Adjoint Mode Computation of Subgradients for McCormick Relaxations

Abstract

In Mitsos et al. (SIAM Journal on Optimization 20(2):573–601, 2009), a method similar to Algorithmic Differentiation (AD) is presented which allows the propagation of, in general nondifferentiable, McCormick relaxations (McCormick, Mathematical Programming 10(2):147–175, 1976; Steihaug, Twelfth Euro AD Workshop, Berlin, 2011) of factorable functions and of the corresponding subgradients in tangent-linear mode. Subgradients are natural extensions of “usual” derivatives which allow the application of derivative-based methods to possibly nondifferentiable convex and concave functions. The software package libMC (Mitsos et al. SIAM Journal on Optimization 20(2):573–601, 2009) performs the automatic propagation of the relaxation and of corresponding subgradients based on the principles of tangent-linear mode AD by overloading. Similar ideas have been ported to Fortran yielding modMC as part of our ongoing collaboration with the authors of Mitsos et al. (SIAM Journal on Optimization 20(2):573–601, 2009). In this article an adjoint method for the computation of subgradients for McCormick relaxations is presented. A corresponding implementation by overloading in Fortran is provided in the form of amodMC. The calculated subgradients are used in a deterministic global optimization algorithm based on a branch-and-bound method. The superiority of adjoint over tangent-linear mode is illustrated by two examples.