No constraints

Remark

Here, we consider a quadratic optimization problem without constraints, where is symmetric and .

Theorem

If is not positive semidefinite, then there are feasible solutions with arbitrarily low cost.

Proof:
There then exists a direction with . Then, for we get which since goes to as .

Theorem

If is positive semidefinite, a point is optimal if and only if .

Proof:
By a previous theorem, the problem is convex. By another theorem, an equivalent statement is that there does not exist a feasible descent direction at . Using another theorem, this is true if and only if there does not exist a such that . Whenever we can choose to get , so indeed is optimal iff .

Theorem

If is positive semidefinite and , then there are feasible solutions with arbitrarily low cost.

Proof:
Since is symmetric, and are orthogonal. Thus for some and . Since we know . It follows that which we can make arbitrarily small since .