Invariants of SDP exactness in quadratic programming

TL;DR

This paper investigates the conditions under which the Shor relaxation of quadratic programs remains exact regardless of the feasible set's generators, providing explicit descriptions and algorithms for binary quadratic programs.

Contribution

It introduces invariance conditions for the exactness region of the Shor relaxation and applies these to binary quadratic programs with explicit solution methods.

Findings

Invariance of the exactness region under generator changes.

Explicit characterization of exactness region for binary quadratic programs.

An algorithm for generating candidate solutions for binary quadratic programs.

Abstract

In this paper we study the Shor relaxation of quadratic programs by fixing a feasible set and considering the space of objective functions for which the Shor relaxation is exact. We first give conditions under which this region is invariant under the choice of generators defining the feasible set. We then describe this region when the feasible set is invariant under the action of a subgroup of the general linear group. We conclude by applying these results to quadratic binary programs. We give an explicit description of objective functions where the Shor relaxation is exact and use this knowledge to design an algorithm that produces candidate solutions for binary quadratic programs.