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

DOI:
https://doi.org/10.1090/S0002-9939-02-06761-8

Published electronically:
November 6, 2002

MathSciNet review:
1948101

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 .

**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**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
26B25,
90C29

Retrieve articles in all journals with MSC (2000): 26B25, 90C29

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:
https://doi.org/10.1090/S0002-9939-02-06761-8

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