# Solver for a MIQP with an indefinite coefficient matrix

Do CPLEX or Gurobi handle MIQPs with indefinite coefficient matrices?

The problem I am dealing with has quadratic terms in which one variable is binary and the other variable is continuous. The coefficient matrix of the quadratic form is far from positive semi-definite. Its entries are data-dependent. Thus, I have no control over the positive semi-definiteness.

Can CPLEX or Gurobi handle such models at all? Note that it's a MIQP with binary variables and continuous variables, not a QP.

Is there other software for such problems?

What is the state of the art, algorithmically?

A typical way to deal with this is to replace products $xy$ where $x$ is binary and $y$ continuous with a new variable $w$, and then add a constraint to ensure $w=0$ when $x=0$ and $w=y$ when $x=1$. This can be accomplished with $-M(1-x) \leq w-y \leq M(1-x)$, $-Mx \leq w \leq Mx$ where $M$ is a sufficiently large (but as small as possible) constant to ensure the feasible set does not change, i.e., an upper bound on the absolute value of any optimal $y$.