Abstract
We investigate how well the graph of a bilinear function \(b{:}\;[0,1]^n\rightarrow \mathbb {R}\) can be approximated by its McCormick relaxation. In particular, we are interested in the smallest number c such that the difference between the concave upper bounding and convex lower bounding functions obtained from the McCormick relaxation approach is at most c times the difference between the concave and convex envelopes. Answering a question of Luedtke, Namazifar and Linderoth, we show that this factor c cannot be bounded by a constant independent of n. More precisely, we show that for a random bilinear function b we have asymptotically almost surely \(c\geqslant \sqrt{n}/4\). On the other hand, we prove that \(c\leqslant 600\sqrt{n}\), which improves the linear upper bound proved by Luedtke, Namazifar and Linderoth. In addition, we present an alternative proof for a result of Misener, Smadbeck and Floudas characterizing functions b for which the McCormick relaxation is equal to the convex hull.
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Acknowledgments
This research was supported by the ARC Linkage Grant no. LP110200524, Hunter Valley Coal Chain Coordinator (hvccc.com.au) and Triple Point Technology (tpt.com).
We thank Jeff Linderoth and James Luedtke for fruitful discussions of the topics presented in this paper, both during a visit of Jeff Linderoth to Newcastle, Australia, and at the 22nd ISMP in Pittsburgh. We also thank Aleksandar Nikolov for pointing us to the “old arguments by Spencer and Erdős” used in the proof of Theorem 3 (see [14]).
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Boland, N., Dey, S.S., Kalinowski, T. et al. Bounding the gap between the McCormick relaxation and the convex hull for bilinear functions. Math. Program. 162, 523–535 (2017). https://doi.org/10.1007/s10107-016-1031-5
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DOI: https://doi.org/10.1007/s10107-016-1031-5