Suppose we have $A \in \mathbb{R}^{m \times n}$, we want to minimize
\begin{align} \min&\quad\|Ax\|_2^2 \\ \text{s.t.}&\quad x^\top C_1x = 1, \\ &\quad x^\top C_2 x = 1, \\ &\quad Dx > 0 \end{align}
Is there any optimization method to handle this?
Actually, the first and second constraints aims to force part of $x$ has the norm of $1$. The third constraint aims to force part of $x$ has the positive value.
Ignore the $Dx > 0$ constraint, can we formulate the optimization into a generalized eigenvalue problem or second-order cone programming?
Thanks in advance.