2005/11/24 by Markus Schweighofer, Schweighofer, Markus · 1 citation
Computer Science · Mathematics · #13J30 #90C26 #Advanced Optimization Algorithms Research #Algebraic Geometry (math.AG) #Commutative Algebra and Its Applications #FOS: Mathematics #Optimization and Control (math.OC) #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.math/0511617
openalex publication_date 2005/11/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this work, the combine the theory of generalized critical values with the theory of iterated rings of bounded elements (real holomorphy rings). We consider the problem of computing the global infimum of a real polynomial in several variables. Every global minimizer lies on the gradient variety. If the polynomial attains a minimum, it is therefore equivalent to look for the greatest lower bound on its gradient variety. Nie, Demmel and Sturmfels proved recently a theorem about the existence of sums of squares certificates for such lower bounds. Based on these certificates, they find arbitrarily tight relaxations of the original problem that can be formulated as semidefinite programs and thus be solved efficiently. We deal here with the more general case when the polynomial is bounded from belo w but does not necessarily attain a minimum. In this case, the method of Nie, Demmel and Sturmfels might yield completely wrong results. In order to overcome this problem, we replace the gradient variety by larger semialgebraic sets which we call gradient tentacles. It now gets substantially harder to prove the existence of the necessary sums of squares certificates.