McCormick Relaxations for Division Operator is incorrect

[SahandPartash]

Hello,

I am PhD student working with Kamil Khan. We encountered an issue with McCormick relaxations for the division operator. The relaxations appear incorrect (the convex relaxation is larger than the concave one).

Here is an example (Julia version 1.10.8, McCormick.jl version 0.13.8, and EAGO.jl version 0.8.2):

julia> using McCormick

julia> xMC = MC{1,NS}(-2.0,-1.0)
MC{1, NS}(-2.0, -1.0, [-2, -1], [0.0], [0.0], true)

julia> yMC = 1/xMC
MC{1, NS}(-0.5, -1.0, [-1, -0.5], [-0.0], [-0.0], true)

julia> yMC.cv <= yMC.cc
false

Thank you,
Sahand

[RXGottlieb]

Hi Sahand,

Thank you very much for pointing this out. I was able to pinpoint the problem as belonging to inv (so, any of 1/x, x^-1, y/x --> y*inv(x), etc.), specifically when the domain of x is negative and the convex and concave relaxations aren't the same. This issue seems to have persisted for a couple reasons:

1. Tests for division and inversion only covered the case where the convex/concave relaxations are the same, which did not return an invalid MC object;
2. In every benchmark problem I checked with division (from MINLPLib: ex5_4_3, ex6_2_10, ex6_2_11, ex6_2_13, ex6_2_14, ex7_2_1, ex7_2_3, ex7_2_4, process), the variables have positive domains and the denominators are always positive. Additionally, it's possible that in any test problem that would've had a negative denominator, the authors may have automatically distributed the negative to the numerator (e.g., 1/(-(x^2 + 0.1)) == -(1/(x^2 + 0.1))). While the left-hand version of this example leads to the problem you observed, the equivalent right-hand version yields valid relaxations. It's also perhaps rare to find problems where the domain of the denominator covers both positive and negative reals due to numerical issues when the denominator is exactly 0; problems may have been intentionally designed to clean up such structures before passing them to the optimizer.

The fix I'm putting in addresses your original problem (that case should now give valid relaxations) and adds a test to cover this particular case so it's covered in the future. Thanks again for bringing this up.

[SahandPartash]

Thank you for the prompt response and for fixing the issue.