Sum of squares generalizations for conic sets

TL;DR

This paper explores generalized sum of squares cones for polynomial optimization, aiming to improve efficiency and scalability of solving structured polynomial constraints through specialized cones.

Contribution

It introduces a novel approach using specialized polynomial cones for sum of squares constraints, enhancing computational efficiency over traditional methods.

Findings

Lower-dimensional formulations achieved

More efficient oracles for interior point methods developed

Faster solving times demonstrated in practice

Abstract

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 smaller parameters. In most cases, these algorithmic advantages also translate to faster solving times in practice.