How to derive the dual of a quadratically-constrained quadratic program (QCQP) with second order cone constraint? Here is the optimization problem I want to handle:
$$ \begin{array}{rl}\min_{\mathbf{x},\mathbf{y}}&\frac{1}{2}\mathbf{x}^{{T}}\mathbf{P}_0\mathbf{x}+\mathbf{q}_0^{{T}}\mathbf{x}+\mathbf{r}_0\\ \text{s.t.}&\mathbf{A}\mathbf{x}+\mathbf{B}\mathbf{y}=\mathbf{f}\\ &\mathbf{C}\mathbf{x}+\mathbf{D}\mathbf{y}\le\mathbf{g}\\ &\frac{1}{2}\mathbf{x}^{{T}}{\mathbf{P}}_{i}\mathbf{x}+\mathbf{q}_{{i}}^{{T}}\mathbf{x}+\mathbf{r}_{{i}}\leq0,{i}=1,\ldots,k\\ & \left\|\mathbf{S}_{i}\mathbf{y}+\mathbf{t}_{i}\right\|_2\leq\mathbf{u}_{i}^{T}\mathbf{y}+{v}_{i},{i}=1,\ldots,l\end{array} $$ where $\mathbf P_0,\ldots,\mathbf P_m$ are $n\times n$ matrices and $ \mathbf x \in \mathbb R^n, \mathbf y \in \mathbb R^m$ is the optimization variable. The optimization variables are separated into $ \mathbf x$ and $ \mathbf y$ as they follow different constraints.
In my setting, $\mathbf P_0,\ldots,\mathbf P_m$ are all positive semidefinite and thus the problem is convex. I wanna ask:
- What is the dual of this problem?
- How to ensure strong duality?