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 ϵ-infeasible points that do not converge to a feasible point as ϵ 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.
@article{arxiv.2001.06511,
title = {A perturbation view of level-set methods for convex optimization},
author = {Ron Estrin and Michael P. Friedlander},
journal= {arXiv preprint arXiv:2001.06511},
year = {2020}
}