If $x_0$ is a minimizer of a minimization problem, then there exist $y\in\partial f(x_0)$ and $\lambda>0$ such that $-y\in\lambda\partial f(x_{0})$

Question

I am trying to solve the following convex minimization problem:

minimize: $f(x)$

subject to: $g(x)\leq 0.$

Where $f, g$ : $\mathbb{R}^{n}\rightarrow \mathbb{R}$ are convex functions (not necessarily differentiable). I want to show that if $x_{0}$ is a minimizer of this problem, then there exist $y\in\partial f(x_0)$ and $\lambda>0$ such that $-y\in\lambda\partial f(x_{0})$.

If $C:=\{x|g(x)\leq 0\}$, I know we have that $0 \in \partial f(x_0) + N_C(x_0)$, and I figure that plays a role here, but I am not sure how to use it.

Answer 1

The result as stated is not correct.

Take $f(x) = x$, $g(x) = x^2$. Then the minimiser is $x_0 = 0$ but $\partial f(x_0) = \{ 1 \}$ and there is no $\lambda >0$ such that $-1 \in \lambda \partial f(x_0)$.