Optimality certificates for convex minimization and Helly numbers

TL;DR

This paper introduces a duality framework for convex minimization over non-convex sets, such as integer points in polytopes, establishing conditions for strong duality that extend previous results.

Contribution

It defines a new family of dual problems for convex minimization over non-convex sets and proves strong duality under natural conditions, advancing duality theory in this area.

Findings

Established a new duality framework for convex minimization over non-convex sets.

Proved strong duality for a more restrictive dual problem under natural conditions.

Extended duality results to cases like integer programming within polytopes.

Abstract

We consider the problem of minimizing a convex function over a subset of R^n that is not necessarily convex (minimization of a convex function over the integer points in a polytope is a special case). We define a family of duals for this problem and show that, under some natural conditions, strong duality holds for a dual problem in this family that is more restrictive than previously considered duals.