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Optimization with Multivalued Mappings pp 265–276Cite as

First and second order optimality conditions in set optimization

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Part of the Springer Optimization and Its Applications book series (SOIA,volume 2)

Summary

By using the second order asymptotic cone two epiderivatives for set-valued maps are proposed and employed to obtain second order necessary optimality conditions in set optimization. These conditions extend some known results in optimization.

Key words

  • Set optimization
  • weak minimality
  • second order epiderivatives
  • second order asymptotic cone
  • optimality conditions

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References

  1. Aubin, J.P. and H. Frankowska, Set valued analysis, Birkhäuser, Boston (1990).

    MATH  Google Scholar 

  2. Bigi, G. and Castellani, M: Second order optimality conditions for differentiable multiobjective problems, RAIRO Oper. Res., 34, 411–426 (2000).

    CrossRef  MATH  MathSciNet  Google Scholar 

  3. Cambini, A. and Martein, L.: First and second order optimality conditions in vector optimization, Preprint, Dept. Stat. Appl. Math., Univ. Pisa, 2000.

    Google Scholar 

  4. Cambini, A., Martein, L. and Vlach, M.: Second order tangent sets and optimality conditions, Math. Japonica, 49, 451–461 (1999).

    MATH  MathSciNet  Google Scholar 

  5. Castellani, M. and Pappalardo, M.: Local second-order approximations and applications in optimization, Optimization, 37, 305–321 (1996).

    MATH  MathSciNet  Google Scholar 

  6. Chen, G.Y. and Jahn, J.: Optimality conditions for set-valued optimization problems, Math. Meth. Oper. Res., 48, 187–200 (1998).

    CrossRef  MATH  MathSciNet  Google Scholar 

  7. Corley, H.W.: Optimality conditions for maximization of set-valued functions, J. Optim. Theory Appl., 58, 1–10 (1988).

    CrossRef  MATH  MathSciNet  Google Scholar 

  8. El Abdouni, B. and Thibault, L.: Optimality conditions for problems with set-valued objectives, J. Appl. Anal., 2, 183–201 (1996).

    CrossRef  MATH  MathSciNet  Google Scholar 

  9. Giannessi, F.: Vector variational inequalities and vector equilibria, Kluwer Academic Publishers, Dordrecht, 2000.

    MATH  Google Scholar 

  10. Götz, A. and Jahn, J.: The Lagrange multiplier rule in set-valued optimization, SIAM J. Optim., 10, 331–344 (1999).

    CrossRef  MATH  Google Scholar 

  11. Isac, G.: Sur l’existence de l’optimum de Pareto, Riv. Mat. Univ. Parma (4) 9, 303–325 (1984).

    MathSciNet  Google Scholar 

  12. Isac, G. and Khan, A.A.: Dubovitski-Milutin approach in set-valued optimization. Submitted to: SIAM J. Cont. Optim. (2004).

    Google Scholar 

  13. Jahn, J.: Vector optimization. Theory, applications, and extensions. Springer-Verlag, Berlin, 2004.

    MATH  Google Scholar 

  14. Jahn, J. and Khan, A.A.: Generalized contingent epiderivatives in set-valued optimization, Numer. Func. Anal. Optim., 27, 807–831 (2002).

    CrossRef  MathSciNet  Google Scholar 

  15. Jahn, J. and Khan, A.A.: Existence theorems and characterizations of generalized contingent epiderivatives, J. Nonl. Convex Anal., 3, 315–330 (2002).

    MATH  MathSciNet  Google Scholar 

  16. Jahn, J; Khan, A.A and Zeilinger, P.: Second order optimality conditions in set-valued optimization, J. Optim. Theory Appl., 125, 331–347 (2005).

    CrossRef  MATH  MathSciNet  Google Scholar 

  17. Jahn, J. and Rauh, R.: Contingent epiderivatives and set-valued optimization, Math. Meth. Oper. Res., 46, 193–211 (1997).

    CrossRef  MATH  MathSciNet  Google Scholar 

  18. Luc, D.T. and Malivert, C.: Invex optimization problems, Bull. Austral. Math. Soc., 46, 47–66 (1992).

    MATH  MathSciNet  Google Scholar 

  19. Jiménez, B. and Novo, V: Second order necessary conditions in set constrained differentiable vector optimization, Math. Meth. Oper. Res., 58, 299–317 (2003).

    CrossRef  MATH  Google Scholar 

  20. Khan, A.A. and Raciti, F.: A multiplier rule in set-valued optimization, Bull. Austral. Math. Soc., 68, 93–100 (2003).

    CrossRef  MATH  MathSciNet  Google Scholar 

  21. Mordukhovich, B.S. and Outrata, J.V.: On second-order subdifferentials and their applications, SIAM J. Optim., 12, 139–169 (2001).

    CrossRef  MATH  MathSciNet  Google Scholar 

  22. Penot, J.P.: Differentiability of relations and differential stability of perturbed optimization problems, SIAM J. Cont. Optim., 22, 529–551 (1984).

    CrossRef  MATH  MathSciNet  Google Scholar 

  23. Penot, J.P.: Second-order conditions for optimization problems with constraints, SIAM J. Cont. Optim., 37, 303–318 (1999).

    CrossRef  MATH  MathSciNet  Google Scholar 

  24. Rockafellar, R.T. and Wets, J.B.: Variational analysis, Springer, Berlin (1997).

    MATH  Google Scholar 

  25. Ward, D.: Calculus for parabolic second-order derivatives, Set-Valued Anal., 1, 213–246 (1993).

    CrossRef  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Departamento de Ingenieria Industrial y de Sistemas, Centro de Calidad ITESM, Campus Monterrey Ave. Eugenio Garza Sada 2501, Sur Monterrey N.L., Mexico, 64849

    V. Kalashnikov

  2. Department of Mathematical Sciences, Michigan Technological University, 1400 Townsend Drive, Houghton, MI, 49931-1295, USA

    B. Jadamba

  3. Department of Mathematics, University of Wisconsin, 1800 College Drive, Rice Lake, WI, 54868, USA

    A. A. Khan

Authors
  1. V. Kalashnikov
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  2. B. Jadamba
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  3. A. A. Khan
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Editor information

Editors and Affiliations

  1. TU Bergakademie Freiberg, Germany

    Stephan Dempe

  2. ITESM, Monterrey, Mexico

    Vyacheslav Kalashnikov

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Kalashnikov, V., Jadamba, B., Khan, A.A. (2006). First and second order optimality conditions in set optimization. In: Dempe, S., Kalashnikov, V. (eds) Optimization with Multivalued Mappings. Springer Optimization and Its Applications, vol 2. Springer, Boston, MA . https://doi.org/10.1007/0-387-34221-4_13

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