A note on some analytic center cutting plane methods for convex feasibility and minimization problems

Recently Goffin, Luo and Ye have analyzed the complexity of an analytic center algorithm for convex feasibility problems defined by a separation oracle. The oracle is called at the (possibly approximate) analytic center of the set given by the linear inequalities which are the previous answers of the oracle. We discuss oracles for the problem of minimizing a convex (possibly nondifferentiable) function subject to box constraints, and give corresponding complexity estimates.