Journal of Optimization Theory and Applications

, Volume 159, Issue 2, pp 547–553 | Cite as

Duality and Weak Efficiency in Vector Variational Problems

  • M. Arana Jiménez
  • F. Ortegón Gallego


We establish weak, strong, and converse duality results for weakly efficient solutions in vector or multiobjective variational problems, which extend and improve recent papers. For this purpose, we consider Kuhn–Tucker optimality conditions, weighting variational problems, and some classes of generalized convex functions, recently introduced, which are extended in this work. Furthermore, a related open question is discussed.


Variational problem Duality Pseudoinvexity Weakly efficient solution Critical point 



This work was partially supported by Ministerio de Economía y Competitividad, under grants MTM2010-15383 and MTM2010-16401 with the participation of FEDER, and Consejería de Educación y Ciencia de la Junta de Andalucía, research groups FQM-243 and FQM-315.


  1. 1.
    Pereira, F.L.: Control design for autonomous vehicles: a dynamic optimization perspective. Eur. J. Control 7, 178–202 (2001) CrossRefGoogle Scholar
  2. 2.
    Pereira, F.L.: A maximum principle for impulsive control problems with state constraints. Commun. Appl. Math. 19, 1–19 (2000) Google Scholar
  3. 3.
    Hanson, M.A.: Bounds for functionally convex optimal control problems. J. Math. Anal. Appl. 8, 84–89 (1964) MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Chinchuluun, A., Pardalos, P.M.: A survey of recent developments in multiobjective optimization. Ann. Oper. Res. 154, 29–50 (2007) MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Arana, M., Ruiz, G., Rufian, A. (eds.): Optimality Conditions in Vector Optimization. Bentham Science Publishers, Bussum (2010) Google Scholar
  6. 6.
    Floudas, C.A., Pardalos, P.M. (eds.): Encyclopedia of Optimization. Springer, Berlin (2009) MATHGoogle Scholar
  7. 7.
    Carosi, L., Martein, L. (eds.): Recent Developments on Mathematical Programming and Applications. ARACNE editrice S.r.l, Roma (2009) Google Scholar
  8. 8.
    Ahmad, I., Gualti, T.R.: Mixed type duality for multiobjective variational problems with generalized (F,ρ)-convexity. J. Math. Anal. Appl. 306, 669–683 (2005) MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Liang, Z.A., Huang, H.X., Pardalos, P.M.: Efficiency conditions and duality for a class of multiobjective fractional programming problems. J. Glob. Optim. 27, 447–471 (2003) MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Arana, M., Rufián, A., Osuna, R., Ruiz, G.: Pseudoinvexity, optimality conditions and efficiency in multiobjective problems; duality. Nonlinear Anal. 68, 24–34 (2008) MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Arana, M., Osuna, R., Ruiz, G., Rojas, M.: On variational problems: characterization of solutions and duality. J. Math. Anal. Appl. 311, 1–12 (2005) MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Arana, M., Ruiz, G., Rufián, A., Osuna, R.: Weak efficiency in multiobjective variational problems under generalized convexity. J. Glob. Optim. 52, 109–121 (2012) CrossRefMATHGoogle Scholar
  13. 13.
    Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, New York (2011) MATHGoogle Scholar
  14. 14.
    Bector, C.R., Chandra, S., Husain, I.: Generalized concavity and duality in continuous programming. Util. Math. 25, 171–190 (1984) MathSciNetMATHGoogle Scholar
  15. 15.
    Hanson, M.A.: On sufficiency of Kuhn–Tucker conditions. J. Math. Anal. Appl. 80, 545–550 (1981) MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Mond, B., Husain, I.: Sufficient optimality criteria and duality for variational problems with generalized invexity. J. Aust. Math. Soc. Ser. B, Appl. Math 31, 108–121 (1989) MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Mond, B., Smart, I.: Duality and sufficiency in control problems with invexity. J. Math. Anal. Appl. 136, 325–333 (1988) MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983) MATHGoogle Scholar
  19. 19.
    Geofrion, A.M.: Proper efficiency and the theory of vector maximization. J. Math. Anal. Appl. 22, 618–630 (1968) MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Statistics and Operational Research, Faculty of SSCC and CommunicationUniversity of CádizJerez de la FronteraSpain
  2. 2.Department of Mathematics, Faculty of SciencesUniversity of CádizPuerto RealSpain

Personalised recommendations