Nonconvex optimization, Global optimization, Convex relaxation, McCormick, AlphaBB, Interval extensions,

This article presents an analysis of the convergence order of Taylor models and McCormick-Taylor models, namely Taylor models with McCormick relaxations as the remainder bounder, for factorable functions. Building upon the analysis of McCormick relaxations by Bompadre and Mitsos (J Glob Optim, 52(1):1-28, 2012), convergence bounds are established for the addition, multiplication and composition operations. It is proved that the convergence orders of both qth-order Taylor models and qth-order McCormick-Taylor models are at least q + 1, under relatively mild assumptions. Moreover, it is verified through simple numerical examples that these bounds are sharp. A consequence of this analysis is that, unlike McCormick relaxations over natural interval extensions, McCormick-Taylor models do not result in increased order of convergence over Taylor models in general. As demonstrated by the numerical case studies however, McCormick-Taylor models can provide tighter bounds or even result in a higher convergence rate.

No abstract

We present lower bounds for the vehicle routing problem (VRP) with and without split deliveries, improving the well known bound of Haimovich and Rinnooy Kan. These bounds are then utilized in a design of best-to-date approximation algorithms.

No abstract