Apologies for a basic question.
For a convex problem $$\min_x f(x) + g(x) ,$$ where assuming $f(x)$ is differentiable with $m$-Lipschitz continuous gradient, while $g(x)$ is closed convex proper.
Then, KKT point $x^*$ satisfies $$0 \in \nabla f(x^*) + \partial g(x^*) .$$
My question is: Can we say $\nabla f(x^*) = 0$ at the KKT point? If yes, then it means, the subgradient $\partial g \in 0$ at the KKT Point? or do I miss any assumption?
No, you cannot say this. To give a simple example, consider the functions $f, g \colon \mathbb R \to \mathbb R$ defined via $f(x) = -g(x) = x$. Then, every $x^* \in \mathbb R$ is a minimizer with $f'(x^*) = 1$ and $\partial g(x^*) = \{-1\}$.