= Solution
For the equality constraint, the <Lagrangian> and dual function are
$$
L(x,z)=f(x)+z^T(Ax-b),
\qquad q(z)=\inf_xL(x,z),
$$
and the <Lagrangian dual problem> is $\sup_{z\in\mathbb R^m}q(z)$. <Weak duality> says $q(z)\leq p^*$ for every $z$. <Strong duality> means the dual supremum equals the primal infimum, usually with a dual maximizer. A sufficient convex constraint qualification is that $f$ be proper, closed and convex and that some $x\in\operatorname{ri}(\operatorname{dom}f)$ satisfy $Ax=b$.
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