Convexifiability of Continuous and Discrete Nonnegative Quadratic Programs for Gap-Free Duality

TL;DR

This paper demonstrates that certain nonconvex quadratic programs with nonnegative variables exhibit a hidden convexifiability property that guarantees zero duality gap, including in discrete and robust cases, under specific conditions.

Contribution

It establishes the convexifiability property for nonnegative quadratic programs, revealing zero duality gaps in classes like mixed integer quadratic programs and their robust counterparts.

Findings

Convexifiability guarantees zero duality gap in nonnegative quadratic programs.

Hidden convexifiability applies to discrete and mixed integer quadratic programs.

Robust mixed integer quadratic programs also exhibit zero duality gaps under certain conditions.

Abstract

In this paper we show that a convexifiability property of nonconvex quadratic programs with nonnegative variables and quadratic constraints guarantees zero duality gap between the quadratic programs and their semi-Lagrangian duals. More importantly, we establish that this convexifiability is hidden in classes of nonnegative homogeneous quadratic programs and discrete quadratic programs, such as mixed integer quadratic programs, revealing zero duality gaps. As an application, we prove that robust counterparts of uncertain mixed integer quadratic programs with objective data uncertainty enjoy zero duality gaps under suitable conditions. Various sufficient conditions for convexifiability are also given.