Automatic differentiation rules for Tsoukalas–Mitsos convex relaxations in global process optimization

Abstract

With increasing digitalization and vertical integration of chemical process systems, nonconvex optimization problems often emerge in chemical engineering applications, yet require specialized optimization techniques. Typical global optimization methods proceed by progressively refining bounds on the unknown optimal value, by strategically employing convex relaxations. This article constructs a general closed-form expression for the convex subdifferentials of recent “multivariate McCormick” convex relaxations of nontrivial composite functions, by solving a previous duality formulation in all cases using nonsmooth Karush–Kuhn–Tucker conditions. Based on this subdifferential expression, new automatic differentiation rules are developed to compute gradients and subgradients for multivariate McCormick relaxations, to ultimately generate useful bounds in global optimization. Unlike established differentiation techniques for these relaxations, our new rules are expressed in closed form, do not require solving separate dual optimization problems, are efficiently carried out, and are compatible with the reverse/adjoint mode of algorithmic differentiation. Our formulations become more straightforward when the relevant functions are either smooth or piecewise smooth.

Keywords

• Convex relaxations
• Automatic differentiation
• Global optimization
• Factorable functions