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453134 Computation of Sensitivities of Dynamic Systems with Lexicographic Linear Programs Embedded

The objectives of a LLP in standard form as a function of its right-hand side are piecewise linear functions [6], and therefore, nonsmooth. This source of nonsmoothness can be propagated to the parametric dependence of the final states of the dynamic system. Therefore, there exist some parameter values for which the Jacobian of the dynamic system may not exist. Computing elements of Clarke’s generalized Jacobian for complex nonsmooth functions is challenging [7], but can be done efficiently for piecewise differentiable functions, such as LLPs parameterized by their right-hand side, with lexicographic-directional (LD) derivatives [8]. LD-derivatives of nonsmooth dynamic systems can be computed to obtain elements of the plenary hull of the generalized Jacobian [9] if the right-hand side of the ODE is abs-factorable, which means it can be factored as analytic and absolute value functions. This is the case for ODE systems with LLPs embedded; however, obtaining explicitly the abs-factorable representation of a large LLP is intractable. Therefore, an algorithm that relies on LLP basis information instead is presented. This algorithm enables the efficient computation of LD-derivatives of DFBA systems, which is critical for their systematic optimization.

This paper first introduces the ODE system whose solution gives the LD-derivatives of ODE systems with LLPs embedded. Next, some complications regarding the computation of these LD-derivatives are illustrated. Next, an algorithm that addresses these challenges is presented. Finally, the theory developed is implemented to optimize a DFBA case study.

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F. H. Clarke, Optimization and Nonsmooth Analysis, Philadelphia: Society for Industrial and Applied Mathematics, 1990. |

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K. A. Khan and P. Barton, "A vector forward mode of automatic differentiation for generalized derivative evaluation," |

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K. A. Khan and P. I. Barton, "Generalized Derivatives for Solutions of Parametric Ordinary Differential Equations with Non-differentiable Right-Hand Sides," |

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