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

Famous quotes containing the words constrained and/or program:

    “An expansive life, one not constrained by four walls, requires as well an expansive pocket.”
    —Anton Pavlovich Chekhov (1860–1904)

    “The story is told of a man who, seeing one of the thoroughbred stables for the first time, suddenly removed his hat and said in awed tones, “My Lord! The cathedral of the horse.””
    —For the State of Kentucky, U.S. public relief program (1935-1943)