Exact augmented Lagrangian duality for mixed integer convex optimization

TL;DR

This paper explores conditions under which mixed integer convex optimization problems have exact augmented Lagrangian dual representations using sharp augmenting functions, extending recent results for MILP and MIQP.

Contribution

It provides a generalizable constructive proof technique for the existence of exact penalty representations in mixed integer convex programs, generalizing prior specific cases.

Findings

Established conditions for exact penalty representations.

Extended recent results from MILP and MIQP.

Quantified finite penalty parameters.

Abstract

Augmented Lagrangian dual augments the classical Lagrangian dual with a non-negative non-linear penalty function of the violation of the relaxed/dualized constraints in order to reduce the duality gap. We investigate the cases in which mixed integer convex optimization problems have an exact penalty representation using sharp augmenting functions (norms as augmenting penalty functions). We present a generalizable constructive proof technique for proving existence of exact penalty representations for mixed integer convex programs under specific conditions using the associated value functions. This generalizes the recent results for MILP (Feizollahi, Ahmed and Sun, 2017) and MIQP (Gu, Ahmed and Dey 2020) whilst also providing an alternative proof for the aforementioned along with quantification of the finite penalty parameter in these cases.