** Quadratically-constrained quadratic programming** (QCQP) problems are optimization problems with a quadratic objective function and quadratic constraints. The general QCQP problem has the following form:

\[ \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.

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