Global optimization of polynomials restricted to a smooth variety using sums of squares

Let f1,…,fp be in , where , that generate a radical ideal and let V be their complex zero-set. Assume that V is smooth and equidimensional. Given bounded below, consider the optimization problem of computing f⋆=infx∈V∩Rnf(x). For , we denote by the polynomial and by the complex zero-set of .We construct families of polynomials in : each is related to the section of a linear subspace with the critical locus of a linear projection. We prove that there exists a non-empty Zariski-open set 풪⊂GLn(C) such that for all , f(x) is positive for all x∈V∩Rn if and only if, can be expressed as a sum of squares of polynomials on the truncated variety generated by the ideal , for 0≤i≤d.Hence, we can obtain algebraic certificates for lower bounds on f⋆ using semi-definite programs. Some numerical experiments are given. We also discuss how to decrease the number of polynomials in .