Computing AD-compatible subgradients of convex relaxations of implicit functions

Abstract: Automatic generation of convex relaxations and subgradients is critical in global optimization, and is typically carried out using variants of automatic/algorithmic differentiation (AD). At previous AD conferences, variants of the forward and reverse AD modes were presented to evaluate accurate subgradients for convex relaxations of supplied composite functions. In a recent approach for generating convex relaxations of implicit functions, these relaxations are constructed as optimal-value functions; this formulation is versatile but complicates sensitivity analysis. We present the first subgradient propagation rules for these implicit function relaxations, based on supplied AD-like knowledge of the residual function. Our new subgradient rules allow implicit function relaxations to be added to the elemental function libraries for the forward AD modes for subgradient propagation of convex relaxations. Proof-of-concept numerical results in Julia are presented.