Journal of Convex Analysis 21 (2014), No. 3, 857--886
Copyright Heldermann Verlag 2014
Proper Approximate Solutions and ε-Subdifferentials in Vector Optimization: Moreau-Rockafellar Type Theorems
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
We provide Moreau-Rockafellar type theorems for a kind of proper ε-subdifferential of vector-valued mappings. For this aim, we introduce and study a new notion of approximate strong solution of a vector optimization problem, a new strong ε-subdifferential based on this type of approximate strong solutions, and a new regularity condition. Finally, as a consequence of the obtained exact sum rules, the gap between the strong and the proper ε-subdifferentials is provided.
Keywords: Vector optimization, proper epsilon-efficiency, proper epsilon-subdifferential, nearly subconvexlikeness, strong epsilon-efficiency, strong epsilon-subdifferential, linear scalarization, epsilon-subdifferential.
MSC: 90C48, 90C25, 90C29, 49J52, 49K27
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