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