This section describes how to use moment matrics and conic programming, semidefinite programming(SDP) and second-order cone programming(SOCP) to construct variational relaxations.

We spent a good amount of time going over the background information in the section. First we discussed the definition of a moment matrix and the property that any valid moment matrix is positive semidefinite. Next we looked at the definition of two different bases, the multinomial base and the indicator base and went trhough the lemma that shows that it is always possible to convert between the two. Lastly, we looked at the new definition of the marginal polytope for hypergraphs in terms of the multinomial base.

Next we went over how the Lasserre sequence(moment matrices) provides a nested hierarchy of outer bounds on the marginal polytope. For any hypergraph on m nodes, the

The last section discussed an alternate second-order cone relaxation technique, which we deffered discussing until next week’s meeting.