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.)