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A basic closed semialgebraic subset of $\mathbb{R}^{n}$ is defined bysimultaneous polynomial inequalities $p_{1}\geq 0,\ldots,p_{m}\geq 0$. Weconsider Lasserre's relaxation hierarchy to solve the problem of minimizing apolynomial over such a set. These relaxations give an increasing sequence oflower bounds of the infimum. In this paper we provide a new certificate for theoptimal value of a Lasserre relaxation be the optimal value of the polynomialoptimization problem. This certificate is that a modified version of an optimalsolution of the Lasserre relaxation is a generalized Hankel matrix. Thiscertificate is more general than the already known certificate of an optimalsolution being flat. In case we have optimality we will extract the potencialminimizers with a truncated version of the Gelfand-Naimark-Segal constructionon the optimal solution of the Lasserre relaxation. We prove also that theoperators of this truncated construction commute if and only if the matrix ofthis modified optimal solution is a generalized Hankel matrix. Thisgeneralization of flatness will bring us to reprove a result of Curto andFialkow on the existence of quadrature rule if the optimal solution is flat anda result of Xu and Mysovskikh on the existance of a Gaussian quadrature rule ifthe modified optimal solution is generalized Hankel matrix. At the end, weprovide a numerical linear algebraic algorithm for dectecting optimality andextracting solutions of a polynomial optimization problem.

Detecting optimality and extracting solutions in polynomial optimization with the truncated GNS construction