Convexity-preserving properties of set-valued ratios of affine functions
DOI:
https://doi.org/10.24193/subbmath.2021.3.14Keywords:
Set-valued affine function, single-valued selection, ratio of affine functions, generalized convexity.Abstract
The aim of this paper is to introduce some special classes of set-valued functions that preserve the convexity of sets by direct and inverse images. In particular, we show that the so-called set-valued ratios of affine functions belong to these classes, by characterizing them in terms of vector-valued selections that are ratios of affine functions in the classical sense of Rothblum.
References
Aubin, J.-P., Frankowska, H., Set-Valued Analysis, Birhauser, Boston, 1990.
Avriel, M., Diewert W.E., Schaible S., Zang, I., Generalized Concavity, Plenum Press, New York, 1988.
Berge, C., Topological Spaces Including a Treatment of Multi-Valued Functions, Vector Spaces and Convexity, Oliver & Boyd, Edinburgh-London, 1963.
Cambini, A., Martein, L., Generalized convexity and optimization: Theory and Applications, Springer-Verlag, Berlin, 2009.
Deutsch, F., Singer, I., On single-valuedness of convex set-valued maps, Set-Valued Anal. 1 (1993), 97-103.
Gopfert, A., Riahi, H., Tammer, C., Zalinescu, C., Variational Methods in Partially Ordered Spaces, Springer-Verlag, New York, 2003.
Kuroiwa, D., Popovici, N., Rocca, M., A characterization of cone-convexity for set-valued functions by cone-quasiconvexity, Set-Valued Var. Anal. 23 (2015), 295-304.
Nikodem, K., K-convex and K-concave set-valued functions, Habilitation dissertation, Scientic Bulletin of Lodz Technical University, Nr. 559, 1989.
Nikodem, K., Popa, D., On single-valuedness of set-valued maps satisfying linear inclusions, Banach J. Math. Anal. 3 (2009), 44-51.
Orzan, A., A new class of fractional type set-valued functions, Carpathian J. Math. 35 (2019), 79-84.
Rothblum, U.G., Ratios of affine functions, Math. Program. 32 (1985), 357-365.
Seto, K., Kuroiwa, D., Popovici, N., A systematization of convexity and quasiconvexity concepts for set-valued maps, defined by l-type and u-type preorder relations, Optimization 67 (2018), 1077-1094.
Stancu-Minasian, I.M., Fractional programming. Theory, methods and applications, Mathematics and its Applications, Kluwer-Dordrecht 409 (1997).
Tan, D.H., A note on multivalued affine mappings, Stud. Univ. Babes-Bolyai Math., Cluj-Napoca 33 (1988), 55-59.
Downloads
Published
Issue
Section
License
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
Transfer of copyright agreement: When the article is accepted for publication, the authors and the representative of the coauthors, hereby agree to transfer to Studia Universitatis Babeș-Bolyai Mathematica all rights, including those pertaining to electronic forms and transmissions, under existing copyright laws, except for the following, which the authors specifically retain: the authors can use the material however they want as long as it fits the NC ND terms of the license. The authors have all rights for reuse according to the license.
