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Sum of squares generalizations for conic sets

Chris Coey
Lea Kapelevich
Mathematical Programming (2022)
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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 2019, 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 l1-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.