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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A characterization of quasiconvex vector-valued functions


Authors: Joël Benoist, Jonathan M. Borwein and Nicolae Popovici
Journal: Proc. Amer. Math. Soc. 131 (2003), 1109-1113
MSC (2000): Primary 26B25; Secondary 90C29
Published electronically: November 6, 2002
MathSciNet review: 1948101
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Abstract: The aim of this paper is to characterize in terms of scalar quasiconvexity the vector-valued functions which are $K$-quasiconvex with respect to a closed convex cone $K$ in a Banach space. Our main result extends a well-known characterization of $K$-quasiconvexity by means of extreme directions of the polar cone of $K$, obtained by Dinh The Luc in the particular case when $K$ is a polyhedral cone generated by exactly $n$ linearly independent vectors in the Euclidean space $\mathbb{R}^n$.


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Additional Information

Joël Benoist
Affiliation: LACO, UPRESSA 6090, Department of Mathematics, University of Limoges, 87060 Limoges, France
Email: benoist@unilim.fr

Jonathan M. Borwein
Affiliation: Department of Mathematics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6
Email: borwein@cecm.sfu.ca

Nicolae Popovici
Affiliation: Faculty of Mathematics and Computer Science, Babeş-Bolyai University of Cluj, 3400 Cluj-Napoca, Romania
Email: popovici@math.ubbcluj.ro

DOI: http://dx.doi.org/10.1090/S0002-9939-02-06761-8
PII: S 0002-9939(02)06761-8
Keywords: Quasiconvex vector-valued functions, scalarization, polar cones
Received by editor(s): July 7, 2001
Published electronically: November 6, 2002
Additional Notes: The second author’s research was supported by NSERC and by the Canada Research Chair Programme
The third author’s research was supported by CNCSIS Romania under Grant no. 1066/2001
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2002 American Mathematical Society