Is every optimization problem with a piece-wise affine objective function the dual of some differentiable problem?

Question

It is well known that a problem can have a $C^1$ objective function and a convex feasible set, while the dual problem can be piece-wise $C^1$ only.

So I'm wondering - if you have a piece-wise affine, concave objective function that you want to maximize over a convex set (call this problem $P$), does there always exist a problem $P'$ such that the objective function is $C^1$, the feasible set is convex, and the dual of $P'$ is $P$?

Answer 1

This is a very good question that I can sort of answer, but not in a 100% satisfactory way.

Since a problem with a piece-wise affine objective function can always be expressed as an equivalent linear program, the dual of this program will fulfill the criteria.

However, this type of duality doesn't have any sort of injectivity - since a nonlinear $C^1$ problem often has a piece-wise affine dual, of which we can construct an LP dual that is different from the original problem.