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Vector Equilibrium Problems Under Asymptotic Analysis

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Abstract

Given a closed convex set K in Rn; a vector function F:K×K → Rm; a closed convex (not necessarily pointed) cone P(x) in ℝm with non-empty interior, PP(x) ≠ Ø, various existence results to the problem

find x∈K such that F(x,y)∉- int P(x) ∀ y ∈K

under P(x)-convexity/lower semicontinuity of F(x,ċ) and pseudomonotonicity on F, are established. Moreover, under a stronger pseudomonotonicity assumption on F (which reduces to the previous one in case m=1), some characterizations of the non-emptiness of the solution set are given. Also, several alternative necessary and/or sufficient conditions for the solution set to be non-empty and compact are presented. However, the solution set fails to be convex in general. A sufficient condition to the solution set to be a singleton is also stated. The classical case P(x)=ℝm + is specially discussed by assuming semi-strict quasiconvexity. The results are then applied to vector variational inequalities and minimization problems. Our approach is based upon the computing of certain cones containing particular recession directions of K and F.

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Authors and Affiliations

  1. Departamento de Ingeniería Matemática, Universidad de Concepción, Facultad de Ciencias Físicas y Matemáticas, Casilla 160-C, Concepción-, Chile

    Fabián Flores-Bazán & Fernando Flores-Bazán

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  1. Fabián Flores-Bazán
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  2. Fernando Flores-Bazán
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Flores-Bazán, F., Flores-Bazán, F. Vector Equilibrium Problems Under Asymptotic Analysis. Journal of Global Optimization 26, 141–166 (2003). https://doi.org/10.1023/A:1023048928834

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  • DOI: https://doi.org/10.1023/A:1023048928834