The quadratic optimization problem

Definition: Quadratic function

A function is called quadratic whenever it is of the form for some symmetric matrix , some and some .

Quadratic optimization deals with optimizing a quadratic function but with linear constraints.

Definition: Quadratic optimization problem

An optimization problem is called quadratic whenever it is of the form for some symmetric matrix , vector and constant , matrices , , some vectors , and some vector .

Theorem

Let be a quadratic function given by Then, for every , and , Proof:
We have

Corollary

For any we have

Remark

is the Hessian of .