Convexity-preserving properties of set-valued ratios of affine functions

Alexandru Orzan, Nicolae Popovici

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.


Keywords


Set-valued affine function; single-valued selection; ratio of affine functions; generalized convexity.

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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.




DOI: http://dx.doi.org/10.24193/subbmath.2021.3.14

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