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On Relatively Solid Convex Cones in Real Linear Spaces

Author

Listed:
  • Vicente Novo

    (Universidad Nacional de Educación a Distancia)

  • Constantin Zălinescu

    (Iaşi Branch of Romanian Academy)

Abstract

Having a convex cone K in an infinite-dimensional real linear space X, Adán and Novo stated (in J Optim Theory Appl 121:515–540, 2004) that the relative algebraic interior of K is nonempty if and only if the relative algebraic interior of the positive dual cone of K is nonempty. In this paper, we show that the direct implication is not true even if K is closed with respect to the finest locally convex topology $$\tau _{c}$$ τ c on X, while the reverse implication is not true if K is not $$\tau _{c}$$ τ c -closed. However, in the main result of this paper, we prove that the latter implication is true if the algebraic interior of the positive dual cone of K is nonempty; the general case remains an open problem. As a by-product, a result about separation of cones is obtained that improves Theorem 2.2 of the work mentioned above.

Suggested Citation

  • Vicente Novo & Constantin Zălinescu, 2021. "On Relatively Solid Convex Cones in Real Linear Spaces," Journal of Optimization Theory and Applications, Springer, vol. 188(1), pages 277-290, January.
    • Handle: RePEc:spr:joptap:v:188:y:2021:i:1:d:10.1007_s10957-020-01773-z
    DOI: 10.1007/s10957-020-01773-z
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    References listed on IDEAS

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    1. Adan, M. & Novo, V., 2003. "Weak efficiency in vector optimization using a closure of algebraic type under cone-convexlikeness," European Journal of Operational Research, Elsevier, vol. 149(3), pages 641-653, September.
    2. M. Adán & V. Novo, 2004. "Proper Efficiency in Vector Optimization on Real Linear Spaces," Journal of Optimization Theory and Applications, Springer, vol. 121(3), pages 515-540, June.
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    Cited by:

    1. Christian Günther & Bahareh Khazayel & Christiane Tammer, 2022. "Vector Optimization w.r.t. Relatively Solid Convex Cones in Real Linear Spaces," Journal of Optimization Theory and Applications, Springer, vol. 193(1), pages 408-442, June.

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