  We consider the linear conic optimization problem with the cone of
nonnegative polynomials. Its dual optimization problem is the generalized
moment problem. Moment-SOS relaxations are powerful for solving them. This
paper studies finite convergence of the Moment-SOS hierarchy when the
constraining set is defined by equations whose ideal may not be real radical.
Under the archimedeanness, we show that the Moment-SOS hierarchy has finite
convergence if some classical optimality conditions hold at every minimizer of
the optimal nonnegative polynomial for the linear conic optimization problem.
When the archimedeanness fails (this is the case for unbounded sets), we
propose a homogenized Moment-SOS hierarchy and prove its finite convergence
under similar assumptions. Furthermore, we also prove the finite convergence of
the Moment-SOS hierarchy with denominators. In particular, this paper resolves
a conjecture posed in the earlier work.
