Definable ellipsoid method, sums-of-squares proofs, and the graph isomorphism problem

The ellipsoid method is an algorithm that solves the (weak) feasibility and linear optimization problems for convex sets by making oracle calls to their (weak) separation problem. We observe that the previously known method for showing that this reduction can be done in fixed-point logic with counting (FPC) for linear and semidefinite programs applies to any family of explicitly bounded convex sets. We further show that the exact feasibility problem for semidefinite programs is expressible in the infinitary version of FPC. As a corollary, we get that, for the graph isomorphism problem, the Lasserre/sums-of-squares semidefinite programming hierarchy of relaxations collapses to the Sherali--Adams linear programming hierarchy, up to a small loss in the degree.