I would consider the dual problem. Suppose your linear equality constraints are given by $Ax = b$. We form the Lagrangian

$$\mathcal{L}(x, v) = c^\top x + \lVert V x \rVert_2^2 + \lVert D x \rVert_2^2 - \lVert W x \rVert_2^2 + v^\top ( Ax - b )$$

The Lagrange dual function is defined to be

$$g(v) = \min_x \mathcal{L}(x, v)$$

We can solve this minimization problem to find

$$g(v) = \mathcal{L}\left( - Q^{-1}( c + A^{\top}v), v \right)$$

which gives rise to the dual problem $\max_v g(v)$. Lastly, by strong duality the optimal objective value of the dual problem is equal to the optimal objective value of the primal problem.

I would consider the dual problem. Suppose your linear equality constraints are given by $Ax = b$. We form the Lagrangian

$$\mathcal{L}(x, v) = c^\top x + \lVert V x \rVert_2^2 + \lVert D x \rVert_2^2 - \lVert W x \rVert_2^2 + v^\top ( Ax - b )$$

The Lagrange dual function is defined to be

$$g(v) = \min_x \mathcal{L}(x, v)$$

We can solve this minimization problem to find

$$g(v) = \mathcal{L}\left( -\frac{1}{2} Q^{-1}( c + A^{\top}v), v \right)$$

which gives rise to the dual problem $\max_v g(v)$. Lastly, by strong duality the optimal objective value of the dual problem is equal to the optimal objective value of the primal problem.