We consider linear optimization over a fixed compact convex feasible region
that is semi-algebraic (or, more generally, "tame"). Generically, we prove that
the optimal solution is unique and lies on a unique manifold, around which the
feasible region is "partly smooth", ensuring finite identification of the
manifold by many optimization algorithms. Furthermore, second-order optimality
conditions hold, guaranteeing smooth behavior of the optimal solution under
small perturbations to the objective