  In polynomial optimization problems, nonnegativity constraints are typically
handled using the sum of squares condition. This can be efficiently enforced
using semidefinite programming formulations, or as more recently proposed by
Papp and Yildiz [18], using the sum of squares cone directly in a nonsymmetric
interior point algorithm. Beyond nonnegativity, more complicated polynomial
constraints (in particular, generalizations of the positive semidefinite,
second order and $\ell_1$-norm cones) can also be modeled through structured
sum of squares programs. We take a different approach and propose using more
specialized polynomial cones instead. This can result in lower dimensional
formulations, more efficient oracles for interior point methods, or
self-concordant barriers with lower parameters. In most cases, these
algorithmic advantages also translate to faster solving times in practice.
