(184d) Computing Elements of Generalized Jacobians for Nonsmooth Systems

Nonsmoothness can enter chemical engineering models through intrinsic nonsmooth phenomena such as flow reversals, incorporation of check and relief valves, thermodynamic phase transitions, and flow regime changes, and through the use of nonsmooth physical property and chemical kinetics models. Conventional derivative-based methods for equation solving and optimization can fail when applied to nonsmooth models, since required derivative information may not be available, and convergence results may depend on differentiability assumptions.

Specialized methods for nonsmooth systems make use of the (Clarke) generalized Jacobian [1], which is a set-valued mapping containing slope information. In particular, semismooth Newton methods for equation solving [2] and bundle methods for local optimization [4] require computation of matrix-valued elements of the generalized Jacobian at each iteration. Computing these generalized Jacobian elements is difficult, however, because the generalized Jacobian does not obey calculus rules sharply.

In this work, a method is presented for evaluating generalized Jacobian elements for a nonsmooth function that is expressed as a finite composition of piecewise differentiable functions. The method is fully-automatable, and is guaranteed to be computationally tractable relative to the cost of a function evaluation. The method works by using a nonsmooth variant of the forward mode of automatic differentiation [3] to compute directional derivatives along certain basis directions. These basis directions are then perturbed in a systematic manner until they provide sufficient information to evaluate a generalized Jacobian element.

A developed implementation of the method in C++ is discussed, in which operator overloading is used to
evaluate generalized Jacobian elements for nonsmooth functions that are provided as subroutines. The
method is applied to several particular nonsmooth systems for illustration, including pipeline network
models and flash models.

[1] F. H. Clarke, Optimization and Nonsmooth Analysis, John Wiley & Sons, Inc., New York, 1983.
[2] F. Facchinei, J.-S. Pang, Finite-dimensional variational inequalities and complementarity
problems, Springer-Verlag New York, Inc., New York, 2003.
[3] A. Griewank, Automatic directional differentiation of nonsmooth composite functions. In:Recent
Developments in Optimization, French-German Conference on Optimization. Dijon (1994)
[4] C. Lemarechal, et al., On a bundle algorithm for nonsmooth optimization. In: O.L. Mangasarian,
R.R. Meyer, S.M. Robinson (eds.) Nonlinear Programming 4. Academic Press, New York, NY (1981)