Say we have a set
$C:= \{y\in \mathbb{R}^n : g_i(y) \leq 0, \space i=1,...,m\}$
where $g_i : \mathbb{R}^n \to \mathbb{R}$ are convex and differentiable functions, then we have
$\tilde C : = \{y: g_i(x) + \langle \nabla g_i(x),y-x \rangle \leq 0, i=1,...,m\}$,
the polyhedral approximation of $C$.
Then why is it that $\tilde C \supset C $ ?
According to the gradient inequality: $g(y) \geq g(x) + \langle \nabla g(x),y-x \rangle $, then doesn't this imply that $\tilde C \subset C$ ?