## Bounding the gap between the McCormick relaxation and the convex hull for bilinear functions.(English)Zbl 1384.90073

Summary: 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 J. Luedtke et al. [ibid. 136, No. 2 (B), 325–351 (2012; Zbl 1286.90117)], 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 [loc. cit.]. In addition, we present an alternative proof for a result of R. Misener et al. [Optim. Methods Softw. 30, No. 1, 215–249 (2015; Zbl 1325.90071)] characterizing functions $$b$$ for which the McCormick relaxation is equal to the convex hull.

### MSC:

 90C26 Nonconvex programming, global optimization 90C20 Quadratic programming

### Keywords:

global optimization; bilinear function; convex hull

### Citations:

Zbl 1286.90117; Zbl 1325.90071

BARON; GloMIQO
Full Text:

### References:

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