We describe a general idea of path-following algorithms for solving
optimization problems. The concept of the universal barrier function (u.b.f) is introduced.
The key result of Yu. Nesterov and A.Nemirovsky allowing to obtain excellent complexity estimates is explained. Finally, we describe new results related to computations of u.b. fs for a very broad class of cones generated by Chebyshev systems. Connections with classical work of M. Krein , A. Nudelman and I. Schoenberg are explained. New results allow one to substantially extend the
domain of applicability of modern interior-point algorithms.
