Journal of Convex Analysis 14 (2007), No. 2, 275--296
Copyright Heldermann Verlag 2007
Weak and Proper Efficiency in Set-Valued Optimization on Real Linear Spaces
Elvira Hernández
Dep. de Matemática Aplicada, E.T.S.I. Industriales, Universidad Nacional de Educación a Distancia, C/ Juan del Rosal 12, 28040 Madrid, Spain
ehernandez@ind.uned.es
Bienvenido Jiménez
Dep. de Matemática Aplicada, E.T.S.I. Industriales, Universidad Nacional de Educación a Distancia, C/ Juan del Rosal 12, 28040 Madrid, Spain
bjimen1@encina.pntic.mec.es
Vicente Novo
Dep. de Matemática Aplicada, E.T.S.I. Industriales, Universidad Nacional de Educación a Distancia, C/ Juan del Rosal 12, 28040 Madrid, Spain
vnovo@ind.uned.es
We extend the notion of cone-subconvexlikeness of set-valued maps on topological linear spaces to set-valued maps on linear spaces, (that is, general linear spaces without any particular topology), and we provide several characterizations. An alternative theorem is also established for this kind of maps. Using the notion of vector closure introduced recently by Adán and Novo, we also provide, in this framework, an adaptation of the proper efficiency in the sense of Benson for set-valued maps. The previous notion and results are then applied to obtain optimality conditions of weak efficiency and a characterization of Benson proper efficiency by means of scalarization and multipliers rules.
Keywords: Set-valued maps, set-valued optimization, proper efficiency, generalized convexity.
MSC: 90C29, 90C46
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