  For an ideal $I\subseteq\mathbb{R}[x]$ given by a set of generators, a new
semidefinite characterization of its real radical $I(V_\mathbb{R}(I))$ is
presented, provided it is zero-dimensional (even if $I$ is not). Moreover we
propose an algorithm using numerical linear algebra and semidefinite
optimization techniques, to compute all (finitely many) points of the real
variety $V_\mathbb{R}(I)$ as well as a set of generators of the real radical
ideal. The latter is obtained in the form of a border or Gr\"obner basis. The
algorithm is based on moment relaxations and, in contrast to other existing
methods, it exploits the real algebraic nature of the problem right from the
beginning and avoids the computation of complex components.
