Consider feasible problems.
\begin{align} \min_{x}\quad & 0 \\ \textrm{s.t.} \quad & \dots \end{align}
so the optimal is either p^{*} = 0 or p^{*} = +\infty. And thus the Lagrange Dual Problem boils down to checking if d^{*} \geq 0, if so, then the original problem is’nt feasible (since dual gives a lower bound.)