# Evaluating Generalized Derivatives of Nonsmooth Dynamic Systems

Kamil Khan* and Paul Barton

The (Clarke) generalized Jacobian is a set-valued mapping which provides useful sensitivity information for nonsmooth functions [1]. Elements of the generalized Jacobian are used in semismooth Newton methods for equation-solving, and in bundle methods for local optimization. In recent years, variants of the forward mode of automatic differentiation have been developed to evaluate Clarke generalized Jacobian elements for any finite composition of simple nonsmooth functions [2,3]. The evaluated elements are also lexicographic derivatives of the composite function, which are essentially slopes of higher-order directional derivatives [4]. However, there are no existing methods for evaluating generalized Jacobian elements for nonsmooth dynamic systems, which would be useful in the local optimization of these systems. Similarly, there are no existing methods for evaluating subgradients of convex solutions to systems of parametric ordinary differential equations (ODEs) with nonsmooth and nonconvex right-hand side functions. Such subgradients would be useful in solving the lower-bounding problems in a branch-and-bound procedure for global nonsmooth dynamic optimization. In this presentation, it is argued that in applications such as semismooth Newton methods and bundle methods, elements of the plenary hull of the generalized Jacobian are as useful as elements of the generalized Jacobian itself. Given a system of nonsmooth parametric ODEs, directional derivatives of any unique solution are described as the unique solutions to corresponding ODE systems. Certain elements of the plenary hull of the generalized Jacobian are also described as the unique solutions to corresponding ODE systems, which are expressed in terms of lexicographic derivatives of the original right-hand side function. Provided that these ODE systems can be solved, the scope of semismooth Newton methods and bundle methods is thereby extended to nonsmooth dynamic systems. \begin{thebibliography}{99} \bibitem{eins} F. H. Clarke, Optimization and Nonsmooth Analysis, SIAM, Philadelphia, PA, 1990. \bibitem{zwei} A. Griewank, On stable piecewise linearization and generalized algorithmic differentiation, Optimization Methods and Software, In press (2013). \bibitem{drei} K. A. Khan and P. I. Barton, Evaluating an element of the Clarke generalized Jacobian of a composite piecewise differentiable function, ACM T. Math. Software, In press (2013). \bibitem{vier} Y. Nesterov, Lexicographic differentiation of nonsmooth functions, Math. Program. B, 104 (2005), pp. 669-700. \end{thebibliography}

Mathematics Subject Classification: 34A12 26B05 46J52

Keywords: Nonsmooth analysis; dynamic systems; sensitivity analysis

Minisymposion: Nonsmooth Optimization