Harry A.J. Watson

Reverse propagation of McCormick relaxations

Constraint propagation techniques have heavily utilized interval arithmetic while the application of convex and concave relaxations has been mostly restricted to the domain of global optimization. Here, reverse McCormick propagation, a method to construct and improve McCormick relaxations using a directed acyclic graph representation of the constraints, is proposed. In particular, this allows the interpretation of constraints as implicitly defining set-valued mappings between variables, and allows the construction and improvement of relaxations of these mappings. Reverse McCormick propagation yields potentially tighter enclosures of the solutions of constraint satisfaction problems than reverse interval propagation. Ultimately, the relaxations of the objective of a non-convex program can be improved by incorporating information about the constraints.

Computationally relevant generalized derivatives: theory, evaluation and applications

A new method for evaluating generalized derivatives in nonsmooth problems is reviewed. Lexicographic directional (LD-)derivatives are a recently developed tool in nonsmooth analysis for evaluating generalized derivative elements in a tractable and robust way. Applicable to problems in both steady-state and dynamic settings, LD-derivatives exhibit a number of advantages over current theory and algorithms. As highlighted in this article, the LD-derivative approach now admits a suitable theory for inverse and implicit functions, nonsmooth dynamical systems and optimization problems, among others. Moreover, this technique includes an extension of the standard vector forward mode of automatic differentiation (AD) and acts as the natural extension of classical calculus results to the nonsmooth case in many ways. The theory of LD-derivatives is placed in the context of state-of-the-art methods in nonsmooth analysis, with an application in multistream heat exchanger modelling and design used to illustrate the usefulness of the approach.

Corrections to: Differentiable McCormick relaxations

These errata correct various errors in the closed-form relaxations provided by Khan, Watson, and Barton in the article “Differentiable McCormick Relaxations” (J Glob Optim, 67:687–729, 2017). Without these corrections, the provided closed-form relaxations may fail to be convex or concave and may fail to be valid relaxations.

Robust Flash Calculations through Nonsmooth Inside-Out Algorithms

Abstract The ability to solve many important problems in process systems engineering is dependent on being able to perform vapor-liquid equilibrium (flash) calculations rapidly and consistently. The flash equations are particularly challenging to solve for non-ideal systems with many components, and many process simulators use some version of the inside-out algorithms ( Boston and Britt, 1978 ) for performing such calculations. However, these methods assume that the calculation result is always a two-phase mixture, which is not guaranteed outside of a range of input parameters that is not generally known a priori . This article continues the development of nonsmooth inside-out algorithms ( Watson et al., 2017 ) which retain the benefits of the original algorithms without the possibility of loss of reliability or performance when only a single phase is present. This article shows how this can be accomplished for the fixed pressure-entropy flash through the use of nonsmooth equations that relax the equilibrium constraints when necessary, automatically allowing convergence to either single-phase or two-phase solutions.

Modeling and simulation of phase change and non-ideality in multistream heat exchangers

Abstract Accurate simulation and design of cryogenic processes is challenging largely due to the presence of multistream heat exchanger (MHEX) unit operations. In important processes such as liquefied natural gas production, the streams in these operations are multicomponent mixtures and undergo phase changes which significantly alter their physical properties. However, the locations of the transitions between phases are complex functions of the process variables and not necessarily known a priori . These challenges are addressed in this article by extending a newly developed modeling strategy for MHEXs to include the detection of phase changes and rigorous flash calculations when the thermodynamic state of the fluids is described by a cubic equation of state (CEOS). The resulting model is nonsmooth and is solved by an extension of the vector forward mode of automatic differentiation in conjunction with a linear programming based Newton method. A simulation of the Poly Refrigerant Integrated Cycle Operations (PRICO) process is presented to demonstrate the model and solution algorithm.