  This paper proposes tight semidefinite relaxations for polynomial
optimization. The optimality conditions are investigated. We show that
generally Lagrange multipliers can be expressed as polynomial functions in
decision variables over the set of critical points. The polynomial expressions
can be determined by linear equations. Based on these expressions, new Lasserre
type semidefinite relaxations are constructed for solving polynomial
optimization. We show that the hierarchy of new relaxations has finite
convergence, or equivalently, the new relaxations are tight for a finite
relaxation order.
