Dempe, S., Kalashnikov, V., Pérez-Valdés, G.A., Kalashnykova, N.: Bilevel Programming Problems—Theory. Algorithms and Applications to Energy Networks. Springer, Heidelberg (2015)Google Scholar
  17. 17.
    Dempe, S., Mordukhovich, B.S., Zemkoho, A.B.: Sensitivity analysis for two-level value functions with applications to bilevel programming. SIAM J. Optim. 22, 1309–1343 (2012)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Dempe, S., Mordukhovich, B.S., Zemkoho, A.B.: Necessary optimality conditions in pessimistic bilevel programming. Optimization 63(4), 505–533 (2014)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Dempe, S., Schmidt, H.: On an algorithm solving two-level programming problems with nonunique lower level solutions. Comput. Opt. Appl. 6, 227–249 (1996)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Dempe, S., Zemkoho, A.B.: On the Karush-Kuhn-Tucker reformulation of the bilevel optimization problem. Nonlinear Anal. Theory Methods Appl. 75, 1202–1218 (2012)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    DeNegre, S.T., Ralphs, T.K.: A branch-and-cut algorithm for integer bilevel linear programs. In: Chinneck, J.W., Kristjansson, B., Saltzman, M.J. (eds.) Operations Research and Cyber-Infrastructure. Operations Research/Computer Science Interfaces, vol. 47, pp. 65–78. Springer, USA (2009)CrossRefGoogle Scholar
  22. 22.
    Franke, S.: The bilevel programming problem: optimal value and Karush-Kuhn-Tucker reformulation. Ph.D. thesis, TU Bergakademie Freiberg (2014)Google Scholar
  23. 23.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman and Co., San Francisco (1979)MATHGoogle Scholar
  24. 24.
    Gauvin, J., Dubeau, F.: Differential properties of the marginal function in mathematical programming. Math. Program. Study 19, 101–119 (1982)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Hoheisel, T., Kanzow, C., Schwartz, A.: Theoretical and numerical comparison of relaxation methods for mathematical programs with complementarity constraints. Math. Program. 137(1–2), 257–288 (2013)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Jongen, H.Th: Weber, G.-W: Nonlinear optimization: characterization of structural optimization. J. Glob. Optim. 1, 47–64 (1991)Google Scholar
  27. 27.
    Kalashnykova, N.I., Kalashnikov, V.V., Dempe, S., Franco, A.A.: Application of a heuristic algorithm to mixed-integer bi-level programming problems. Int. J. Innovative Compu. Inf. Control 7(4), 1819–1829 (2011)Google Scholar
  28. 28.
    Klatte, D., Kummer, B., Stability properties of infima and optimal solutions of parametric optimization problems. In: Demyanov, V.F. (ed.) Nondifferentiable Optimization: Motivations and Applications, Proceedings of the IIASA Workshop, Sopron, 1984. Lecture Notes in Economics and Mathematical Systems, vol. 255, pp. 215–229. Springer, Berlin (1984)Google Scholar
  29. 29.
    Kleniati, P.-M., Adjiman, C.S.: Branch-and-sandwich: a deterministic global optimization algorithm for optimistic bilevel programming problems. Part II: convergence analysis and numerical results. J. Glob. Optim. 60(3), 459–481 (2014)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Kojima, M.: Strongly stable stationary solutions in nonlinear programs. In: Robinson, S.M. (ed.) Analysis and Computation of Fixed Points, pp. 93–138. Academic Press, New York (1980)CrossRefGoogle Scholar
  31. 31.
    Kummer, B.: Newton’s method for non-differentiable functions. Advances in Mathematical Optimization, Mathematical Research, vol. 45. Akademie-Verlag, Berlin (1988)Google Scholar
  32. 32.
    Loridan, P., Morgan, J.: Weak via strong Stackelberg problem: new results. J. Glob. Optim. 8, 263–287 (1996)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Mersha, A.G.: Solution methods for bilevel programming problems. Ph.D. thesis, TU Bergakademie Freiberg (2008)Google Scholar
  34. 34.
    Mersha, A.G., Dempe, S.: Feasible direction method for bilevel programming problem. Optimization 61(4–6), 597–616 (2012)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Minchenko, L., Stakhovski, S.: Parametric nonlinear programming problems under the relaxed constant rank condition. SIAM J. Optim. 21(1), 314332 (2011)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Mirrlees, J.A.: The theory of moral hazard and unobservable behaviour: part I. Rev. Econ. Stud. 66, 3–21 (1999)CrossRefMATHGoogle Scholar
  37. 37.
    Mitsos, A., Chachuat, B., Barton, P.I.: Towards global bilevel dynamic optimization. J. Glob. Optim. 45(1), 63–93 (2009)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Mordukhovich, B.S., Nam, N.M., Yen, N.D.: Subgradients of marginal functions in parametric mathematical programming. Math. Program. 116, 369–396 (2009)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Nožička, E., Guddat, J., Hollatz, H., Bank, B.: Theorie der linearen parametrischen Optimierung. Akademie-Verlag, Berlin (1974)MATHGoogle Scholar
  40. 40.
    Outrata, J., Kočvara, M., Zowe, J.: Nonsmooth Approach to Optimization Problems with Equilibrium Constraints. Kluwer Academic Publishers, Dordrecht (1998)CrossRefMATHGoogle Scholar
  41. 41.
    Pilecka, M.: Combined reformulation of bilevel programming problems. Master’s thesis, TU Bergakademie Freiberg, Fakultät für Mathematrik und Informatik (2011)Google Scholar
  42. 42.
    Ralph, D., Dempe, S.: Directional derivatives of the solution of a parametric nonlinear program. Math. Program. 70, 159–172 (1995)MathSciNetMATHGoogle Scholar
  43. 43.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)CrossRefMATHGoogle Scholar
  44. 44.
    Scheel, H., Scholtes, S.: Mathematical programs with equilibrium constraints: stationarity, optimality, and sensitivity. Math. Oper. Res. 25, 1–22 (2000)MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Scholtes, S.: Convergence properties of a regularization scheme for mathematical programs with complementarity constraints. SIAM J. Optim. 11, 918–936 (2001)MathSciNetCrossRefMATHGoogle Scholar
  46. 46.
    Scholtes, S.: Introduction to Piecewise Differentiable Equations. Springer, New York (2012)CrossRefMATHGoogle Scholar
  47. 47.
    Stackelberg, H.V.: Marktform und Gleichgewicht [english translation: The Theory of the Market Economy]. Springer, Berlin (1934), Oxford University Press (1952)Google Scholar
  48. 48.
    Topkis, D.M., Veinott, A.F.: On the convergence of some feasible direction algorithms for nonlinear programming. SIAM J. Control 5, 268–279 (1967)MathSciNetCrossRefMATHGoogle Scholar
  49. 49.
    Ye, J.J., Zhu, D.L.: Optimality conditions for bilevel programming problems. Optimization 33, 9–27 (1995)MathSciNetCrossRefMATHGoogle Scholar
  50. 50.
    Zemkoho, A.B.: Bilevel programming: Reformulations, regularity, and stationarity. Ph.D. thesis, TU Bergakademie Freiberg (2012)Google Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2017

Authors and Affiliations

  1. 1.TU Bergakademie FreibergFreibergGermany

Personalised recommendations