On tightness and anchoring of McCormick and other relaxations

Abstract

We say that a convex relaxation of a function is anchored at a particular point in their domains if the values of the function and the relaxation at this point are equal. The opposite of anchoring is offset, i.e., a positive difference between the function and its convex relaxation values over the entire domain. We present theoretical results supported by theoretical and numerical examples showing that anchoring (at corner points) is a useful property but neither necessary nor sufficient for favorable Hausdorff and pointwise convergence order of a relaxation-based bounding scheme. Next, we investigate the tightness and convergence behavior of McCormick relaxations in specific cases. McCormick relaxations have favorable convergence orders, but a positive offset may still slow down the convergence within a simple branch-and-bound algorithm. We demonstrate that use of tighter underlying interval extensions can help reduce the offset and accelerate convergence.

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Acknowledgements

We would like to thank the late Prof. Floudas who motivated us to look into anchoring. We appreciate the thorough review and helpful comments provided by the anonymous reviewers and editors which resulted in a significantly improved manuscript. This project has received funding from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) Improved McCormick Relaxations for the efficient Global Optimization in the Space of Degrees of Freedom MA 1188/34-1.

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Correspondence to Alexander Mitsos.

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Najman, J., Mitsos, A. On tightness and anchoring of McCormick and other relaxations. J Glob Optim 74, 677–703 (2019). https://doi.org/10.1007/s10898-017-0598-6

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Keywords

  • Global optimization
  • Nonconvex optimization
  • Convergence order
  • Convex relaxation
  • McCormick
  • Interval analysis

Mathematics Subject Classification

  • 49M20
  • 49M37
  • 65K05
  • 90C26