Journal of Convex Analysis 23 (2016), No. 3, 661--690
Copyright Heldermann Verlag 2016
Henig Approximate Proper Efficiency and Optimization Problems with Difference of Vector Mappings
César Gutiérrez
Dep. de Matemática Aplicada, Universidad de Valladolid, Paseo de Belén 15, Campus Miguel Delibes, 47011 Valladolid, Spain
cesargv@mat.uva.es
Lidia Huerga
Dep. de Matemática Aplicada, Universidad Nacional de Educación a Distancia, Calle Juan del Rosal 12, Ciudad Universitaria, 28040 Madrid, Spain
lhuerga@bec.uned.es
Bienvenido Jiménez
Dep. de Matemática Aplicada, Universidad Nacional de Educación a Distancia, Calle Juan del Rosal 12, Ciudad Universitaria, 28040 Madrid, Spain
bjimenez@ind.uned.es
Vicente Novo
Dep. de Matemática Aplicada, Universidad Nacional de Educación a Distancia, Calle Juan del Rosal 12, Ciudad Universitaria, 28040 Madrid, Spain
vnovo@ind.uned.es
This work focuses on approximate proper solutions of vector optimization problems. A concept of Henig approximate proper efficiency is introduced and analyzed from several points of view. First, its main properties are stated and the limit behavior in multiobjective problems of the whole Henig approximate proper efficient set is deduced. These results show that the introduced concept is suitable to approximate the efficient solution set of the problem. After that, the Henig approximate proper efficient solutions are characterized by linear scalarizations under convexity assumptions, and by ε-subgradients in optimization problems dealing with difference of vector mappings. For this last objective, a notion of ε-subdifferential is introduced and studied, obtaining, in particular, a Moreau-Rockafellar type theorem.
Keywords: Vector optimization, proper epsilon-efficiency, optimization of difference of vector mappings, epsilon-subdifferential, nearly cone-subconvexlikeness, linear scalarization.
MSC: 90C48, 90C25, 90C29, 90C46, 49K27
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