Computation formulas and multiplier rules for graphical derivatives in separable Banach spaces

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.