G. P. McCormick [Math Prog 1976] provides the framework for convex/concave relaxations of factorable functions, via rules for the product of functions and compositions of the form F(f(z)), where F is a univariate function. Herein, the composition theorem is generalized to allow multivariate outer functions F, and theory for the propagation of subgradients is presented. In addition to extending the framework, the new result provides a tool for the proof of relaxations. Moreover, a direct consequence is an improved relaxation for the product of two functions, at least as tight as McCormick’s result, and often tighter. The result also allows the direct relaxation of multilinear products of functions. Furthermore, the composition result is applied to obtain improved convex underestimators for the minimum/maximum and the division of two functions for which current relaxations are often weak. Finally, our approach interprets the McCormick relaxation approach as a decomposition method for the auxiliary variable method, and suggests ideas for hybrid methods combining the advantages of both approaches.
Journal of Global Optimization 59:633-662, 2014 http://dx.doi.org/10.1007/s10898-014-0176-0 (open access)