Journal of Optimization Theory and Applications

, Volume 167, Issue 3, pp 783–795 | Cite as

Vectorization in Set Optimization

  • Johannes JahnEmail author


In analogy to the scalarization principle in vector optimization, this paper presents a new vectorization approach for set optimization problems. Vectorization means the replacement of a set optimization problem by a suitable vector optimization problem. This approach is developed for the set less order relation used by Kuroiwa and the minmax less order relation introduced by Ha and Jahn.


Set optimization Order relations Scalarization Optimality conditions 


  1. 1.
    Borwein, J.M.: Multivalued convexity and optimization: a unified approach to inequality and equality constraints. Math. Program. 13, 183–199 (1977) zbMATHCrossRefGoogle Scholar
  2. 2.
    Oettli, W.: Optimality conditions for programming problems involving multivalued mappings. In: Korte, B. (ed.) Modern Applied Mathematics, Optimization and Operations Research. North-Holland, Amsterdam (1980) Google Scholar
  3. 3.
    Kuroiwa, D.: Natural criteria of set-valued optimization. Manuscript, Shimane University, Japan (1998) Google Scholar
  4. 4.
    Kuroiwa, D., Tanaka, T., Ha, X.T.D.: On cone convexity of set-valued maps. Nonlinear Anal. 30, 1487–1496 (1997) zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Young, R.C.: The algebra of many-valued quantities. Math. Ann. 104, 260–290 (1931) MathSciNetCrossRefGoogle Scholar
  6. 6.
    Nishnianidze, Z.G.: Fixed points of monotonic multiple-valued operators. Bull. Georgian Acad. Sci. 114, 489–491 (1984) (in Russian) zbMATHMathSciNetGoogle Scholar
  7. 7.
    Chiriaev, A., Walster, G.W.: Interval Arithmetic Specification. Technical Report (1998) Google Scholar
  8. 8.
    Sun, M.: In: Inc.: Interval Arithmetic Programming Reference, Palo Alto, USA (2000) Google Scholar
  9. 9.
    Jahn, J., Ha, T.X.D.: New order relations in set optimization. J. Optim. Theory Appl. 148, 209–236 (2011) zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Neukel, N.: Private Communication. University of Erlangen–Nuremberg, Erlangen (2011) Google Scholar
  11. 11.
    Hamel, A.H., Heyde, F.: Duality for set-valued measures of risk. SIAM J. Financ. Math. 1, 66–95 (2010) zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Hamel, A.H., Schrage, C.: Notes on extended real- and set-valued functions. J. Convex Anal. 19, 355–384 (2012) zbMATHMathSciNetGoogle Scholar
  13. 13.
    Jahn, J.: Vector Optimization—Theory, Applications, and Extensions. Springer, Berlin (2011) zbMATHCrossRefGoogle Scholar
  14. 14.
    Löhne, A., Schrage, C.: An algorithm to solve polyhedral convex set optimization problems. Optimization 62, 131–141 (2013) zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Rubinov, A.M.: Sublinear operators and their applications. Russ. Math. Surv. 32(4), 115–175 (1977) zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Gerstewitz (Tammer), C.: Nichtkonvexe Dualität in der Vektoroptimierung. Wissenschaftl. Z. Tech. Hochsch. Leuna-Merseburg 25, 357–364 (1983) Google Scholar
  17. 17.
    Pascoletti, A., Serafini, P.: Scalarizing vector optimization problems. J. Optim. Theory Appl. 42, 499–524 (1984) zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Xiao, G., Yu, G., Xiao, H.: Characterizations of scalarization function and set optimization problems with various orders. Manuscript (2012) Google Scholar
  19. 19.
    Gutiérrez, C., Jiménez, B., Miglierina, E., Molho, E.: Scalarization in set optimization with solid and nonsolid ordering cones. Manuscript (2012) Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department MathematikUniversität Erlangen-NürnbergErlangenGermany

Personalised recommendations