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

Use multiplier $-z$ for $Ax-b-s=0$. The Lagrangian isThe infimum over $s$ is finite exactly when $z\ge 0$. The infimum over $x$ occurs at $x=Q^{-1}{A}^{T}z$, and henceThe dual is ${\max }_{z\ge 0}h(z)$. Since the primal objective is coercive, the explicit Slater conditionis sufficient for feasibility, attainment, and equality of primal and dual values.