Generalized Derivatives for Solutions of Parametric Ordinary Differential Equations with Non-differentiable Right-Hand Sides

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• 作者：Kamil A. Khan ; Paul I. Barton
• 关键词：Generalized Jacobians ; Sensitivity analysis ; Nonsmooth analysis ; Ordinary differential equations ; 49J52 ; 34A12 ; 90C31
• 刊名：Journal of Optimization Theory and Applications
• 出版年：2014
• 出版时间：November 2014
• 年：2014
• 卷：163
• 期：2
• 页码：355-386
• 全文大小：336 KB
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• 作者单位：Kamil A. Khan (1)
  Paul I. Barton (1)
  
  1. Process Systems Engineering Laboratory, Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA
  
• ISSN：1573-2878

文摘

Sensitivity analysis provides useful information for equation-solving, optimization, and post-optimality analysis. However, obtaining useful sensitivity information for systems with nonsmooth dynamic systems embedded is a challenging task. In this article, for any locally Lipschitz continuous mapping between finite-dimensional Euclidean spaces, Nesterov’s lexicographic derivatives are shown to be elements of the plenary hull of the (Clarke) generalized Jacobian whenever they exist. It is argued that in applications, and in several established results in nonsmooth analysis, elements of the plenary hull of the generalized Jacobian of a locally Lipschitz continuous function are no less useful than elements of the generalized Jacobian itself. Directional derivatives and lexicographic derivatives of solutions of parametric ordinary differential equation (ODE) systems are expressed as the unique solutions of corresponding ODE systems, under Carathéodory-style assumptions. Hence, the scope of numerical methods for nonsmooth equation-solving and local optimization is extended to systems with nonsmooth parametric ODEs embedded.