We present a new algorithm for computing the real radical of an ideal I and, more generally, the S-radical of I, which is based on convex moment optimization. A truncated positive generic linear functional σ vanishing on the generators of I is computed solving a Moment Optimization Problem (MOP). We show that, for a large enough degree of truncation, the annihilator of σ generates the real radical of I. We give an effective, general stopping criterion on the degree to detect when the prime ideals lying over the annihilator are real and compute the real radical as the intersection of real prime ideals lying over I.