An exact Jacobian SDP relaxation for polynomial optimization

Given polynomials f (x), g _i (x), h _j (x), we study how to minimize f (x) on the set $$S = left{ x in mathbb{R}^n:, h_1(x) = cdots = h_{m_1}(x) = 0, g_1(x)geq 0, ldots, g_{m_2}(x) geq 0 right}.$$ Let f _min be the minimum of f on S. Suppose S is nonsingular and f _min 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 ${varphi_1, ldots, varphi_r}$ , by using the Jacobian of f, h _i , g _j , such that the above problem is equivalent to $$begin{gathered}underset{xinmathbb{R}^n}{min} f(x) hfill , , {rm s.t.}; h_i(x) = 0, , varphi_j(x) = 0, , 1leq i leq m_1, 1 leq j leq r, hfill quad , , , g_1(x)^{nu_1}cdots g_{m_2}(x)^{nu_{m_2}}geq 0, , quadforall, nu ,in {0,1}^{m_2} .hfill end{gathered}$$ 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 f _min. Some variations and examples are also shown.