2 research outputs found
Tight Sum-of-Squares lower bounds for binary polynomial optimization problems
We give two results concerning the power of the Sum-of-Squares(SoS)/Lasserre
hierarchy. For binary polynomial optimization problems of degree and an
odd number of variables , we prove that levels of the
SoS/Lasserre hierarchy are necessary to provide the exact optimal value. This
matches the recent upper bound result by Sakaue, Takeda, Kim and Ito.
Additionally, we study a conjecture by Laurent, who considered the linear
representation of a set with no integral points. She showed that the
Sherali-Adams hierarchy requires levels to detect the empty integer hull,
and conjectured that the SoS/Lasserre rank for the same problem is . We
disprove this conjecture and derive lower and upper bounds for the rank
Matchings, hypergraphs, association schemes, and semidefinite optimization
We utilize association schemes to analyze the quality of semidefinite
programming (SDP) based convex relaxations of integral packing and covering
polyhedra determined by matchings in hypergraphs. As a by-product of our
approach, we obtain bounds on the clique and stability numbers of some regular
graphs reminiscent of classical bounds by Delsarte and Hoffman. We determine
exactly or provide bounds on the performance of Lov\'{a}sz-Schrijver SDP
hierarchy, and illustrate the usefulness of obtaining commutative subschemes
from non-commutative schemes via contraction in this context