  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.

Optimization and Control