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New Optimality Conditions for the Semivectorial Bilevel Optimization Problem

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Abstract

The paper is concerned with the optimistic formulation of a bilevel optimization problem with multiobjective lower-level problem. Considering the scalarization approach for the multiobjective program, we transform our problem into a scalar-objective optimization problem with inequality constraints by means of the well-known optimal value reformulation. Completely detailed first-order necessary optimality conditions are then derived in the smooth and nonsmooth settings while using the generalized differentiation calculus of Mordukhovich. Our approach is different from the one previously used in the literature and the conditions obtained are new. Furthermore, they reduce to those of a usual bilevel program, if the lower-level objective function becomes single-valued.

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References

  1. Bonnel, H., Morgan, J.: Semivectorial bilevel optimization problem: penalty approach. J. Optim. Theory Appl. 131, 365–382 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  2. Zheng, Y., Wan, Z.: A solution method for semivectorial bilevel programming problem via penalty method. J. Appl. Math. Comput. 37, 207–219 (2011)

    Article  MathSciNet  Google Scholar 

  3. Eichfelder, G.: Multiobjective bilevel optimization. Math. Program. 123, 419–449 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bonnel, H.: Optimality conditions for the semivectorial bilevel optimization problem. Pac. J. Optim. 2, 447–467 (2006)

    MathSciNet  MATH  Google Scholar 

  5. Dempe, S., Dutta, J., Mordukhovich, B.S.: New necessary optimality conditions in optimistic bilevel programming. Optimization 56, 577–604 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ehrgott, M.: Multicriteria Optimization. Springer, Berlin (2005)

    MATH  Google Scholar 

  7. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation. I: Basic Theory. II: Applications. Springer, Berlin (2006)

    Google Scholar 

  8. Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998)

    Book  MATH  Google Scholar 

  9. Henrion, R., Outrata, J.V.: A subdifferential condition for calmness of multifunctions. J. Math. Anal. Appl. 258, 110–130 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  10. Outrata, J.V.: A note on the usage of nondifferentiable exact penalties in some special optimization problems. Kybernetika 24, 251–258 (1988)

    MathSciNet  MATH  Google Scholar 

  11. Dempe, S., Zemkoho, A.B.: On the Karush–Kuhn–Tucker reformulation of the bilevel optimization problem. Nonlinear Anal. 75, 1202–1218 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. Mordukhovich, B.S., Outrata, J.V.: Coderivative analysis of quasi-variational inequalities with applications to stability and optimization. SIAM J. Optim. 18, 389–412 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  13. Ye, J.J., Ye, X.Y.: Necessary optimality conditions for optimization problems with variational inequality constraints. Math. Oper. Res. 22, 977–997 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  14. Bonnel, H., Morgan, J.: Semivectorial bilevel convex optimal control problems: an existence result. Working paper

  15. Dempe, S., Zemkoho, A.B.: The generalized Mangasarian–Fromowitz constraint qualification and optimality conditions for bilevel programs. J. Optim. Theory Appl. 148, 433–441 (2011)

    Article  MathSciNet  Google Scholar 

  16. Mordukhovich, B.S., Nam, M.N., Yen, N.D.: Subgradients of marginal functions in parametric mathematical programming. Math. Program. 116, 369–396 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Mordukhovich, B.S., Nam, M.N., Phan, H.M.: Variational analysis of marginal function with applications to bilevel programming problems. J. Optim. Theory Appl. 152, 557–586 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  18. Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1984)

    Google Scholar 

  19. Ye, J.J., Zhu, D.L.: Optimality conditions for bilevel programming problems. Optimization 33, 9–27 (1995). With Erratum in Optimization 39, 361–366 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  20. Dempe, S., Zemkoho, A.B.: The bilevel programming problem: reformulations, constraint qualifications and optimality conditions. Math. Program. (2012). doi:10.1007/s10107-011-0508-5

    Google Scholar 

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Correspondence to S. Dempe.

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Communicated by Patrice Marcotte.

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Dempe, S., Gadhi, N. & Zemkoho, A.B. New Optimality Conditions for the Semivectorial Bilevel Optimization Problem. J Optim Theory Appl 157, 54–74 (2013). https://doi.org/10.1007/s10957-012-0161-z

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  • DOI: https://doi.org/10.1007/s10957-012-0161-z

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