A unified minimal solution in set optimization

  • KhushbooEmail author
  • C. S. Lalitha


In this paper we extend the notion of minimal solutions for a vector optimization problem considered by Flores-Bazán et al. (J Optim Theory Appl 164:455–478, 2015) to a set-valued optimization problem, with both vector and set solution criteria. Also, we extend the Gerstewitz function proposed by Hernández and Rodríguez-Marín (J Math Anal Appl 325:1–18, 2007) and use it to scalarize minimal solutions with respect to set criterion. We also provide an existence result of minimal solutions with set criterion. Finally, we investigate links between the minimal solutions with respect to vector criterion and set criterion.


Set-valued optimization Nonlinear scalarization Generalized Gerstewitz’s function 

Mathematics Subject Classification

54C60 90C26 90C29 90C31 



The authors wish to thank referee and associate editor for their suggestions which helped to improve the presentation of this article. This research, for the second author, was supported by MATRIC scheme of Department of Science and Technology, India.


  1. 1.
    Araya, Y.: Four types of nonlinear scalarizations and some applications in set optimization. Nonlinear Anal. 75, 3821–3835 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bao, T.Q., Mordukhovich, B.S.: Set-valued optimization in welfare economics. Adv. Math. Econ. 13, 113–153 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Dhingra, M., Lalitha, C.: Set optimization using improvement sets. Yugosl. J. Oper. Res. 27, 153–167 (2017)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Flores-Bazán, F., Hernández, E.: A unified vector optimization problem: complete scalarizations and applications. Optimization 60, 1399–1419 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Flores-Bazán, F., Flores-Bazán, F., Laengle, S.: Characterizing efficiency on infinite-dimensional commodity spaces with ordering cones having possibly empty interior. J. Optim. Theory Appl. 164, 455–478 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Flores-Bazán, F., Gutiérrez, C., Novo, V.: A Brézis-Browder principle on partially ordered spaces and related ordering theorems. J. Math. Anal. Appl. 375, 245–260 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Flores-Bazán, F., Hernández, E., Novo, V.: Characterizing efficiency without linear structure: a unified approach. J. Glob. Optim. 41, 43–60 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Geoffroy, M.H., Marcelin, Y., Nedelcheva, D.: Convergence of relaxed minimizers in set optimization. Optim. Lett. 11, 1677–1690 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Gerth, C., Weidner, P.: Nonconvex separation theorems and some applications in vector optimization. J. Optim. Theory Appl. 67, 297–320 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gerstewitz, C., Iwanow, E.: Dualität für nichtkonvexe Vektoroptimierungsprobleme. Wiss. Z. Tech. Hochsch. Ilmenau 31, 61–81 (1985)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Hamel, A., Löhne, A.: Minimal element theorems and Ekeland’s principle with set relations. J. Nonlinear Convex Anal. 7, 19–37 (2006)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Hamel, A.H., Heyde, F., Rudloff, B.: Set-valued risk measures for conical market models. Math. Financ. Econ. 5, 1–28 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Han, Y., Huang, N.-J.: Continuity and convexity of a nonlinear scalarizing function in set optimization problems with applications. J. Optim. Theory Appl. 177, 679–695 (2018)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Hernández, E., Rodríguez-Marín, L., Sama, M.: Some equivalent problems in set optimization. Oper. Res. Lett. 37, 61–64 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Hernández, E., Rodríguez-Marín, L., Sama, M.: On solutions of set-valued optimization problems. Comput. Math. Appl. 60, 1401–1408 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Hiriart-Urruty, J.-B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms, vol. 305, 1st edn. Springer, Berlin (1993)zbMATHGoogle Scholar
  18. 18.
    Khan, A.A., Tammer, C., Zălinescu, C.: Set-Valued Optimization: An Introduction with Applications. Springer, Berlin (2015)CrossRefzbMATHGoogle Scholar
  19. 19.
    Khoshkhabar-amiranloo, S., Khorram, E.: Scalarization of Levitin–Polyak well-posed set optimization problems. Optimization 66, 113–127 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Khoshkhabar-amiranloo, S., Khorram, E.: Pointwise well-posedness and scalarization in set optimization. Math. Methods Oper. Res. 82, 195–210 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Khushboo, Lalitha, C.S.: Scalarizations for a unified vector optimization problem based on order representing and order preserving properties. J. Glob. Optim. 70, 903–916 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Kuroiwa, D.: Existence theorems of set optimization with set-valued maps. J. Inf. Optim. Sci. 24, 73–84 (2003)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Makarov, V.L., Levin, M.J., Rubinov, A.M.: Mathematical Economic Theory: Pure and Mixed Types of Economic Mechanisms. Advanced Textbooks in Economics, vol. 33. North-Holland Publishing Co., Amsterdam (1995)zbMATHGoogle Scholar
  24. 24.
    Rubinov, A.M., Singer, I.: Topical and sub-topical functions, downward sets and abstract convexity. Optimization 50, 307–351 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Rubinov, A.M., Gasimov, R.N.: Scalarization and nonlinear scalar duality for vector optimization with preferences that are not necessarily a pre-order relation. J. Glob. Optim. 29, 455–477 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Sach, P.H.: New nonlinear scalarization functions and applications. Nonlinear Anal. 75, 2281–2292 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Sach, P.H., Tuan, L.A.: New scalarizing approach to the stability analysis in parametric generalized Ky Fan inequality problems. J. Optim. Theory Appl. 157, 347–364 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Shimizu, A., Nishizawa, S., Tanaka, T.: Optimality conditions in set-valued optimization using nonlinear scalarization methods. In: Proceedings of the 4th International Conference on Nonlinear Analysis and Convex Analysis (Okinawa, 2005), Yokohama Publishers, Yokohama, 565–574 (2007)Google Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of DelhiDelhiIndia
  2. 2.Department of MathematicsUniversity of Delhi South CampusNew DelhiIndia

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