Given a problem: $$\min_x f(x)$$ subject to $$g(x) \le C$$
In general, when it is equivalent to the problem $$\min_x f(x) + \lambda g(x)$$ for certain $\lambda$?
Here my equivalence means : the solution for the latter given a $\lambda$ must be the solution of the former for some $C$, though the mapping between $\lambda$ and $C$ can not be easily found. And vice versa.
Clearly, for convex problem where $g$ and $f$ are convex functions, this is always true due to the strong duality. But in general, is it true? For example, we $f$ is convex but $g$ is not (or vice versa), do we still have the equivalence? Seems like it is easy to claim that the solution set of the second problem (solutions for $\lambda \in [0, \infty)$) must be contained in the solution set of the first problem (solutions for $C \in [0, \infty)$). How about the reverse?