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

Let f(1), ... , f(p) be in Q|x| where X = (X-1, ... , X-n)(1), that generate a radical ideal and let V be their complex zero-set. Assume that V is smooth and equidimensional. Given f is an element of Q|x| bounded below, consider the optimization problem of computing f* = inf(x is an element of V boolean AND Rn) f (x). For A is an element of GL(n)(C), we denote by f(A) the polynomial f(AX) and by V-A the complex zero-set of f(1)(A) ... , f(p)(A). We construct families of polynomials M-0(A), ... , M-d(A) in Q|x|: each M-i(A) 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 (sic) subset of GL(n)(C) such that for all A is an element of (sic) boolean AND GL(n)(Q), f (x) is positive for all x is an element of V boolean AND R-n if and only if, f(A) can be expressed as a sum of squares of polynomials on the truncated variety generated by the ideal (M-i(A)), 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 M-i(A). (C) 2011 Elsevier Ltd. All rights reserved.