2009/01/23 by J. Maurice Rojas, Rojas, J. Maurice, Swaminathan Sethuraman +1
Computer Science · Mathematics · #Advanced Optimization Algorithms Research #Algebraic Geometry (math.AG) #FOS: Mathematics #Numerical Methods and Algorithms #Optimization and Control (math.OC) #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.0901.3786
openalex publication_date 2009/01/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
To prove that a polynomial is nonnegative on Rn one can try to show that it is a sum of squares of polynomials (SOS). The latter problem is now known to be reducible to a semidefinite programming (SDP) computation much faster than classical algebraic methods, thus enabling new speed-ups in algebraic optimization. However, exactly how often nonnegative polynomials are in fact sums of squares of polynomials remains an open problem. Blekherman was recently able to show that for degree k polynomials in n variables -- with k>=4 fixed -- those that are SOS occupy a vanishingly small fraction of those that are nonnegative on Rn, as n tends to infinity. With an eye toward the case of small n, we refine Blekherman's bounds by incorporating the underlying Newton polytope, simultaneously sharpening some of his older bounds along the way. Our refined asymptotics show that certain Newton polytopes may lead to families of polynomials where efficient SDP can still be used for most inputs.