Algebraic boundaries of convex semi-algebraic sets are closely related to polynomial optimization problems. Building upon Rainer Sinn’s work, we refine the stratification of iterated singular loci to a Whitney (a)-regular stratification, which gives a list of candidates of varieties whose dual is an irreducible component of the algebraic boundary of the dual convex body. We also present an algorithm based on Teissier’s criterion to compute Whitney (a) stratifications, which employs ideal saturation and prime decompositions of conormal spaces.