Nonlinear Analysis 71 (2009) 4241–4250 Contents lists available at ScienceDirect Nonlinear Analysis journal homepage: www.elsevier.com/locate/na Computation formulas and multiplier rules for graphical derivatives in separable Banach spaces ✩ E. Hernández a , A.A. Khan b , L. Rodríguez-Marín a , M. Sama a,∗ a Departamento de Matemática Aplicada, E.T.S.I. Industriales, Universidad Nacional de Educación a Distancia, calle Juan del Rosal, 12, 28040 Madrid, Spain b School of Mathematical Sciences, Rochester Institute of Technology, 85 Lomb Memorial Drive, Rochester, NY 14623, USA article info Article history: Received 19 November 2008 Accepted 25 February 2009 MSC: 90C29 90C30 49J52 49K27 Keywords: Ordered spaces Schauder bases Set-valued analysis Contingent epiderivatives Vector optimization Lagrange multipliers Weak minimizers Proper minimizers abstract In this paper, we present new computation formulas for the contingent epiderivative and hypoderivative of a set-valued map taking values in a Banach space with a shrinking Schauder basis. These formulas are established in terms of the Fourier coefficients, and, in particular, in terms of the derivatives of the component maps associated with the Schauder basis. As an application, we obtain multiplier rules for vector optimization problems in terms of the derivatives of the component maps, extending classical results from smooth multiobjective optimization problems. © 2009 Elsevier Ltd. All rights reserved. 1. Introduction and preliminaries The theory of graphical derivatives of set-valued maps is a useful tool to derive optimality conditions for set-valued optimization problems, and in particular for vector optimization problems [1–7]. For a real normed space X , and two real Banach spaces Y and Z being partially ordered by closed convex and pointed cones C and K , respectively; the general problem of vector optimization (P)  minimize f (x) subject to g (x) ∈−K , x ∈ S , where f , g are maps from a nonempty subset S to Y and Z , respectively, is a natural extension of the classical problem of multiobjective optimization with constraints. The primary objective of this paper is to develop a scalarization technique for the computation of the contingent epi/hypoderivative of set-valued maps that allows us to extend various results ranging from smooth multiobjective optimization to the general vector optimization Problem (P). ✩ For E. Hernández and M. Sama, this work is partially supported by MEC (Spain), proj. MTM2006-02629 and (i-MATH) CSD2006-00032 (Consolider - Ingenio 2010). ∗ Corresponding author. Tel.: +34 913987927. E-mail addresses: ehernandez@ind.uned.es (E. Hernández), aaksma@rit.edu (A.A. Khan), lromarin@ind.uned.es (L. Rodríguez-Marín), msama@ind.uned.es (M. Sama). 0362-546X/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2009.02.114