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 use this observation to 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 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.

翻译：椭球法是一种通过对其（弱）分离问题执行预言机调用来求解凸集（弱）可行性和线性优化问题的算法。我们观察到，先前已知的用于证明在线性规划和半定规划中这种归约可以在带计数的定点逻辑（FPC）中实现的方法，适用于任意有显式有界凸集族。利用这一观察，我们证明半定规划的精确可行性问题在FPC的无穷版本中是可表达的。作为推论，对于同构问题，Lasserre/平方和半定规划松弛层次结构在度数上的微小损失下坍缩至Sherali-Adams线性规划层次结构。