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Journal of Convex Analysis 14 (2007), No. 2, 433--454
Copyright Heldermann Verlag 2007

Infinite Dimensional Clarke Generalized Jacobian

Zsolt Páles
Institute of Mathematics, University of Debrecen, 4010 Debrecen Pf. 12, Hungary

Vera Zeidan
Department of Mathematics, Michigan State University, East Lansing, MI 48824, U.S.A.

We extend for a locally Lipschitz function the notion of Clarke's generalized Jacobian to the setting where the domain lies in an infinite dimensional normed space. When the function is real-valued this notion reduces to the Clarke's generalized gradient. Using this extension, we obtain an exact smooth-nonsmooth chain rule from which the sum rule and the product rule follow. Also an exact formula for the generalized Jacobian of piecewise differentiable functions will be provided.

Keywords: Generalized Jacobian, chain rule, sum rule, piecewise smooth functions.

MSC: 49A52, 58C20

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