I want to minimize the function
$$\begin{aligned} & \underset{x}{\text{minimize}} & & \left[\underset{i}{\Sigma}(x_i - w_i)^2 + (x^TPx - 1)^2\right] \\ & \text{subject to} & & \Sigma_ix_i=0 \\ &&& \Sigma_i|x_i| \leq c_1 \\ &&& |x_i| \leq c_2 \forall i \end{aligned}$$
where $w$ is a known desired vector and P is a PSD matrix, $c_1$ and $c_2$ are constants. Both expressions are convex, but the second expression is not DCP. Is there a way to convert the second expression, $(x^TPx - 1)^2$, to a dual expression that is DCP and can be solved by an optimizer such as CVX?