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.
Google mostly lists proprietary software instead of papers to read.
Any better suggestions that are explicitly made for QCMILP (quadratically constrained mixed integer linear programs?