Evaluating an Element of the Clarke Generalized Jacobian of a Piecewise Differentiable Function

  • Kamil A. Khan
  • Paul I. Barton
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 87)


The (Clarke) generalized Jacobian of a locally Lipschitz continuous function is a derivative-like set-valued mapping that contains slope information. Several methods for optimization and equation solving require evaluation of generalized Jacobian elements. However, since the generalized Jacobian does not satisfy calculus rules sharply, this evaluation can be difficult. In this work, a method is presented for evaluating generalized Jacobian elements of a nonsmooth function that is expressed as a finite composition of absolute value functions and continuously differentiable functions. The method makes use of the principles of automatic differentiation and the theory of piecewise differentiable functions, and is guaranteed to be computationally tractable relative to the cost of a function evaluation.


Forward mode Generalized gradient Piecewise differentiable functions Nonsmooth analysis 



This work has been funded by the MIT-BP Conversion Program.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Process Systems Engineering Laboratory, Department of Chemical EngineeringMassachusetts Institute of TechnologyCambridgeUSA

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