Abstract
A classical result by Crouzeix (1977) states that a real-valued function is convex if and only if any function obtained from it by adding a linear functional is quasiconvex. The principal aim of this paper is to present a similar characterization for certain cone-convex set-valued functions by means of cone-quasiconvex and affine set-valued functions.
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References
Aubin, J.-P., Frankowska, H.: Set-Valued Analysis. Birhäuser, Boston (1990)
Benoist, J., Borwein, J.M., Popovici, N.: A characterization of quasiconvex vector-valued functions. Proc. Amer. Math. Soc 131, 1109–1113 (2003)
Borwein, J.M.: Multivalued convexity and optimization: a unified approach to inequality and equality constraints. Math. Program 13, 183–199 (1977)
Boţ, R.I., Csetnek, E.R.: Regularity conditions via generalized interiority notions in convex optimization: new achievements and their relation to some classical statements. Optimization 61, 35–65 (2012)
Crouzeix, J.-P.: Contribution à l’étude des fonctions quasi-convexes (in French), Doctoral Thesis, University of Clermont-Ferrand II (1977)
Gautier, S.: Affine and eclipsing multifunctions. Numer. Funct. Anal. Appl. 11, 679–699 (1990)
Göpfert, A., Riahi, H., Tammer, Chr., Zălinescu, C.: Variational Methods in Partially Ordered Spaces. Springer-Verlag, New York (2003)
Gorokhovik, V.V.: Representations of affine multifunctions by affine selections. Set-Valued Anal. 16, 185–198 (2008)
Gorokhovik, V.V., Zabreiko, P.P.: On Fréchet differentiability of multifunctions. Optimization 54, 391–409 (2005)
Holmes, R.B.: Geometric Functional Analysis and its Applications. Springer-Verlag, Berlin (1975)
Kuroiwa, D.: Convexity for set-valued maps. Appl. Math. Lett. 9, 97–101 (1996)
La Torre, D., Popovici, N., Rocca, M.: Scalar characterizations of weakly cone-convex and weakly cone-quasiconvex functions. Nonlinear Anal. 72, 1909–1915 (2010)
Lemaréchal, C., Zowe, J.: The eclipsing concept to approximate a multi-valued map**. Optimization 22, 3–37 (1991)
Luc, D.T.: Theory of Vector Optimization. Springer-Verlag, Berlin (1989)
Nikodem, K., Popa, D.: On single-valuedness of set-valued maps satisfying linear inclusions. Banach J. Math. Anal. 3, 44–51 (2009)
Rådström, H.: An embedding theorem for spaces of convex sets. Proc. Amer. Math. Soc. 3, 165–169 (1952)
Urbański, R.: A generalization of the Minkowski-Rådström-Hörmander theorem. Bull. Polish Acad. Sci. Math. 24, 709–715 (1976)
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Kuroiwa, D., Popovici, N. & Rocca, M. A Characterization of Cone-Convexity for Set-Valued Functions by Cone-Quasiconvexity. Set-Valued Var. Anal 23, 295–304 (2015). https://doi.org/10.1007/s11228-014-0307-2
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DOI: https://doi.org/10.1007/s11228-014-0307-2