A basic closed semialgebraic subset of Rn is defined by
simultaneous polynomial inequalities p1≥0,…,pm≥0. We
consider Lasserre's relaxation hierarchy to solve the problem of minimizing a
polynomial over such a set. These relaxations give an increasing sequence of
lower bounds of the infimum. In this paper we provide a new certificate for the
optimal value of a Lasserre relaxation be the optimal value of the polynomial
optimization problem. This certificate is that a modified version of an optimal
solution of the Lasserre relaxation is a generalized Hankel matrix. This
certificate is more general than the already known certificate of an optimal
solution being flat. In case we have optimality we will extract the potencial
minimizers with a truncated version of the Gelfand-Naimark-Segal construction
on the optimal solution of the Lasserre relaxation. We prove also that the
operators of this truncated construction commute if and only if the matrix of
this modified optimal solution is a generalized Hankel matrix. This
generalization of flatness will bring us to reprove a result of Curto and
Fialkow on the existence of quadrature rule if the optimal solution is flat and
a result of Xu and Mysovskikh on the existance of a Gaussian quadrature rule if
the modified optimal solution is generalized Hankel matrix. At the end, we
provide a numerical linear algebraic algorithm for dectecting optimality and
extracting solutions of a polynomial optimization problem