Certifying Polynomial Nonnegativity via Hyperbolic Optimization

Abstract

We describe a new approach to certifying the global nonnegativity of multivariate polynomials by solving hyperbolic optimization problems-a class of convex optimization problems that generalize semidefinite programs. We show how to produce families of nonnegative polynomials (which we call hyperbolic certificates of nonnegativity) from any hyperbolic polynomial. We investigate the pairs (n, d) for which there is a hyperbolic polynomial of degree d in n variables such that an associated hyperbolic certificate of nonnegativity is not a sum of squares. If d ≥ 4 we show that this occurs whenever n ≥ 4. In the degree three case, we find an explicit hyperbolic cubic in 43 variables that gives hyperbolic certificates that are not sums of squares. As a corollary, we obtain the first known hyperbolic cubic no power of which has a definite determinantal representation. Our approach also allows us to show that, given a cubic p, and a direction e, the decision problem "Is p hyperbolic with respect to e?" is co-NP hard.

Linear Principal Minor Polynomials: Hyperbolic Determinantal Inequalities and Spectral Containment

“…is not a sum of squares, this has interesting implications for determinantal representations as well as a hyperbolic certificate of nonnegativity of ∆ v,a (p) which cannot be recovered by sums of squares. Saunderson [27] characterized all pairs (d, n) for which there exists such a hyperbolic polynomial p ∈ R[x 1 , . .…”

“…Therefore, any instance where ∆ a,b (p) is not a sum of squares gives an example of a hyperbolic polynomial none of whose powers has a definite symmetric determinantal representation. Another source of interest in such examples comes from the point of view taken in [27], as these give rise to families of polynomials that are not sums of squares but whose nonnegativity can be certified via hyperbolic programming. Saunderson [27] characterized all pairs (d, n) for which there exists such a hyperbolic polynomial p ∈ R[x] = R[x 1 , .…”

“…Another source of interest in such examples comes from the point of view taken in [27], as these give rise to families of polynomials that are not sums of squares but whose nonnegativity can be certified via hyperbolic programming. Saunderson [27] characterized all pairs (d, n) for which there exists such a hyperbolic polynomial p ∈ R[x] = R[x 1 , . .…”

“…Remark 6.2. In [27] Saunderson constructs a hyperbolic cubic in 43 variables whose Bézout matrix is not a matrix sum of squares. This is the smallest such example that has been known so far.…”

“…Using lpm polynomials we construct a hyperbolic cubic in 6 variables which has a Rayleigh difference that is not a sum of squares. The previously smallest known example with 43 variables was contructed by Saunderson in [27]. Finally, we study whether the eigenvalue vector λ of a matrix X lying in the hyperbolicity cone of a stable lpm polynomial P (X) lies in the hyperbolicity cone of p(x) and show how this is related to a potential generalization of the classical Schur-Horn theorem [28,14].…”