Consider this specific optimization problem: \begin{equation*} \begin{aligned} & \underset{x}{\text{minimize}} & & f_0(x) \\ & \text{subject to} & & f_i(x) \leq 0, \; i = 1, \ldots, m. & & h_i(x) = 0, \; i = 1, \ldots, p. \end{aligned} \end{equation*} where $f_0(x)$ is differentiable, $X$ is the feasible set, $X$ is a convex set, and $X$ = {$x$|$f_i(x)$ $\leq$ $0$, $i = 1, \ldots, m, h_i(x)$=$0$, $i$ = $1$,...,$p$}.
From the Convex Optimization book of Stephen Boyd, I know how to prove that:
$x$ is the optimal point if and only if $x \in X$ and $\nabla f_0(x)^T(y - x) \geq 0\text{ for all } y\in X$, $y \neq x$.
However, when X is a polyhedral set, I know that $x$ is the optimal point if and only if
min($\nabla f_0(x)^T(y - x)) = 0\text{ for all } y\in X$, $y \neq x$ , but I don't know how to prove it.