Minimizing Polynomials via Sum of Squares over the Gradient Ideal

Sums of Squares, Moment Matrices and Optimization Over Polynomials

“…they lie in V R (I grad p ), the (real) gradient variety of p, the unconstrained minimization problem (1.3) can be p − p min is a sum of squares modulo I grad p . Summarizing, the above results of Nie et al [99] show that the parameter p grad can be approximated via converging moment/SOS bounds; when p has a minimum, then p min = p grad and thus p min too can be approximated.…”

“…Yet asymptotic convergence does hold and sometimes even finite convergence. Nie et al [99] show the representation results from Theorems 7.15-7.16 below, for positive (nonnegative) polynomials on their gradient variety as sums of squares modulo their gradient ideal. As an immediate application, there is asymptotic convergence (moreover, finite convergence when I grad p is radical) of the moment/SOS bounds from the programs (6.3), (6.2) (applied to the polynomial constraints ∂p/∂x i = 0 (i = 1, .…”

“…Nie, Demmel and Sturmfels [99] propose an alternative way of transforming (1.3) into a constrained problem when p attains its infimum. Define the gradient ideal of p 8) as the ideal generated by the partial derivatives of p. Since all global minimizers of p are critical points, i.e.…”

“…The polynomial p is nonnegative over R 3 , thus over V R (I grad p ), but it is not a sum of squares modulo I grad p . Indeed one can verify that p − M/4 ∈ I grad p and that M is not a sum of squares modulo I grad p (see [99] for details); thus I grad p is not radical.…”

“…Sums of squares over the gradient ideal. We give here the proofs for Theorems 7.15 and 7.16 about sums of squares representations modulo the gradient ideal, following Nie et al [99] (although our proof slightly differs at some places). We begin with the following lemma which can be seen as an extension of Lemma 2.3 about existence of interpolation polynomials.…”