We analyze the convergence and the complexity of a potential reduction column generation algorithm for solving general convex or quasiconvex feasibility problems defined by a separation oracle. The oracle is called at the analytic center of the set given by the intersection of the linear inequalities which are the previous answers of the oracle. We show that the algorithm converges in finite time and is in fact a fully polynomial approximation algorithm, provided that the feasible region has an nonempty interior. This result is based on the works of Ye~[10] and Nesterov~[7].