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A characterization of quasiconvex vector-valued functions
Author(s):
Joël
Benoist;
Jonathan
M.
Borwein;
Nicolae
Popovici
Journal:
Proc. Amer. Math. Soc.
131
(2003),
1109-1113.
MSC (2000):
Primary 26B25;
Secondary 90C29
Posted:
November 6, 2002
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Abstract:
The aim of this paper is to characterize in terms of scalar quasiconvexity the vector-valued functions which are  -quasiconvex with respect to a closed convex cone  in a Banach space. Our main result extends a well-known characterization of  -quasiconvexity by means of extreme directions of the polar cone of  , obtained by Dinh The Luc in the particular case when  is a polyhedral cone generated by exactly  linearly independent vectors in the Euclidean space  .
References:
-
- 1.
- J. M. Borwein, A Lagrange multiplier theorem and a sandwich theorem for convex relations, Math. Scand. 48 (1981), no. 2, 189-204. MR 83d:49027
- 2.
- -, Adjoint process duality, Math. Oper. Res. 8 (1983), no. 3, 403-434. MR 85h:90092
- 3.
- -, Norm duality for convex processes and applications, J. Optim. Theory Appl. 48 (1986), no. 1, 53-64. MR 87d:90126
- 4.
- A. Cambini and L. Martein, Generalized concavity in multiobjective programming, Generalized convexity, generalized monotonicity: recent results (Luminy, 1996), Kluwer Acad. Publ., Dordrecht, 1998, pp. 453-467. MR 99g:90105
- 5.
- G. Choquet, Lectures on analysis. Vol. I: Integration and topological vector spaces, Vol. II: Representation theory, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 40:3252, MR 40:3253
- 6.
- F. Ferro, Minimax type theorems for
 -valued functions, Ann. Mat. Pura Appl. (4) 132 (1982), 113-130. MR 84i:49045 - 7.
- J. Jahn, Mathematical vector optimization in partially ordered linear spaces, Verlag Peter D. Lang, Frankfurt am Main, 1986. MR 87f:90095
- 8.
- V. Jeyakumar, W. Oettli, and M. Natividad, A solvability theorem for a class of quasiconvex mappings with applications to optimization, J. Math. Anal. Appl. 179 (1993), no. 2, 537-546. MR 94i:90094
- 9.
- D. T. Luc, Connectedness of the efficient point sets in quasiconcave vector maximization, J. Math. Anal. Appl. 122 (1987), no. 2, 346-354. MR 88f:90154
- 10.
- -, Theory of vector optimization, Springer-Verlag, Berlin, 1989. MR 92e:90003
- 11.
- P. H. Sach, Characterization of scalar quasiconvexity and convexity of locally Lipschitz vector-valued maps, Optimization 46 (1999), no. 3, 283-310. MR 2001d:90069
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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, Babes-Bolyai University of Cluj, 3400 Cluj-Napoca, Romania
Email:
popovici@math.ubbcluj.ro
DOI:
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
Posted:
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
Copyright of article:
Copyright
2002,
American Mathematical Society
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