Strong duality for nonconvex quadratic program (with multiple constraints)

Question

Consider the following optimization \begin{eqnarray} P_1: \quad &\underset{x\in\mathbb{C}^N}{\mathrm{minimize}}&\; f_0(x) \\ &\mathrm{subject\;to}&\; f_i(x) \leq 0, i=1,\ldots,m \\ &&\; h_i(x) \leq 0, i=1,\ldots,n. \end{eqnarray} Assume that

1. $f_i(x)$, $i=0,\ldots,m$ are convex quadratic functions and $h_i(x)$, $i=1,\ldots,n$ are nonconvex quadratic functions.
2. The optimization problem is feasible.
3. For any given $\lambda = [\lambda_1,\ldots,\lambda_m]^T \in \mathbb{R}_+^{m}$, strong duality holds for the following nonconvex optimization \begin{eqnarray} P_2(\lambda):\quad &\underset{x\in\mathbb{C}^N}{\mathrm{minimize}}&\; f_0(x) + \sum_{i=1}^m \lambda_i f_i(x) \\ &\mathrm{subject\;to}&\; h_i(x) \leq 0, i=1,\ldots,n. \end{eqnarray}

Will strong duality hold for the optimization $P_1$ as well? I have thought for this theorem for 2 weeks and could not find the answer. Your hints to the proof would be much appreciated.