Quadratically constrained quadratic program

A Quadratically constrained quadratic program (QCQP) is a convex optimization problem of the form

minimize $(1/2)x^{T}P_{0}x+q_{0}^{T}x+r_{0}$ subject to

$x^{T}P_{i}x+q_{i}^{T}x+r_{i}\leq 0$, $i\in \{1,\dots ,m\}$

$Ax=b$

where $x$ is the optimization variable and $P_{i}\in \mathbb {S} _{+}^{n}$ for $i\in \{0,\dots ,m\}$. If any $P_{i}$ is not positive definite, solution of the problem may be NP-hard; for example, the constraint $x_{1}(1-x_{1})\leq 0$ and $x_{1}(1-x_{1})\geq 0$ is equivalent to the constraint $x_{1}(1-x_{1})=0$, which is in turn equivalent to the constraint $x_{1}\in \{0,1\}$. This is a constraint in integer programming.