From discriminants to optimization: deciding copositivity

Abstract

Deciding whether a polynomial is nonnegative on the positive orthant, that is, whether it is copositive, is a basic question in real algebraic geometry, with applications in polynomial optimization.

In this talk I will present joint work with Elisenda Feliu and Máté L. Telek, in which we approach this question through discriminants over the positive orthant. For a polynomial with prescribed support and coefficient signs, this discriminant is a semialgebraic set that separates coefficient space into chambers on which the positive zero set has constant topology, so that copositivity becomes a question of chamber membership. We decide this membership by following a path in coefficient space until it first intersects this discriminant; the position of this intersection point, computable with homotopy continuation techniques, provides a criterion for deciding copositivity. As an application we show that under a combinatorial condition on the signed supports called nonseparability, the copositive polynomials with that signed support are exactly the sums of nonnegative circuits.