Equivariant semidefinite lifts of regular polygons

Abstract: Given a polytope P in $\mathbb{R}^n$, we say that P has a positive semidefinite lift (psd lift) of size d if one can express P as the linear projection of an affine slice of the positive semidefinite cone $\mathbf{S}^d_+$. If a polytope P has symmetry, we can consider equivariant psd lifts, i.e. those psd lifts that respect the symmetry of P. One of the simplest families of polytopes with interesting symmetries are regular polygons in the plane, which have played an important role in the study of linear programming lifts (or extended formulations). In this paper we study equivariant psd lifts of regular polygons. We first show that the standard Lasserre/sum-of-squares hierarchy for the regular N-gon requires exactly ceil(N/4) iterations and thus yields an equivariant psd lift of size linear in N. In contrast we show that one can construct an equivariant psd lift of the regular 2^n-gon of size 2n-1, which is exponentially smaller than the psd lift of the sum-of-squares hierarchy. Our construction relies on finding a sparse sum-of-squares certificate for the facet-defining inequalities of the regular 2^n-gon, i.e., one that only uses a small (logarithmic) number of monomials. Since any equivariant LP lift of the regular 2^n-gon must have size 2^n, this gives the first example of a polytope with an exponential gap between sizes of equivariant LP lifts and equivariant psd lifts. Finally we prove that our construction is essentially optimal by showing that any equivariant psd lift of the regular N-gon must have size at least logarithmic in N.