An Exact Jacobian SDP Relaxation for Polynomial Optimization

Understand

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.