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Journal of Optimization Theory and Applications

, Volume 164, Issue 2, pp 455–478 | Cite as

Characterizing Efficiency on Infinite-dimensional Commodity Spaces with Ordering Cones Having Possibly Empty Interior

  • Fabián Flores-BazánEmail author
  • Fernando Flores-Bazán
  • Sigifredo Laengle
Article

Abstract

Some production models in finance require infinite-dimensional commodity spaces, where efficiency is defined in terms of an ordering cone having possibly empty interior. Since weak efficiency is more tractable than efficiency from a mathematical point of view, this paper characterizes the equality between efficiency and weak efficiency in infinite-dimensional spaces without further assumptions, like closedness or free disposability. This is obtained as an application of our main result that characterizes the solutions to a unified vector optimization problem in terms of its scalarization. Standard models as efficiency, weak efficiency (defined in terms of quasi-relative interior), weak strict efficiency, strict efficiency, or strong solutions are carefully described. In addition, we exhibit two particular instances and compute the efficient and weak efficient solution set in Lebesgue spaces.

Keywords

Vector optimization Scalarization Efficiency Infinite-dimensional commodity space Quasi-relative interior 

Mathematics Subject Classication (2000)

90C26 90C29 90C30 90C46 

Notes

Acknowledgments

This research, for the first author, was supported in part by CONICYT-Chile through FONDECYT 112-0980 and BASAL projects, CMM, Universidad de Chile. Support by Ecos-Conicyt CE08-2010 and MATH-AmSud 13math-01 are also greatly acknowledged. The authors also wants to express their great gratitude to Nicolas Hadjisavvas for very stimulating discussions during the last stage of this final version.

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Fabián Flores-Bazán
    • 1
    Email author
  • Fernando Flores-Bazán
    • 2
  • Sigifredo Laengle
    • 3
  1. 1.Departamento de Ingeniería Matemática, CI²MAUniversidad de ConcepciónConcepciónChile
  2. 2.Departamento de Matemática, Facultad de CienciasUniversidad del Bío BíoConcepciónChile
  3. 3.Departamento de Control de GestiónUniversidad de ChileSantiago de ChileChile

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