An Exact Jacobian SDP Relaxation for Polynomial Optimization

Given polynomials f(x), g_i(x), h_j(x), we study how to minimize f on the semialgebraic set S = { x \in R^n: h_1(x)=...=h_{m_1}(x) =0, g_1(x) >= 0, ..., g_{m_2}(x) >= 0}. Let f_{min} be the minimum of f on S. Suppose S is nonsingular and f_{min} is achievable on S,which is true generically. The paper proposes a new semidefinite programming (SDP) relaxation for this problem. First we construct a set of new polynomials \varphi_1(x), \ldots, \varphi_r(x), by using the Jacobian of f,h_i,g_j, such that the above problem is unchanged by adding new equations \varphi_j(x)=0. Then 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, it returns a lower bound that is equal to f_{min}. Some variations and examples are also shown.