I'm working with an optimization problem where we are using Lagrangian relaxation to relax the constraints on a problem, however, there are some cases where the constraints of the problem cannot be met, $\nexists x ~ \text{s.t.} ~ f_i(x) \leq 0 ~\forall i$). (It is for this exact reason that we want to use Lagrangian relaxation)
Unless I am missing something, when looking at the dual problem, it seems that in the case that our feasible set is empty then $\sup_{\lambda \geq 0} \min_x \mathcal{L}(x, \lambda)$ will always return $\infty$. If this is true, is there anything to be gained by looking at the primal and dual forms of the Lagrangian of our problem? In other words, is $\exists x \in \text{dom}(f) ~ \text{s.t.} ~ f_i(x) \leq 0 ~ \forall i$, a prerequisite for the primal and dual forms of a problem to be useful?