Quadratically Constrained Quadratic Program

In mathematical optimization, a quadratically constrained quadratic program (QCQP) is an optimization problem in which both the objective function and the constraints are quadratic functions. It has the form

\begin{align} & \text{minimize} && \tfrac12 x^\mathrm{T} P_0 x + q_0^\mathrm{T} x \\ & \text{subject to} && \tfrac12 x^\mathrm{T} P_i x + q_i^\mathrm{T} x + r_i \leq 0 \quad \text{for } i = 1,\dots,m, \\ &&& Ax = b, \end{align}

where P0, … Pm are n-by-n matrices and x ∈ Rn is the optimization variable.

If P0, … Pm are all positive semidefinite, then the problem is convex. If these matrices are neither positive or negative semidefinite, the problem is non-convex. If P1, … Pm are all zero, then the constraints are in fact linear and the problem is a quadratic program.

Read more about Quadratically Constrained Quadratic Program:  Hardness, Relaxation, Example, Solvers and Scripting (programming) Languages

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