(6bn) Optimization of Nonsmooth Chemical Process Models
Nonsmoothness can enter chemical process models through sources such as transitions in phase or flow regime, activation of safety mechanisms, and switches between discrete operating modes. Established numerical methods for simulation, sensitivity analysis, and optimization typically require derivative information, and have limited applicability to nonsmooth systems. Specialized numerical methods exist for nonsmooth problems, such as bundle methods for local optimization, and semismooth Newton methods for equation solving. These nonsmooth methods require local sensitivity information in the form of the generalized derivatives developed by Clarke and Nesterov. However, established methods for evaluation of these generalized derivatives are limited either in scope or in the quality of the obtained derivatives. Nonsmooth dynamic systems introduce particular difficulties, since little is known a priori about the behavior of state variables as functions of system parameters, and since even theoretical descriptions of generalized derivatives are lacking for dynamic systems.
For my PhD work with Paul Barton in the Department of Chemical Engineering at the Massachusetts Institute of Technology, I have focused on describing and evaluating generalized derivatives for nonsmooth systems. I developed and implemented a variant of automatic differentiation that evaluates generalized derivatives for composite nonsmooth functions. This method is the first computationally tractable numerical method for accurate generalized derivative evaluation, and is fully automatable.
Several new results were also obtained for nonsmooth dynamic systems that are represented as systems of parametric ordinary differential equations (ODEs) with nonsmooth right-hand side functions. Firstly, I obtained sufficient conditions under which the state variables of such a nonsmooth dynamic system are in fact smooth functions of system parameters. These conditions are computationally inexpensive and simple to verify a posteriori during a simulation run, and can sometimes be shown to hold a priori. Secondly, I provided the first theoretical description of a useful generalized derivative for a general nonsmooth dynamic system. As in classical sensitivity analysis approaches for smooth dynamic systems, this generalized derivative is expressed in terms of an auxiliary dynamic system. These nonsmooth dynamic sensitivity results were generalized to certain hybrid discrete/continuous systems, which can be used to model systems with discontinuous jumps.