Quadratic programming (QP) is a special type of mathematical optimization problem.
The quadratic programming problem can be formulated like this:
Assume x belongs to Rn space. The (n x n) matrix (math) E is positive semidefinite and h is any (n x 1) vector.
Minimize (with respect to x)
f(x) = 0.5 x' E x + h' x
with at least one instance of the following kind of constraints (if there exists an answer then it satisfies these):
(1) A*x <= b (inequality constraint) (2) C*x = d (equality contraint)
If E is positive definite then f(x) is a convex function , and constraints are linear functions, we have from optimization theory that for point x to be an optimum point it is necessary and sufficient that x is a Karush-Kuhn-Tucker (KKT) point.
If there are only equality constraints, then the QP can be solved by a linear system. Otherwise, the most common method of solving a QP is an interior point method, such as LOQO. Active set methods are also commonly used.