Journal of Global Optimization

, Volume 67, Issue 4, pp 687–729 | Cite as

Differentiable McCormick relaxations

  • Kamil A. Khan
  • Harry A. J. Watson
  • Paul I. Barton


McCormick’s classical relaxation technique constructs closed-form convex and concave relaxations of compositions of simple intrinsic functions. These relaxations have several properties which make them useful for lower bounding problems in global optimization: they can be evaluated automatically, accurately, and computationally inexpensively, and they converge rapidly to the relaxed function as the underlying domain is reduced in size. They may also be adapted to yield relaxations of certain implicit functions and differential equation solutions. However, McCormick’s relaxations may be nonsmooth, and this nonsmoothness can create theoretical and computational obstacles when relaxations are to be deployed. This article presents a continuously differentiable variant of McCormick’s original relaxations in the multivariate McCormick framework of Tsoukalas and Mitsos. Gradients of the new differentiable relaxations may be computed efficiently using the standard forward or reverse modes of automatic differentiation. Extensions to differentiable relaxations of implicit functions and solutions of parametric ordinary differential equations are discussed. A C++ implementation based on the library MC++ is described and applied to a case study in nonsmooth nonconvex optimization.


Nonconvex optimization Convex underestimators McCormick relaxations Implicit functions 

Mathematics Subject Classification

49M20 90C26 65G40 26B25 


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Kamil A. Khan
    • 1
  • Harry A. J. Watson
    • 2
  • Paul I. Barton
    • 2
  1. 1.Mathematics and Computer Science DivisionArgonne National LaboratoryLemontUSA
  2. 2.Process Systems Engineering LaboratoryMassachusetts Institute of TechnologyCambridgeUSA

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