  Lasserre's hierarchy is a sequence of semidefinite relaxations for solving
polynomial optimization problems globally. This paper studies the relationship
between optimality conditions in nonlinear programming theory and finite
convergence of Lasserre's hierarchy. Our main results are: i) Lasserre's
hierarchy has finite convergence when the constraint qualification, strict
complementarity and second order sufficiency conditions hold at every global
minimizer, under the standard archimedean assumption; the proof uses a result
of Marshall on boundary hessian conditions. ii) these optimality conditions are
all satisfied at every local minimizer if a finite set of polynomials, which
are in the coefficients of input polynomials, do not vanish at the input data
(i.e., they hold in a Zariski open set). This implies that Lasserre's hierarchy
has finite convergence generically.
