SOS Is Not Obviously Automatizable, Even Approximately

Abstract

Suppose we want to minimize a polynomial p(x) = p(x_1,...,x_n), subject to some polynomial constraints q_1(x),...,q_m(x) >_ 0, using the Sum-of-Squares (SOS) SDP hierarachy. Assume we are in the "explicitly bounded" ("Archimedean") case where the constraints include x_i^2 <_ 1 for all 1 <_ i <_ n. It is often stated that the degree-d version of the SOS hierarchy can be solved, to
high accuracy, in time n^O(d). Indeed, I myself have stated this in several previous works.

The point of this note is to state (or remind the reader) that this is not obviously true. The difficulty comes not from the "r" in the Ellipsoid Algorithm, but from the "R"; a priori, we only know an exponential upper bound on the number of bits needed to write down the SOS solution. An explicit example is given of a degree-2 SOS program illustrating the difficulty.