Solver for quadratically constrained mixed-integer linear programs

I have an optimization problem with vectors $$x$$, $$y$$, and $$z$$, where $$x$$ is an integer vector. My objective function is linear (i.e. $$\|y\|_1$$), but one of my constraints is quadratic ($$x^Ty \leq z$$). The other constraints are all linear. In addition, I have equality and inequality constraints. What methods are there to optimize this?

I see two options:

• Convex optimization methods (but they might be inefficient, given that my loss is linear.
• Use methods for quadratic objectives and just use them with linear objective functions.

Any better suggestions that are explicitly made for QCMILP (quadratically constrained mixed integer linear programs?

• Having a linear objective won't particularly help. Actually, you can always reformulate a QCQP to have a linear objective by minimizing $z$ subject to $z = \mathrm{obj}$. However, having a single quadratic constraint should favor methods based on linear approximations, which are the ones usually implemented by exact MIQCQP solvers (Gurobi, Cplex...) Commented Mar 1, 2023 at 8:33
• Maybe not useful, but $x^Ty\le z$ can be linearized (with some effort) using the fact that $x$ is integer-valued. (Hint: Expand using binary variables and then linearize the terms). Commented Mar 1, 2023 at 15:46
• I don't think solvers like Cplex and Gurobi will automatically linearize quadratic terms with general integer variables, but only with binary variables. Commented Mar 1, 2023 at 16:03
• The quadratic constraint, $x^T y \leq z$, is not convex in general as the Hessian is indefinite. Please update your question to reflect this. Commented Mar 2, 2023 at 12:50