Conjugates and Subdifferentials

  • Akhtar A. Khan
  • Christiane Tammer
  • Constantin Zălinescu
Part of the Vector Optimization book series (VECTOROPT)


To each type of efficiency for optimization problems it is possible to associate notions of conjugate and subdifferential for vector valued functions or set-valued maps. In this chapter we study the conjugate and the subdifferential corresponding to the strong efficiency as well as the subdifferentials corresponding to the weak and Henig type efficiencies. For the strong conjugate and subdifferential we establish similar results to those in the convex scalar case, while for the other types of subdifferential we establish formulas for the subdifferentials of the sum and the composition of functions and set-valued maps.


  1. 76.
    Bourbaki, N.: Eléments de mathématique. XVIII. Première partie: Les structures fondamentales de l’analyse. Livre V: Espaces vectoriels topologiques. Chapitre III: Espaces d’applications linéaires continues. Chapitre IV: La dualité dans les espaces vectoriels topologiques. Chapitre V: Espaces hilbertiens. Actualités Sci. Ind., no. 1229. Hermann & Cie, Paris (1955)Google Scholar
  2. 91.
    Chen, G.Y., Huang, X., Yang, X.: Vector optimization. Lecture Notes in Economics and Mathematical Systems, vol. 541. Springer, Berlin (2005)Google Scholar
  3. 92.
    Chen, G.Y., Jahn, J.: Optimality conditions for set-valued optimization problems. Math. Methods Oper. Res. 48(2), 187–200 (1998)MathSciNetCrossRefMATHGoogle Scholar
  4. 102.
    Combari, C., Laghdir, M., Thibault, L.: Sous-différentiels de fonctions convexes composées. Ann. Sci. Math. Québec 18(2), 119–148 (1994)MathSciNetMATHGoogle Scholar
  5. 174.
    El Maghri, M.: Pareto-Fenchel ε-subdifferential sum rule and ε-efficiency. Optim. Lett. 6(4), 763–781 (2012)MathSciNetCrossRefMATHGoogle Scholar
  6. 175.
    El Maghri, M., Laghdir, M.: Pareto subdifferential calculus for convex vector mappings and applications to vector optimization. SIAM J. Optim. 19(4), 1970–1994 (2008)MathSciNetCrossRefGoogle Scholar
  7. 358.
    Kusraev, A.G., Kutateladze, S.S.: Subdifferentials: theory and applications. In: Mathematics and Its Applications, vol. 323. Kluwer Academic, Dordrecht (1995). Translated from the RussianGoogle Scholar
  8. 359.
    Laghdir, M.: Some remarks on subdifferentiability of convex functions. Appl. Math. E-Notes 5, 150–156 (electronic) (2005)Google Scholar
  9. 386.
    Li, T., Xu, Y.: The strictly efficient subgradient of set-valued optimisation. Bull. Aust. Math. Soc. 75(3), 361–371 (2007)CrossRefMATHGoogle Scholar
  10. 389.
    Lin, L.J.: Optimization of set-valued functions. J. Math. Anal. Appl. 186(1), 30–51 (1994)MathSciNetCrossRefMATHGoogle Scholar
  11. 485.
    Raffin, C.: Sur les programmes convexes définis dans des espaces vectoriels topologiques. Ann. Inst. Fourier (Grenoble) 20(fasc. 1), 457–491 (1970)Google Scholar
  12. 533.
    Silverman, R.J., Yen, T.: The Hahn-Banach theorem and the least upper bound property. Trans. Am. Math. Soc. 90, 523–526 (1959)MathSciNetCrossRefMATHGoogle Scholar
  13. 535.
    Song, W.: Conjugate duality in set-valued vector optimization. J. Math. Anal. Appl. 216(1), 265–283 (1997)MathSciNetCrossRefMATHGoogle Scholar
  14. 537.
    Song, W.: Characterizations of some remarkable classes of cones. J. Math. Anal. Appl. 279(1), 308–316 (2003)MathSciNetCrossRefMATHGoogle Scholar
  15. 549.
    Taa, A.: Subdifferentials of multifunctions and Lagrange multipliers for multiobjective optimization. J. Math. Anal. Appl. 283(2), 398–415 (2003)MathSciNetCrossRefMATHGoogle Scholar
  16. 550.
    Taa, A.: ε-subdifferentials of set-valued maps and ε-weak Pareto optimality for multiobjective optimization. Math. Methods Oper. Res. 62(2), 187–209 (2005)Google Scholar
  17. 551.
    Taa, A.: On subdifferential calculus for set-valued mappings and optimality conditions. Nonlinear Anal. 74(18), 7312–7324 (2011)MathSciNetCrossRefMATHGoogle Scholar
  18. 566.
    Tanino, T.: Conjugate duality in vector optimization. J. Math. Anal. Appl. 167(1), 84–97 (1992)MathSciNetCrossRefMATHGoogle Scholar
  19. 569.
    Tanino, T., Sawaragi, Y.: Conjugate maps and duality in multiobjective optimization. J. Optim. Theory Appl. 31(4), 473–499 (1980)MathSciNetCrossRefMATHGoogle Scholar
  20. 576.
    Ursescu, C.: Multifunctions with convex closed graph. Czechoslovak Math. J. 25(100)(3), 438–441 (1975)Google Scholar
  21. 580.
    Valadier, M.: Sous-différentiabilité de fonctions convexes à valeurs dans un espace vectoriel ordonné. Math. Scand. 30, 65–74 (1972)MathSciNetMATHGoogle Scholar
  22. 609.
    Zălinescu, C.: The Fenchel–Rockafellar duality theory for mathematical programming in oreder-complete vector lattices and applications. Tech. Rep. 45, INCREST, Bucharest (1980)Google Scholar
  23. 610.
    Zălinescu, C.: Duality for vectorial nonconvex optimization by convexification and applications. An. Ştiinţ. Univ. Al. I. Cuza Iaşi Secţ. I a Mat. 29(3, suppl.), 15–34 (1983)Google Scholar
  24. 611.
    Zălinescu, C.: Optimality conditions in infinite dimensional convex programming (Romanian). Ph.D. thesis, University Alexandru Ioan Cuza Iasi, Iasi, Romania (1983)Google Scholar
  25. 614.
    Zălinescu, C.: Convex Analysis in General Vector Spaces. World Scientific, River Edge (2002)CrossRefMATHGoogle Scholar
  26. 616.
    Duality for vectorial convex optimization, conjugate operators and subdifferentials. The continuous case. Illmenau University, Eisenach (1984), Internationale Tagung Mathematische Optimierung – Theorie und AnwendungenGoogle Scholar
  27. 623.
    Zhou, Z.A., Yang, X.M., Peng, J.W.: ε-strict subdifferentials of set-valued maps and optimality conditions. Nonlinear Anal. 75(9), 3761–3775 (2012)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Akhtar A. Khan
    • 1
  • Christiane Tammer
    • 2
  • Constantin Zălinescu
    • 3
  1. 1.Rochester Institute of Technology School of Mathematical SciencesRochesterUSA
  2. 2.HalleGermany
  3. 3.Faculty of MathematicsUniversity “Al. I. Cuza” IasiIasiRomania

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