Suppose I have an objective function $$ \min_{\bf x} f({\bf x}) + \lambda \|{\bf x}\|_1 $$ where $f({\bf x})$ is a differentiable non-convex function. I am trying to derive an upper bound for $\lambda$ similar to the procedure used in the lasso (as given here). Is it correct to state that
$$ {\bf 0} \in f'({\bf x}) + \lambda \partial\|{\bf x}\|_1 $$ which leads $$ \lambda > max(|f'({\bf 0})|) $$ for ${\bf x} = {\bf 0}$ to be a stationary solution.