Scalarization Functionals with Uniform Level Sets in Set Optimization
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We use the original form of Gerstewitz’s nonlinear scalarization functional to characterize upper and lower set-less minimizers of set-valued maps acting from a nonempty set into a real linear space with respect to the lower (resp. upper) set-less relation introduced by Kuroiwa. Our main results are as follows: An upper set-less minimizer to a set-valued map (with respect to the image space) is an upper set-less minimal solution to a scalarization of the set-valued map (with respect to the space of real numbers), where the hypergraphical multifunction is involved in the scalarization and vice versa, a lower set-less minimizer to a set-valued map (with respect to the image space) is an upper set-less minimal solution to an appropriate scalarization of the set-valued map (in the space of real numbers), where the epigraphical multifunction is involved in the scalarization and vice versa, and a lower set-less minimizer to a set-valued map becomes a (Pareto) minimizer to the same map provided that the map enjoys a domination property.
KeywordsNonlinear scalarization functional Set optimization Set-less relations Scalarization characteristics
Mathematics Subject Classification49J53 65K10 90C26
This research was supported by Alexander von Humboldt Foundation. The first author wishes to thank the Institute of Mathematics for the very warm hospitality and providing the excellent facilities during his Humboldt fellowship at Martin-Luther-University Halle-Wittenberg. The authors would like to thank the anonymous referees for their helpful remarks, which allowed us to improve the original presentation.
- 9.Krasnoselskii, M.A.: Positive solutions of operator equations (translated from the Russian by Richard E. Flaherty; edited by Leo F. Boron). P. Noordhoff Ltd. Groningen (1964)Google Scholar
- 13.Kuwano, I., Tanaka, T., Yamada, S.: Characterization of nonlinear scalarizing functions for set-valued maps. In: Akashi, S., Takahashi, W., Tanaka, T. (eds.) Nonlinear Analysis and Optimization, pp. 193–204. Yokohama Publishers, Yokohama (2009)Google Scholar
- 24.Köbis, E., Tammer, C.: Characterization of set relations by means of a nonlinear scalarization functional. In: Le Thi, H.A., Pham Dinh, T., Nguyen, N. (eds.) Modelling, Computation and Optimization in Information Systems and Management Sciences. Advances in Intelligent Systems and Computing, vol. 359. Springer, Cham (2015)Google Scholar