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 \(\tau _{c}\) on X, while the reverse implication is not true if K is not \(\tau _{c}\)-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.
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Acknowledgements
The authors thank Professors M. Durea and B. Jiménez for their careful reading and useful remarks on the manuscript, as well as the reviewers for their comments. This work, for the first author, was partially supported by Ministerio de Ciencia, Innovación y Universidades (MCIU), Agencia Estatal de Investigación (AEI) (Spain) and Fondo Europeo de Desarrollo Regional (FEDER) under Project PGC2018-096899-B-I00 (MCIU/AEI/FEDER, UE).
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Communicated by Marc Teboulle.
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Novo, V., Zălinescu, C. On Relatively Solid Convex Cones in Real Linear Spaces. J Optim Theory Appl 188, 277–290 (2021). https://doi.org/10.1007/s10957-020-01773-z
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DOI: https://doi.org/10.1007/s10957-020-01773-z
Keywords
- Algebraic relative interior
- Algebraic dual cone
- Algebraic and vectorial closures
- Topological dual cone
- Convex core topology