CMX Lunch Seminar: James Saunderson

Certifying polynomial nonnegativity via hyperbolic optimization

Certifying nonnegativity of multivariate polynomials is fundamental to solving optimization problems modeled with polynomials. One well-known way to certify nonnegativity is to express a polynomial as a sum of squares. Furthermore, the search for such a certificate can be carried out via semidefinite optimization. An interesting generalization of semidefinite optimization, that retains many of its good algorithmic properties, is hyperbolic optimization. Are there natural certificates of nonnegativity that we can search for via hyperbolic optimization, and that are not obviously captured by sums of squares? If so, these could have the potential to generate hyperbolic optimization-based relaxations of optimization problems with that may be stronger, in some sense, than semidefinite optimization-based relaxations.

In this talk, I will describe one candidate for such "hyperbolic certificates of nonnegativity," and discuss what is known about their relationship with sums of squares.