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
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.