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Optimal Elements in Vector Optimization with a Variable Ordering Structure

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Optimality concepts for vector optimization problems with a variable ordering structure are examined. These considerations are motivated by an application in medical image registration, where the preferences vary depending on the element in the image space.

The idea of variable ordering structures was first introduced by Yu (J. Opt. Theory Appl. 14:319–377, [1974]) in terms of domination structures. Variable ordering structures mean that there is a set-valued map with cone values that associates to each element an ordering. A candidate element is called a nondominated element iff it is not dominated by other reference elements w.r.t. their corresponding ordering. In addition to nondominated elements, another notion of optimal elements, called minimal elements, has also been discussed. For that notion, only the ordering of the candidate element itself is considered. This paper shows that these two different optimality concepts are connected by duality properties. Characterizations and existence results of the above two solution concepts are also given.

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  1. Yu, P.L.: Cone convexity, cone extreme points, and nondominated solutions in decision problems with multiobjectives. J. Optim. Theory Appl. 14, 319–377 (1974)

    Article  MATH  Google Scholar 

  2. Bergstresser, K., Charnes, A., Yu, P.L.: Generalization of domination structures and nondominated solutions in multicriteria decision making. J. Optim. Theory Appl. 18, 3–13 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  3. Wacker, W.: Multikriterielle Optimierung bei der Registrierung medizinischer Daten. Master Thesis, University of Erlangen-Nürnberg (2008)

  4. Fischer, B., Haber, E., Modersitzki, J.: Mathematics meets medicine—an optimal alignment. SIAG/OPT Views News 19(2), 1–7 (2008)

    Google Scholar 

  5. Karaskal, E.K., Michalowski, W.: Incorporating wealth information into a multiple criteria decision making model. Eur. J. Oper. Res. 150, 204–219 (2003)

    Article  Google Scholar 

  6. Wiecek, M.M.: Advances in cone-based preference modeling for decision making with multiple criteria. Decis. Mak. Manuf. Serv. 1, 153–173 (2007)

    MathSciNet  MATH  Google Scholar 

  7. Baatar, D., Wiecek, M.M.: Advancing equitability in multiobjective programming. Comput. Math. Appl. 52, 225–234 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  8. Engau, A.: Variable preference modeling with ideal-symmetric convex cones. J. Glob. Optim. 42, 295–311 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Yu, P.L.: Multiple-criteria Decision Making: Concepts, Techniques and Extensions. Plenum Press, New York (1985)

    MATH  Google Scholar 

  10. Chew, K.L.: Domination structures in abstract spaces. In: Southeast Asian Bulletin of Mathematics. Proc. First Franco-Southeast Asian Math. Conference, vol. 2, pp. 190–204 (1979)

    Google Scholar 

  11. Huang, N.J., Yang, X.Q., Chan, W.K.: Vector complementarity problems with a variable ordering relation. Eur. J. Oper. Res. 176, 15–26 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chen, G.Y.: Existence of solutions for a vector variational inequality: an extension of the Hartmann-Stampacchia Theorem. J. Optim. Theory Appl. 74, 445–456 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  13. Chen, G.Y., Yang, X.Q.: Characterizations of variable domination structures via nonlinear scalarization. J. Optim. Theory Appl. 112, 97–110 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  14. Chen, G.-Y., Huang, X., Yang, X.: Vector Optimization, Set-Valued and Variational Analysis. Springer, Berlin (2005)

    MATH  Google Scholar 

  15. Jahn, J.: Vector Optimization—Theory, Applications, and Extensions. Springer, Berlin (2004)

    MATH  Google Scholar 

  16. Borwein, J.M.: The geometry of Pareto efficiency over cones. Optimization 11, 235–248 (1980)

    MATH  Google Scholar 

  17. Weidner, P.: Problems in scalarizing multicriteria approaches. In: Köksalen, M., Zionts, S. (eds.) Multiple Criteria Decision Making in the New Millenium, pp. 199–209. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  18. Weidner, P.: Problems in models and methods of vector optimization. Wiss. Schr.reihe TU Karl-Marx-Stadt 5, 47–57 (1989)

    Google Scholar 

  19. Sawaragi, Y., Nakayama, H., Tanino, T.: Theory of Multiobjective Optimization. Academic Press, London (1985)

    MATH  Google Scholar 

  20. Weidner, P.: Dominanzmengen und Optimalitätsbegriffe in der Vektoroptimierung. Wiss. Z. Ilmenau 31, 133–146 (1985)

    MathSciNet  MATH  Google Scholar 

  21. Li, S.J., Li, M.H.: Levitin-Polyak well-posedness of vector equilibrium problems. Math. Methods Oper. Res. 69, 125–140 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Gerstewitz (Tammer), C.: Nichtkonvexe Dualität in der Vektoroptimierung. Wiss. Z. TH Leuna-Merseburg 25, 357–364 (1983)

    Google Scholar 

  23. Weidner, P.: Ein Trennungskonzept und seine Anwendung auf Vektoroptimierungsverfahren. Habilitation Thesis, Martin-Luther-Universität Halle-Wittenberg (1990)

  24. Eichfelder, G.: Adaptive Scalarization Methods in Multiobjective Optimization. Springer, Berlin (2008)

    Book  MATH  Google Scholar 

  25. Giannessi, F.: Constrained Optimization and Image Space Analysis. Springer, Berlin (2005)

    MATH  Google Scholar 

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Correspondence to Gabriele Eichfelder.

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Communicated by Po-Lung Yu.

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Eichfelder, G. Optimal Elements in Vector Optimization with a Variable Ordering Structure. J Optim Theory Appl 151, 217–240 (2011).

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