Duality for Mixed-Integer Convex Minimization

TL;DR

This paper extends classical optimality conditions to mixed-integer convex problems using lattice-free polyhedra, enabling the definition of an exact dual problem for such complex optimization scenarios.

Contribution

It introduces a novel extension of Karush-Kuhn-Tucker conditions for mixed-integer convex problems based on lattice-free polyhedra, facilitating exact duality.

Findings

Extended KKT conditions for mixed-integer convex problems

Defined an exact dual problem for these problems

Utilized lattice-free polyhedra in optimality conditions

Abstract

We extend in two ways the standard Karush-Kuhn-Tucker optimality conditions to problems with a convex objective, convex functional constraints, and the extra requirement that some of the variables must be integral. While the standard Karush-Kuhn-Tucker conditions involve separating hyperplanes, our extension is based on lattice-free polyhedra. Our optimality conditions allow us to define an exact dual of our original mixed-integer convex problem.