An exact Jacobian SDP relaxation for polynomial optimization

Given polynomials f(x), gi(x), hj(x), we study how to minimize f(x) on the set S = {x ∈ R : h1(x) = · · · = hm1(x) = 0, g1(x) ≥ 0, . . . , gm2(x) ≥ 0} . Let fmin be the minimum of f on S. Suppose S is nonsingular and fmin is achievable on S, which are true generically. This paper proposes a new type semidefinite programming (SDP) relaxation which is the first one for solving this problem exactly. First, we construct new polynomials φ1, . . . , φr, by using the Jacobian of f, hi, gj , such that the above problem is equivalent to min x∈Rn f(x) s.t. hi(x) = 0, φj(x) = 0, 1 ≤ i ≤ m1, 1 ≤ j ≤ r, g1(x) ν1 · · · gm2(x)2 ≥ 0, ∀ν ∈ {0, 1}m2 . Second, we prove that for all N big enough, the standard N -th order Lasserre’s SDP relaxation is exact for solving this equivalent problem, that is, its optimal value is equal to fmin. Some variations and examples are also shown.