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Unboundedness of the images of set-valued mappings having closed graphs: application to vector optimization

Author

Listed:
  • V. T. Hieu

    (Norwegian University of Science and Technology
    Center for Advanced Intelligence Project, RIKEN)

  • E. A. S. Köbis

    (Norwegian University of Science and Technology)

  • M. A. Köbis

    (Norwegian University of Science and Technology)

  • P. H. Schmölling

    (Norwegian University of Science and Technology)

Abstract

In this paper, we propose criteria for unboundedness of the images of set-valued mappings having closed graphs in Euclidean spaces. We focus on mappings whose domains are non-closed or whose values are connected. These criteria allow us to see structural properties of solutions in vector optimization, where solution sets can be considered as the images of solution mappings associated to specific scalarization methods. In particular, we prove that if the domain of a certain solution mapping is non-closed, then the weak Pareto solution set is unbounded. Furthermore, for a quasi-convex problem, we demonstrate two criteria to ensure that if the weak Pareto solution set is disconnected then each connected component is unbounded.

Suggested Citation

  • V. T. Hieu & E. A. S. Köbis & M. A. Köbis & P. H. Schmölling, 2025. "Unboundedness of the images of set-valued mappings having closed graphs: application to vector optimization," Journal of Global Optimization, Springer, vol. 91(1), pages 217-234, January.
    • Handle: RePEc:spr:jglopt:v:91:y:2025:i:1:d:10.1007_s10898-024-01438-5
    DOI: 10.1007/s10898-024-01438-5
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    References listed on IDEAS

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    1. X. H. Gong, 2007. "Connectedness of the Solution Sets and Scalarization for Vector Equilibrium Problems," Journal of Optimization Theory and Applications, Springer, vol. 133(2), pages 151-161, May.
    2. Vu Trung Hieu, 2019. "On the Numbers of Connected Components in the Solution Sets of Polynomial Vector Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 181(1), pages 95-100, April.
    3. Vu Trung Hieu, 2019. "Numbers of the connected components of the solution sets of monotone affine vector variational inequalities," Journal of Global Optimization, Springer, vol. 73(1), pages 223-237, January.
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