  Level-set methods for convex optimization are predicated on the idea that
certain problems can be parameterized so that their solutions can be recovered
as the limiting process of a root-finding procedure. This idea emerges time and
again across a range of algorithms for convex problems. Here we demonstrate
that strong duality is a necessary condition for the level-set approach to
succeed. In the absence of strong duality, the level-set method identifies
$\epsilon$-infeasible points that do not converge to a feasible point as
$\epsilon$ tends to zero. The level-set approach is also used as a proof
technique for establishing sufficient conditions for strong duality that are
different from Slater's constraint qualification.
