Minimize $f(x)$ subject to $x \in C$, $C$ is convex set
Necessary and sufficient condition for $\bar x$ to be global minima of the above problem is $<\nabla f(\bar x), x- \bar x> \geq 0 $ $\forall x \in C$
proof in textbook:
$ x \in C, \bar x, \implies \bar x + \lambda (x- \bar x) \in C \forall \lambda \in (0,1) \\ f(\bar x + \lambda (x- \bar x)) \geq f(\bar x) \\ $ [what if global min is not in C? as showin below in diagram]
by taylors expansion we get
$<\nabla f(\bar x), x- \bar x> \geq 0 $ $\forall x \in C$
It also looks intuitive when i think as below
BUt what if $\bar x$ is not in C ie., when i consider C not containing global min then $\bar x + \lambda (x- \bar x) $ may not be in C right?
