(28b) Computing Sensitivities for Nonsmooth Differential-Algebraic Equations

Differential-algebraic equations (DAEs) have become a widely applied modeling tool. In particular, nonsmooth DAEs provide a natural framework for dynamic models of chemical processes such as campaign continuous pharmaceutical manufacturing [1]. Sources of nonsmoothness include thermodynamic phase changes (e.g., flash evaporation), flow transitions (e.g., nonreturn valves), and safety devices (e.g., relief valves). In analyzing such models, sensitivity analysis provides useful information in nonsmooth equation-solving (e.g., semismooth Newton methods) and optimization problems (e.g., bundle methods for local optimization). However, these algorithms require computation of an element of some class of generalized derivative (e.g., Clarkeâ??s generalized Jacobian [2]), for which theoretical and computational approaches are currently lacking.

In this talk, we detail some new results on obtaining computationally relevant generalized derivatives of DAEs for which existing methods fail. More specifically, a tractable method is provided to furnish lexicographic derivatives of solutions of nonsmooth parametric DAEs using recent advancements in nonsmooth analysis [3-5]. These generalized derivatives are no less useful than elements of the generalized Jacobian in many important applications, including the nonsmooth problem-solving methods detailed above. As in the classical case, the desired parametric sensitivities are given in terms of the solution of an auxiliary nonsmooth DAE system.

References

[1] B. Benyahia, R. Lakerveld and P. I. Barton, â??A plant-wide dynamic model of a continuous pharmaceutical process,â?Â Industrial and Engineering Chemistry Research, vol. 51, no. 47, pp. 15393-15412, 2012.

[2] F. H. Clarke, Optimization and Nonsmooth Analysis, Philadelphia: SIAM, 1990.

[3] Y. Nesterov, â??Lexicographic differentiation of nonsmooth functions,â?Â Mathematical Programming, vol. 104, no. 2, pp. 669-700, 2005.

[4] K. A. Khan and P. I. Barton, â??Generalized derivatives for solutions of parametric ordinary differential equations with non-differentiable right-hand sides,â?Â Journal of Optimization Theory and Applications, vol. 163, no. 2, pp. 355-386, 2014.

[5] K. A. Khan and P. I. Barton, â??A vector forward mode of automatic differentiation for generalized derivative evaluation,â? Optimization Methods and Software, vol. 30, no. 6, pp. 1185-1212, 2015.