Adjoint mode computation of subgradients for McCormick relaxations In [{it A. Mitsos}, {it B. Chachuat} and {it P. I. Barton}, SIAM J. Optim. 20, No. 2, 573--601 (2009; Zbl 1192.65083)], a method similar to Algorithmic Differentiation (AD) is presented which allows the propagation of, in general nondifferentiable, McCormick relaxations [see, e.g., {it G. P. McCormick}, Math. Program. 10, 147--175 (1976; Zbl 0349.90100)] 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., loc. cit.] 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. [loc. cit.]. par 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.

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