Variational forms of non-convex functions

I am trying to understand what kind of variational forms exist for non-convex functions. Alternatively, are there conjugate forms which attain strong duality? For a non-convex function $f$, I am interested in finding an $f^*$ which satisfy the following relation

$$\min\limits_u f(u)=\min\limits_u\sup\limits_v~~(u\cdot v)-f^*(v)$$

Are there results stating under what conditions does the strong duality hold for the Fenchel biconjugate of non-convex functions? $u$ here need not be a finite dimensional vector.

Thanks!