Journal of Optimization Theory and Applications

, Volume 148, Issue 2, pp 209–236 | Cite as

New Order Relations in Set Optimization

  • Johannes JahnEmail author
  • Truong Xuan Duc Ha


In this paper we study a set optimization problem (SOP), i.e. we minimize a set-valued objective map F, which takes values on a real linear space Y equipped with a pre-order induced by a convex cone K. We introduce new order relations on the power set \(\mathcal{P}(Y)\) of Y (or on a subset of it), which are more suitable from a practical point of view than the often used minimizers in set optimization. Next, we propose a simple two-steps unifying approach to studying (SOP) w.r.t. various order relations. Firstly, we extend in a unified scheme some basic concepts of vector optimization, which are defined on the space Y up to an arbitrary nonempty pre-ordered set \((\mathcal{Q},\preccurlyeq)\) without any topological or linear structure. Namely, we define the following concepts w.r.t. the pre-order \(\preccurlyeq\): minimal elements, semicompactness, completeness, domination property of a subset of \(\mathcal{Q}\), and semicontinuity of a set-valued map with values in \(\mathcal{Q}\) in a topological setting. Secondly, we establish existence results for optimal solutions of (SOP), when F takes values on \((\mathcal{Q},\preccurlyeq)\) from which one can easily derive similar results for the case, when F takes values on \(\mathcal{P}(Y)\) equipped with various order relations.


Set optimization Order relations Existence results 


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  1. 1.
    Luc, D.T.: Vector Optimization. Lectures Notes in Economics and Mathematical Systems, vol. 319. Springer, Berlin (1989) Google Scholar
  2. 2.
    Jahn, J.: Vector Optimization—Theory, Applications, and Extensions. Springer, Berlin (2004) zbMATHGoogle Scholar
  3. 3.
    Kuroiwa, D.: Some criteria in set-valued optimization. Investigations on nonlinear analysis and convex analysis (Japanese) (Kyoto, 1996). Surikaisekikenkyusho Kokyuroku No. 985, 171–176 (1997) Google Scholar
  4. 4.
    Kuroiwa, D.: Natural criteria of set-valued optimization. Manuscript, Shimane University, Japan (1998) Google Scholar
  5. 5.
    Kuroiwa, D.: The natural criteria in set-valued optimization. Research on nonlinear analysis and convex analysis (Japanese) (Kyoto, 1997). Surikaisekikenkyusho Kokyuroku No. 1031, 85–90 (1998) Google Scholar
  6. 6.
    Alonso, M., Rodríguez-Marín, L.: Set-relations and optimality conditions in set-valued maps. Nonlinear Anal. 63(8), 1167–1179 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Alonso, M., Rodríguez-Marín, L.: Optimality conditions for set-valued maps with set optimization. Nonlinear Anal. 70(9), 3057–3064 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Ha, T.X.D.: Some variants of the Ekeland variational principle for a set-valued map. J. Optim. Theory Appl. 124(1), 187–206 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Hamel, A.: Variational principles on metric and uniform spaces. Habilitation thesis, University of Halle-Wittenberg, Germany (2005) Google Scholar
  10. 10.
    Hamel, A., Löhne, A.: Minimal element theorems and Ekeland’s principle with set relations. J. Nonlinear Convex Anal. 7(1), 19–37 (2006) MathSciNetzbMATHGoogle Scholar
  11. 11.
    Hernández, E., Rodríguez-Marín, L.: Lagrangian duality in set-valued optimization. J. Optim. Theory Appl. 134(1), 119–134 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Hernández, E., Rodríguez-Marín, L.: Existence theorems for set optimization problems. Nonlinear Anal. 67(6), 1726–1736 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Hernández, E., Rodríguez-Marín, L.: Nonconvex scalarization in set optimization with set-valued maps. J. Math. Anal. Appl. 325(1), 1–18 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Hernández, E., Rodríguez-Marín, L., Sama, M.: Some equivalent problems in set optimization. Oper. Res. Lett. 37(1), 61–64 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Kuroiwa, D.: Some duality theorems of set-valued optimization with natural criteria. In: Nonlinear Analysis and Convex Analysis, Niigata, 1998, pp. 221–228. World Scientific, River Edge (1999) Google Scholar
  16. 16.
    Kuroiwa, D.: On set-valued optimization. Nonlinear Anal. 47(2), 1395–1400 (2001) MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Kuroiwa, D.: Existence theorems of set optimization with set-valued maps. J. Inf. Optim. Sci. 24(1), 73–84 (2003) MathSciNetzbMATHGoogle Scholar
  18. 18.
    Kuroiwa, D.: Existence of efficient points of set optimization with weighted criteria. J. Nonlinear Convex Anal. 4(1), 117–123 (2003) MathSciNetzbMATHGoogle Scholar
  19. 19.
    Kuroiwa, D., Tanaka, T., Ha, T.X.D.: On cone convexity of set-valued maps. In: Proceedings of the Second World Congress of Nonlinear Analysts, Part 3, Athens (1996) Google Scholar
  20. 20.
    Löhne, A.: Optimization with set relations: conjugate duality. Optimization 54(3), 265–282 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Rodríguez-Marín, L., Sama, G.: (Λ,C)-contingent derivatives of set-valued maps. J. Math. Anal. Appl. 335(2), 974–989 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Schröder, B.S.W.: Ordered Sets: An Introduction. Birkhäuser, Boston (2001) Google Scholar
  23. 23.
    Young, R.C.: The algebra of many-valued quantities. Math. Ann. 104, 260–290 (1931) MathSciNetCrossRefGoogle Scholar
  24. 24.
    Nishnianidze, Z.G.: Fixed points of monotonic multiple-valued operators. Bull. Georgian Acad. Sci. 114, 489–491 (1984) (in Russian) MathSciNetzbMATHGoogle Scholar
  25. 25.
    Chiriaev, A., Walster, G.W.: Interval arithmetic specification. Technical Report (1998) Google Scholar
  26. 26.
    Sun Microsystems, Inc.: Interval Arithmetic Programming Reference. Palo Alto, USA (2000) Google Scholar
  27. 27.
    Moore, R.E., Kearfott, R.B., Cloud, M.J.: Introduction to Interval Analysis. SIAM, Philadelphia (2009) zbMATHCrossRefGoogle Scholar
  28. 28.
    Chinaie, M., Zafarani, J.: Image space analysis and scalarization of multivalued optimization. J. Optim. Theory Appl. 142, 451–467 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Wu, H.-C.: Duality theory in interval-valued linear programming problems. Manuscript, National Kaohsiung Normal University, Taiwan (2010) Google Scholar

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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department MathematikUniversität Erlangen-NürnbergErlangenGermany
  2. 2.Institute of MathematicsHanoiVietnam

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