Convergence rate of McCormick relaxations

Theory for the convergence order of the convex relaxations by McCormick (Math Program 10(1):147-175, 1976) for factorable functions is developed. Convergence rules are established for the addition, multiplication and composition operations. The convergence order is considered both in terms of pointwise convergence and of convergence in the Hausdorff metric. The convergence order of the composite function depends on the convergence order of the relaxations of the factors. No improvement in the order of convergence compared to that of the underlying bound calculation, e.g., via interval extensions, can be guaranteed unless the relaxations of the factors have pointwise convergence of high order. The McCormick relaxations are compared with the alpha BB relaxations by Floudas and coworkers (J Chem Phys, 1992, J Glob Optim, 1995, 1996), which guarantee quadratic convergence. Illustrative and numerical examples are given.