Chain Rules for a Proper $$\varepsilon $$ ε -Subdifferential of Vector Mappings

César Gutiérrez (), Lidia Huerga (), Vicente Novo () and Lionel Thibault ()
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César Gutiérrez: Universidad de Valladolid
Lidia Huerga: Universidad Nacional de Educación a Distancia
Vicente Novo: Universidad Nacional de Educación a Distancia
Lionel Thibault: Université Montpellier 2

Journal of Optimization Theory and Applications, 2015, vol. 167, issue 2, No 5, 502-526

Abstract: Abstract In this paper, we derive exact chain rules for a proper epsilon-subdifferential in the sense of Benson of extended vector mappings, recently introduced by ourselves. For this aim, we use a new regularity condition and a new strong epsilon-subdifferential for vector mappings. In particular, we determine chain rules when one of the mappings is linear, obtaining formulations easier to handle in the finite-dimensional case by considering the componentwise order. This Benson proper epsilon-subdifferential generalizes and improves several of the most important proper epsilon-subdifferentials of vector mappings given in the literature and, consequently, the results presented in this work extend known chain rules stated for the last ones. As an application, we derive a characterization of approximate Benson proper solutions of implicitly constrained convex Pareto problems. Moreover, we estimate the distance between the objective values of these approximate proper solutions and the set of nondominated attained values.

Keywords: Vector optimization; Proper $$\varepsilon $$ ε -efficiency; Proper $$\varepsilon $$ ε -subdifferential; Strong $$\varepsilon $$ ε -efficiency; Strong $$\varepsilon $$ ε -subdifferential; Nearly cone-subconvexlikeness; Linear scalarization; 90C48; 90C25; 90C29; 49J52; 49K27 (search for similar items in EconPapers)
Date: 2015
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DOI: 10.1007/s10957-015-0763-3

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