On tightness and anchoring of McCormick and other relaxations

Jaromił Najman and Alexander Mitsos ()
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Jaromił Najman: RWTH Aachen University, AVT - Aachener Verfahrenstechnik, Process Systems Engineering
Alexander Mitsos: RWTH Aachen University, AVT - Aachener Verfahrenstechnik, Process Systems Engineering

Journal of Global Optimization, 2019, vol. 74, issue 4, No 5, 677-703

Abstract: 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.

Keywords: Global optimization; Nonconvex optimization; Convergence order; Convex relaxation; McCormick; Interval analysis; 49M20; 49M37; 65K05; 90C26 (search for similar items in EconPapers)
Date: 2019
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DOI: 10.1007/s10898-017-0598-6

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