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I have a constraint of the following form

$$x^2 \leq yz$$

where $z$ is binary, $y \geq 0$, and $x$ is free. Can Gurobi handle this constraint?

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  • $\begingroup$ They do quadratically constrained optimization. $\endgroup$ – logarithm May 15 at 23:09
  • $\begingroup$ This term "quadratically constrained optimization" unfortunately does not imply a specific type of optimization model. As far as I can see, the terminology is rather ambiguous in this domain. $\endgroup$ – user2512443 May 16 at 17:18
  • $\begingroup$ Binary as in $\{0,1\}$? Or binary as in $\{\pm 1\}$? Are $x$ and $y$ real? $\endgroup$ – Rodrigo de Azevedo 2 days ago
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This is a (mixed integer) rotated quadratic cone. This can be handled in Gurobi. See for instance http://www.gurobi.com/documentation/8.1/refman/c_grbaddqconstr.html and https://www.gurobi.com/documentation/8.1/examples/qcp_py.html .

Alternatively, presuming there is a known upper bound for y, the right-hand side can be linearized per section 2.8 of "FICO MIP formulations and linearizations Quick reference" https://www.gurobi.com/documentation/8.1/examples/qcp_py.html, in which case it can be handled as a quadratic inequality constraint, plus the linearization constraints.

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