 Solution

For $y\in {\mathbb{R}}^m$ and $\gamma \ge 0$, define the optimization LagrangianThe Lagrangian primal isFor feasible $x$, the supremum is $c^{T}x$: equality terms vanish and the nonpositive inequality term has maximum zero at $\gamma =0$. If $Ax\ne b$, scale $y$ in the direction $b-Ax$ to make the supremum infinite. If the inequality fails, send $\gamma \to \infty$. Thus the inner supremum is exactly the objective extended to $+\infty$ off the feasible set. This proves equivalence with the original problem, including its infeasible case.