I have the following minimization problem in $z \in \mathbb R^n$, which contains $x\_1, \dots, x\_t, y \in \mathbb R$.


$$\begin{array}{ll} \text{minimize} & y\\ \text{subject to} & xQx'= y\\ & 0 \leq x\_i \leq1\\ & Az \leq b\end{array}$$


where $Q$ is diagonal and has positive diagonal integer values, $A \in \mathbb Z^{m \times n}$ and $b \in \mathbb Z^m$ are given.



> 
> Is this problem $NP$-hard or solvable in polynomial time?
> 
> 
> 



Linear optimization with one positive definite quadratic equality condition in P?
The following conditions


$$
\begin{array}{l}
y=\sum x\_i^2\\
0\leq x\_i\leq 1\\
y=\sum x\_i
\end{array}
$$


are equivalent to $x\_i\in \{0,1\}$, which means your construction allows you to introduce binary variables. Then you can use the minimization objective to encode something NP-hard, like the smallest dominating set in a graph.


