Consider the following constrained optimization: Min $f(x)$, $x\in C\subset R^n$, where C is convex.
We know that one characterization of a local minimum (necessary condition) is the following: STANDARD CHARACTERIZATION: If $a$ is a local minimum, then $\langle x-a,\nabla f(a)\rangle \geq 0 \:\: \forall x\in \bar C$
I saw in paper that an alternative condition was used:
ALTERNATIVE CHARACTERIZATION:
If $a$ is a local minimum, then $\langle x-a,\nabla f(x)\rangle \geq 0 \:\: \forall x\in \bar C$ I wasn't able to convince myself that this is true. Can anyone comment on this alternative definition?
If the function $f$ is convex, its gradient is monotone, hence, $$\langle x - y, \nabla f(x) - \nabla f(y)\rangle \ge 0.$$ You can apply this with $y = a$ and this shows that $$\langle x -a, \nabla f(a) \rangle \ge 0 \quad\forall x \in \bar C$$ implies $$\langle x -a, \nabla f(x) \rangle \ge 0 \quad\forall x \in \bar C.$$