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Characterization of the Nonemptiness and Compactness of Solution Sets in Convex and Nonconvex Vector Optimization

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Abstract

As a consequence of an abstract theorem proved elsewhere, a vector Weierstrass theorem for the existence of a weakly efficient solution without any convexity assumption is established. By using the notion (recently introduced in an earlier paper) of semistrict quasiconvexity for vector functions and assuming additional structure on the space, new existence results encompassing many results appearing in the literature are derived. Also, when the cone defining the preference relation satisfies some mild assumptions (but including the polyhedral and icecream cones), various characterizations for the nonemptiness and compactness of the weakly efficient solution set to convex vector optimization problems are given. Similar results for a class of nonconvex problems on the real line are established as well.

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Authors and Affiliations

  1. Departamento de Ingeniería Matemática, Facultad de Ciencias Físicas y Matemáticas, Universidad de Concepción, Concepción, Chile

    F. Flores-Bazán (Professor) & C. Vera PhD Student

Authors
  1. F. Flores-Bazán
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  2. C. Vera PhD Student
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Additional information

Communicated by H. P. Benson

Research supported in part by Conicyt-Chile through FONDECYT 104-0610 and FONDAP-Matemáticas Aplicadas II.

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Flores-Bazán, F., Vera, C. Characterization of the Nonemptiness and Compactness of Solution Sets in Convex and Nonconvex Vector Optimization. J Optim Theory Appl 130, 185–207 (2006). https://doi.org/10.1007/s10957-006-9098-4

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  • DOI: https://doi.org/10.1007/s10957-006-9098-4

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