Abstract

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.

Author information

Contact details are reproduced from the original publication and may be historical.

Zsolt Páles

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

pales@math.klte.hu

Vera Zeidan

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

zeidan@math.msu.edu

Keywords

  • Generalized Jacobian
  • chain rule
  • sum rule
  • piecewise smooth functions

Mathematics Subject Classification

49A52, 58C20

Suggested citation

Z. Páles, V. Zeidan. “Infinite Dimensional Clarke Generalized Jacobian.” Journal of Convex Analysis 14 (2007), No. 2, 433–454.

Copyright Heldermann Verlag 2007