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Optimal Elements in Vector Optimization with a Variable Ordering Structure

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  • Gabriele Eichfelder

    (University of Erlangen-Nuremberg)

Abstract

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.

Suggested Citation

  • Gabriele Eichfelder, 2011. "Optimal Elements in Vector Optimization with a Variable Ordering Structure," Journal of Optimization Theory and Applications, Springer, vol. 151(2), pages 217-240, November.
    • Handle: RePEc:spr:joptap:v:151:y:2011:i:2:d:10.1007_s10957-011-9928-x
    DOI: 10.1007/s10957-011-9928-x
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    References listed on IDEAS

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    1. G. Y. Chen & X. Q. Yang, 2002. "Characterizations of Variable Domination Structures via Nonlinear Scalarization," Journal of Optimization Theory and Applications, Springer, vol. 112(1), pages 97-110, January.
    2. Karasakal, Esra Koktener & Michalowski, Wojtek, 2003. "Incorporating wealth information into a multiple criteria decision making model," European Journal of Operational Research, Elsevier, vol. 150(1), pages 204-219, October.
    3. Huang, N.J. & Yang, X.Q. & Chan, W.K., 2007. "Vector complementarity problems with a variable ordering relation," European Journal of Operational Research, Elsevier, vol. 176(1), pages 15-26, January.
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    Cited by:

    1. Shokouh Shahbeyk & Majid Soleimani-damaneh & Refail Kasimbeyli, 2018. "Hartley properly and super nondominated solutions in vector optimization with a variable ordering structure," Journal of Global Optimization, Springer, vol. 71(2), pages 383-405, June.
    2. Truong Q. Bao & Lidia Huerga & Bienvenido Jiménez & Vicente Novo, 2020. "Necessary Conditions for Nondominated Solutions in Vector Optimization," Journal of Optimization Theory and Applications, Springer, vol. 186(3), pages 826-842, September.
    3. J. Y. Bello Cruz & G. Bouza Allende, 2014. "A Steepest Descent-Like Method for Variable Order Vector Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 162(2), pages 371-391, August.
    4. T. Q. Bao & B. S. Mordukhovich & A. Soubeyran, 2015. "Variational Analysis in Psychological Modeling," Journal of Optimization Theory and Applications, Springer, vol. 164(1), pages 290-315, January.
    5. Elena-Andreea Florea, 2018. "Vector Optimization Problems with Generalized Functional Constraints in Variable Ordering Structure Setting," Journal of Optimization Theory and Applications, Springer, vol. 178(1), pages 94-118, July.
    6. Truong Q. Bao & Boris S. Mordukhovich, 2014. "Necessary Nondomination Conditions in Set and Vector Optimization with Variable Ordering Structures," Journal of Optimization Theory and Applications, Springer, vol. 162(2), pages 350-370, August.
    7. Behnam Soleimani, 2014. "Characterization of Approximate Solutions of Vector Optimization Problems with a Variable Order Structure," Journal of Optimization Theory and Applications, Springer, vol. 162(2), pages 605-632, August.
    8. Zhiang Zhou & Wenbin Wei & Fei Huang & Kequan Zhao, 2024. "Approximate weak efficiency of the set-valued optimization problem with variable ordering structures," Journal of Combinatorial Optimization, Springer, vol. 48(3), pages 1-13, October.
    9. Glaydston Carvalho Bento & Gemayqzel Bouza Allende & Yuri Rafael Leite Pereira, 2018. "A Newton-Like Method for Variable Order Vector Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 177(1), pages 201-221, April.
    10. Christian Sommer, 2014. "Geometrical and Topological Properties of a Parameterized Binary Relation in Vector Optimization," Journal of Optimization Theory and Applications, Springer, vol. 163(3), pages 815-840, December.
    11. Gabriele Eichfelder & Maria Pilecka, 2016. "Set Approach for Set Optimization with Variable Ordering Structures Part II: Scalarization Approaches," Journal of Optimization Theory and Applications, Springer, vol. 171(3), pages 947-963, December.
    12. Bettina Zargini, 2022. "Multiobjective Location Problems with Variable Domination Structures and an Application to Select a New Hub Airport," Logistics, MDPI, vol. 6(2), pages 1-13, March.
    13. Gabriele Eichfelder & Refail Kasimbeyli, 2014. "Properly optimal elements in vector optimization with variable ordering structures," Journal of Global Optimization, Springer, vol. 60(4), pages 689-712, December.

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