Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 339 / 2 / a / Solution

For the equality constraint, the Lagrangian and dual function areand the Lagrangian dual problem is ${\sup }_{z\in {\mathbb{R}}^m}q(z)$. Weak duality says $q(z)\le p^{\ast }$ 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 $x\in \mathrm{ri}(\mathrm{dom}f)$ satisfy $Ax=b$.