# Characterizations of robust and stable duality for linearly perturbed uncertain optimization problems

## TL;DR

This paper develops a robust optimization framework with duality relations that account for uncertainty, providing characterizations of zero-duality gap using epsilon-minima and epsilon-subdifferentials, applicable to conic and infinite optimization.

## Contribution

It introduces a novel robust duality model for uncertain optimization problems, linking duality properties to epsilon-minima and epsilon-subdifferentials, and analyzes extreme cases.

## Key findings

- Characterizes robust zero-duality gap using epsilon-minima.
- Provides duality relations for uncertain conic and infinite optimization.
- Analyzes classical and extreme duality cases in detail.

## Abstract

We introduce a robust optimization model consisting in a family of perturbation functions giving rise to certain pairs of dual optimization problems in which the dual variable depends on the uncertainty parameter. The interest of our approach is illustrated by some examples, including uncertain conic optimization and infinite optimization via discretization. The main results characterize desirable robust duality relations (as robust zero-duality gap) by formulas involving the epsilon-minima or the epsilon-subdifferentials of the objective function. The two extreme cases, namely, the usual perturbational duality (without uncertainty), and the duality for the supremum of functions (duality parameter vanishing) are analyzed in detail.