Non-SOS Putinar-like certificates of non-negativity: semi- and fully-sparse

Certificates of non-negativity are fundamental tools for polynomial optimization. Most of the current literature on certificates of non-negativity have been concentrated on certificates based on Sum-of-squares (SOS) polynomials. We propose a framework for constructing certificates of non-negativity based on any class of non-negative polynomials satisfying some mild assumptions. These certificates are similarly structured as Putinar's certificate. In addition to classic certificates of non-negativity, this framework can be used to obtain sparse certificates of non-negativity. For instance, we construct sparse certificates based on other polynomials such as SDSOS-, SAGE-, SONC- and SOS-polynomials. 

Sparse certificates are often much more efficient to compute than non-sparse certificates of non-negativity, and we expect our work to close the gap between the applicability of SOS-based and other types of certificates of non-negativity.

This is a joint work with J.C. Vera Lizcano and L.F. Zuluaga

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