I have a sparse optimization problem of the form:

$$\min_x c^T x + \| D x\|^2_2 + \| V x\|^2_2  - \| W x\|^2_2$$

$D$ is diagonal and $V$, $W$ are non-square matrices. I know that:
$$Q = D^2 + V V^T - WW^T \succ 0$$

is positive definite. Ensuring convexity of the problem, but optimizing with dense matrix $Q$ is very slow.

Can we rewrite this problem so that it is accepted by a DCP solver like CVXPY, while keeping sparse structure of the problem ?

PS: problem also includes some constraints, so is non-quadratic