  We revisit the foundations of gauge duality and demonstrate that it can be
explained using a modern approach to duality based on a perturbation framework.
We therefore put gauge duality and Fenchel-Rockafellar duality on equal
footing, including explaining gauge dual variables as sensitivity measures, and
showing how to recover primal solutions from those of the gauge dual. This
vantage point allows a direct proof that optimal solutions of the
Fenchel-Rockafellar dual of the gauge dual are precisely the primal solutions
rescaled by the optimal value. We extend the gauge duality framework to the
setting in which the functional components are general nonnegative convex
functions, including problems with piecewise linear quadratic functions and
constraints that arise from generalized linear models used in regression.
