Quadratic Equality Constrained Quadratic Program and Convexity There are a few questions on this topic already. However, none of them really answer my question.
The most relevant are these:
Quadratic optimisation with quadratic equality constraints
Quadratic Equality Constraints via SDP
I have a quadratic problem with quadratic constraints, and my constraints are equalities:
$$ \text{mimize}\quad x^TQ_0x+q_0^Tx\\
\text{s. t.}\quad x^TQ_ix+q_i^Tx=0$$
which can be rewritten as:
$$ \text{mimize}\quad x^TQ_0x+q_0^Tx\\
\text{s. t.}\quad x^TQ_ix+q_i^Tx\le 0\\
\quad\quad\quad\quad x^T(-Q_i)x-q_i^Tx\le 0$$
On the literature I've read, the only restriction for the problem to be convex are that the matrices $Q$ have to be positive semi-definite, which is satisfied (in my case) for both restrictions.
Does the equality make the problem nonconvex and can someone give me some references about this?
Or, since the semi-definiteness is preserved, is my problem is still convex?
Lastlty, if it is nonconvex, does going to a SOCP or SDP help me?
Thank you
 A: Quadratic equalities are never convex. Simply consider the trivial scalar case $x^2=1$, which has two distinct feasible points $-1$ and $1$.
As it is nonconvex, you cannot convert it to an SOCP, so your final question doesn't really make any sense.
A: For the constraints, we normally require that the feasible set be a convex set, while the objective function should be convex. In general, the set of points (or vectors) satisfying a quadratic equality constraint may not be a convex set. 
For example, take the scalar case where Q = 1 (positive definite) and the quadratic equation is
\begin{equation*}
x^T(1)x = 9 \\
\Rightarrow x^2 = 9
\end{equation*}
The feasible set, in this case, is {-3,3}. The line joining the elements -3 and 3 does not satisfy the quadratic equality. In particular, taking $\alpha$ as 0.5, the square of zero is clearly not 9. Hence, despite Q being positive definite, the constraint is not convex.
See this answer: Michael Grant (https://math.stackexchange.com/users/52878/michael-grant), Why is the constraint $\|w\| = 1$ non-convex?, URL (version: 2015-05-28): https://math.stackexchange.com/q/1301592.
