A quadratic programming (QP) problem is to minimize a quadratic function subject to some equality and/or inequality constraints:

(233) |

We first consider the special case where the QP problem is only subject to equality constraints and we assume , i.e., the number of constraints is smaller than the number of unknowns in . Then the solution has to satisfy , i.e., it has to be on hyper planes in the N-D space.

The Lagrangian function of the QP problem is:

(234) |

(235) |

These two equations can be combined and expressed in matrix form as:

(236) |

**Example**

If , i.e., the number of equality constraints is the same as the number of variables, then the variable is uniquely determined by the linear system , as the intersect of hyper planes, independent of the objective function . Further if , i.e., the system is over constrained, and its solution does not exist in general. It is therefore more interesting to consider QP problems subject to both equality and inequality constraints: