Let be a quadratic function given by Given a point and direction , if , then is a descent direction at .
Proof:
For we get which is negative in some interval since and .
Theorem
Let be a convex quadratic function given by A direction is a descent direction at a point if and only if Proof:
"if" is given by the previous theorem. Now suppose . We know from a previous theoremprevious theorem that . Thus so is not a descent direction.