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Journal of Global Optimization

, Volume 52, Issue 1, pp 1–28 | Cite as

Convergence rate of McCormick relaxations

  • Agustín Bompadre
  • Alexander MitsosEmail author
Article

Abstract

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 αBB relaxations by Floudas and coworkers (J Chem Phys, 1992, J Glob Optim, 1995, 1996), which guarantee quadratic convergence. Illustrative and numerical examples are given.

Keywords

Nonconvex optimization Global optimization Convex relaxation McCormick AlphaBB Interval extensions 

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Copyright information

© Springer Science+Business Media, LLC. 2011

Authors and Affiliations

  1. 1.Graduate Aerospace LaboratoriesCalifornia Institute of TechnologyPasadenaUSA
  2. 2.Department of Mechanical EngineeringMassachusetts Institute of TechnologyCambridgeUSA

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