# Foundations of gauge and perspective duality

## TL;DR

This paper revisits gauge duality, establishing a modern, unified framework with Fenchel-Rockafellar duality, and extends it to general nonnegative convex functions, enhancing understanding and applicability in convex optimization.

## Contribution

It provides a modern, unified explanation of gauge duality using a perturbation framework and extends the theory to broader classes of convex functions and models.

## Key findings

- Gauge duality can be explained via a perturbation-based duality approach.
- Primal solutions can be recovered from gauge dual solutions through rescaling.
- The framework applies to general nonnegative convex functions, including piecewise linear quadratic functions.

## Abstract

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.