Polynomial Optimization with Real Varieties

We consider the optimization problem of minimizing a polynomial f(x) subject to polynomial constraints h(x)=0, g(x)>=0. Lasserre's hierarchy is a sequence of sum of squares relaxations for finding the global minimum. Let K be the feasible set. We prove the following results: i) If the real variety V_R(h) is finite, then Lasserre's hierarchy has finite convergence, no matter the complex variety V_C(h) is finite or not. This solves an open question in Laurent's survey. ii) If K and V_R(h) have the same vanishing ideal, then the finite convergence of Lasserre's hierarchy is independent of the choice of defining polynomials for the real variety V_R(h). iii) When K is finite, a refined version of Lasserre's hierarchy (using the preordering of g) has finite convergence.