The equivalence between doubly nonnegative relaxation and semidefinite relaxation for binary quadratic programming problems

TL;DR

This paper proves the equivalence between doubly nonnegative relaxation and semidefinite relaxation for binary quadratic programming, showing that the former is a tighter relaxation and providing numerical comparisons.

Contribution

It establishes the theoretical equivalence between doubly nonnegative and semidefinite relaxations for BQP, including the special case of max-cut, and offers numerical insights.

Findings

Doubly nonnegative relaxation is equivalent to a tighter semidefinite relaxation for BQP.

For max-cut, the relaxation reduces to the standard semidefinite relaxation.

Numerical results compare the effectiveness of these relaxations.

Abstract

It has recently been shown (Burer, Math. Program Ser. A 120:479-495, 2009) that a large class of NP-hard nonconvex quadratic programming problems can be modeled as so called completely positive programming problems, which are convex but still NP-hard in general. A basic tractable relaxation is gotten by doubly nonnegative relaxation, resulting in a doubly nonnegative programming. In this paper, we prove that doubly nonnegative relaxation for binary quadratic programming (BQP) problem is equivalent to a tighter semidifinite relaxation for it. When problem (BQP) reduces to max-cut (MC) problem, doubly nonnegative relaxation for it is equivalent to the standard semidifinite relaxation. Furthermore, some compared numerical results are reported.