## Equivariant Semidefinite Lifts of Regular Polygons

##### Author(s)

Fawzi, Hamza; Saunderson, James F; Parrilo, Pablo A.
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Given a polytope P n , we say that P has a positive semidefinite lift (psd lift) of size d if one can express P as the projection of an affine slice of the d×d positive semidefinite cone. Such a representation allows us to solve linear optimization problems over P using a semidefinite program of size d and can be useful in practice when d is much smaller than the number of facets of P. If a polytope P has symmetry, we can consider equivariant psd lifts, i.e., those psd lifts that respect the symmetries of P. One of the simplest families of polytopes with interesting symmetries is regular polygons in the plane. In this paper, we give tight lower and upper bounds on the size of equivariant psd lifts for regular polygons. We give an explicit construction of an equivariant psd lift of the regular 2n-gon of size 2n - 1, and we prove that our construction is essentially optimal by proving a lower bound on the size of any equivariant psd lift of the regular N-gon that is logarithmic in N. Our construction is exponentially smaller than the (equivariant) psd lift obtained from the Lasserre/sum-of-squares hierarchy, and it also gives the first example of a polytope with an exponential gap between equivariant psd lifts and equivariant linear programming 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.

##### Department

Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science; Massachusetts Institute of Technology. Laboratory for Information and Decision Systems##### Journal

Mathematics of Operations Research

##### Publisher

Institute for Operations Research and the Management Sciences (INFORMS)

##### Citation

Fawzi, Hamza, James Saunderson and Pablo A. Parrilo. "Equivariant semidefinite lifts of regular polygons." Mathematics of Operations Research 42, no. 2 (2016): 472-494.

Version: Original manuscript

##### ISSN

0364-765X