Feasible directions and descent directions

Here, we consider a general optimization problemoptimization problem where and is any function.

A nonzero is a feasible direction at if there exists an such that for every . is a descent direction at if there exists an such that for every .

If is both a feasible direction and a descent direction it is called a feasible descent direction for at .

Suppose is a local optimal solution. Then there does not exist a feasible descent direction at .

Proof:
Suppose is a feasible descent direction at . Let be a global optimal solution in its neighbourhood of radius . Then there exists a such that and , contradicting local optimality.