$\begin{array}{ll} \mbox{minimize} & q_0(y) \\ \mbox{subject to} & q_i(y) \leq 0 \, \forall i = 1, \cdots, m \end{array}$ where $$q_i(y) = \frac{1}{2} y^t Q_iy + y^tb_i + c_i, \, y \in R^n$$ for all $$i = 0, 1, \cdots, m$$. The problem is convex if $$Q_i$$ is positive, semidefinite ($$Q_i \succeq 0$$) for all $$i$$, in which case an elegant duality structure is available.