I'm working on a quadratic constrained linear program (QCLP) which I'm trying to obtain its dual formulation. I would like this formulation to be linear, or at least convex. Here are my results so far.
Let $q_0 \in \mathbb{R}^n$ be a column vector of parameters, $x \in \mathbb{R}^n$ be the vector of decision variables, $Q\in \mathbb{R}^{nxn}$ be a positive semidefinite matrix, and $p \in \mathbb{R}$ a constant. The primal formulation of the problem is,
$$ \min q_0^t x $$ subject to $$ x^t Q x + p \leq 0 : \lambda$$ where $\lambda$ is the dual variable associated to the conic constraint. To obtain the dual formulation I write the Lagrangian function $\mathcal{L}(x,\lambda)$ as $\mathcal{L}(x,\lambda) = q_0^t x + \lambda(x^t Q x + p)$. Then the dual function $D(\lambda)$ is
$$ D(\lambda) = \inf_x \mathcal{L}(x,\lambda) $$
which can be found reorganising terms from the lagrangian function as follows $$ D(\lambda) = \lambda p + \inf_x (q_0^t + \lambda x^t Q)x = \begin{cases} \lambda p \quad \text{if} \quad q_0^t + \lambda x^t Q = 0 \\ -\infty \quad \text{otherwise} \end{cases}$$
Making the equality explicit, the dual problem is $$ \max \lambda p$$ subject to $$ q_0 + Q^t x \lambda = 0 \\ \lambda \geq 0 $$
The term $Q^t x \lambda$ is bilinear as it contains primal and dual variables. Is there a way to get a dual problem with linear or convex constraints? Or have I made a mistake in the derivation?
Any assistance would be greatly appreciated.
Thank you very much.