Lagrange Multipliers in Convex Set Optimization with the Set and Vector Criteria


In the setting of normed spaces ordered by a convex cone, we define a set-valued optimization problem where the feasible set is given by a cone-constraint and we derive Lagrange optimality conditions for weakly efficient solutions of the problem following both the set and the vector solution criterion. For the set criterion, a lower set less preorder is considered. The optimality conditions are expressed in terms of coderivatives and are obtained through scalarization, by using a nonlinear scalarizing functional based on the oriented distance.

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  1. 1.

    Amahroq, T., Oussarhan, A., Syam, A.: On Lagrange multiplier rules for set-valued optimization problems in the sense of set criterion. Numer. Funct. Anal. Optim. 41, 710–729 (2020)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Araya, Y.: Four types of nonlinear scalarizations and some applications in set optimization. Nonlinear Anal. 75, 3821–3835 (2012)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Aubin, J.-P., Ekeland, I.: Applied Nonlinear Analysis. Wiley, New York (1984)

    Google Scholar 

  4. 4.

    Bao, T.Q., Mordukhovich, B.S.: Relative Pareto minimizers to multiobjective problems: existence and optimality conditions. Math. Program. 122, 301–347 (2010)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Bouza, G., Quintana, E., Tammer, C.: A unified characterization of nonlinear scalarizing functionals in optimization. Vietnam J. Math. 47, 683–713 (2019)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Crespi, G. P., Ginchev, I., Rocca, M.: First-order optimality conditions in set-valued optimization. Math. Methods Oper. Res. 63, 87–106 (2006)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Göpfert, A., Riahi, H., Tammer, C., Zălinescu, C.: Variational Methods in Partially Ordered Spaces. CMS Books in Mathematics. Springer, New York (2003)

    Google Scholar 

  8. 8.

    Dutta, J., Vetrivel, V.: On approximate minima in vector optimization. Numer. Funct. Anal. Optim. 22, 845–859 (2001)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Ha, T. X. D.: Lagrange multipliers for set-valued optimization problems associated with coderivatives. J. Math. Anal. Appl. 311, 647–663 (2005)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Ha, T.X.D.: Estimates of error bounds for some sets of efficient solutions of a set-valued optimization problem. In: Hamel, A., Heyde, F., Löhne, A., Rudloff, B., Schrage, C (eds.) Set Optimization and Applications - The State of the Art. Springer Proceedings in Mathematics & Statistics, vol. 151, pp 249–273. Springer, Berlin (2015)

  11. 11.

    Ha, T.X.D.: A Hausdorff-type distance, a directional derivative of a set-valued map and applications in set optimization. Optimization 67, 1031–1050 (2018)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Ha, T.X.D.: A new concept of slope for set-valued maps and applications in set optimization studied with Kuroiwa’s set approach. Math. Methods Oper. Res. 91, 137–158 (2020)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Hernández, E., López, R.: Some useful set-valued maps in set optimization. Optimization 66, 1273–1289 (2017)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Hernández, E., Rodríguez-Marín, L.: Nonconvex scalarization in set optimization with set-valued maps. J. Math. Anal. Appl. 325, 1–18 (2007)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Hiriart-Urruty, J.-B.: Tangent cones, generalized gradients and mathematical programming in Banach spaces. Math. Oper. Res. 4, 79–97 (1979)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Hiriart-Urruty, J.-B.: New Concepts in Nondifferentiable Programming. Bull. Soc. Mat. France Mém. 60, 57–85 (1979)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Hiriart-Urruty, J.-B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms, vol. I. Springer, Berlin (1996)

    Google Scholar 

  18. 18.

    Huerga, L., Jiménez, B., Novo, V., Vílchez, A.: Six set scalarizations based on the oriented distance: continuity, convexity and application to convex set optimization. Submitted (2019)

  19. 19.

    Jahn, J., Ha, T.X.D.: New order relations in set optimization. J. Optim. Theory Appl. 148, 209–236 (2011)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Jiménez, B., Novo, V., Vílchez, A.: A set scalarization function based on the oriented distance and relations with other set scalarizations. Optimization 67, 2091–2116 (2018)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Jiménez, B., Novo, V., Vílchez, A.: Characterization of set relations through extensions of the oriented distance. Math. Methods Oper. Res. 91, 89–115 (2020)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Jiménez, B., Novo, V., Vílchez, A.: Six set scalarizations based on the oriented distance: properties and application to set optimization. Optimization 69, 437–470 (2020)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Klein, E., Thompson, A. C.: Theory of Correspondences. Including Applications to Mathematical Economics. Wiley-Interscience/John Wiley and Sons, New York (1984)

    Google Scholar 

  24. 24.

    Kuroiwa, D.: Some criteria in set-valued optimization. Surikaisekikenkyusho Kokyuroku 985, 171–176 (1997)

    MathSciNet  MATH  Google Scholar 

  25. 25.

    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)

    Google Scholar 

  26. 26.

    Schirotzek, W.: Nonsmooth Analysis. Springer, Berlin (2007)

    Google Scholar 

  27. 27.

    Tuan, V. A., Tammer, C., Zălinescu, C.: The Lipschitzianity of convex vector and set-valued functions. TOP 24, 273–299 (2016)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Tikhomirov, V. M.: Analysis II. Convex Analysis and Approximation Theory. Gamkrelidze, R.V (Ed.) Encyclopaedia of Mathematical Sciences, vol. 14. Springer, Berlin (1990)

    Google Scholar 

  29. 29.

    Zaffaroni, A.: Degrees of efficiency and degrees of minimality. SIAM J. Control Optim. 42, 1071–1086 (2003)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Zălinescu, C.: Convex Analysis in General Vector Spaces. World Scientific, Singapore (2002)

    Google Scholar 

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The authors are grateful to the anonymous referees for their useful suggestions and remarks.

This work was partially supported by the Ministerio de Ciencia, Innovación y Universidades (MCIU), Agencia Estatal de Investigación (AEI) (Spain) and Fondo Europeo de Desarrollo Regional (FEDER) under project PGC2018-096899-B-I00 (MCIU/AEI/FEDER, UE).

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Correspondence to V. Novo.

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Dedicated to Marco A. López on the occasion of his 70th birthday.

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Huerga, L., Jiménez, B. & Novo, V. Lagrange Multipliers in Convex Set Optimization with the Set and Vector Criteria. Vietnam J. Math. 48, 345–362 (2020).

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  • Set optimization
  • Lagrange multipliers
  • Lower set preorder relation
  • Scalarization in set optimization
  • Oriented distance

Mathematics Subject Classification (2010)

  • 06A75
  • 49J53
  • 90C29