A comparison of the Sherali–Adams, Lovász-Schrijver, and Lasserre relaxations for 0–1 programming

Sherali and Adams, Lovász and Schrijver and, recently, Lasserre have constructed hierarchies of successive linear or semidefinite relaxations of a 0–1 polytope PRⁿ converging to P in n steps. Lasserre's approach uses results about representations of positive polynomials as sums of squares and the dual theory of moments. We present the three methods in a common elementary framework and show that the Lasserre construction provides the tightest relaxations of P. As an application this gives a direct simple proof for the convergence of the Lasserre's hierarchy. We describe applications to the stable set polytope and to the cut polytope.