Algebra and Combinatorics Seminar

Date: February 17, 2017

Time: 3:00PM - 3:50PM

Location: BLOC 628

Speaker: Timo de Wolff, Texas A&M University

Title: A Positivstellensatz for Sums of Nonnegative Circuit Polynomials

Abstract: Deciding nonnegativity of real polynomials is a fundamental problem in real algebraic geometry and polynomial optimization. Since this problem is NP-hard, one is interested in finding sufficient conditions (certificates) for nonnegativity, which are easier to check. The standard certificates for nonnegativity are sums of squares (SOS). In practice, SOS based semidefinite programming (SDP) is the standard method to solve polynomial optimization problems.

In 2014, Iliman and I introduced an entirely new nonnegativity certificate based on sums of nonnegative circuit polynomials (SONC), which are independent of sums of squares. We successfully applied SONCs to global nonnegativity problems.

In Summer 2016, Dressler, Iliman, and I proved a Positivstellensatz for SONCs, which provides a converging hierarchy of lower bounds for constrained polynomial optimization problems. These bounds can be computed efficiently via relative entropy programming.

In this first of two talks on the topic I will give a brief overview about SONCs and Positivstellensätze in general and then introduce and prove our Positivstellensatz.

The second, corresponding talk will occur directly afterwards in the geometry seminar.