Augmented self-concordant barriers and nonlinear optimization problems with finite complexity

In this paper we study special barrier functions for the convex cones, which are the sum of a self-concordant barrier for the cone and a positive-semidefinite quadratic form. We show that the central path of these augmented barrier functions can be traced with linear speed. We also study the complexity of finding the analytic center of the augmented barrier. This problem itself has some interesting applications. We show that for some special classes of quadratic forms and some convex cones, the computation of the analytic center requires an amount of operations independent on the particular data set. We argue that these problems form a class that is endowed with a property which we call finite polynomial complexity.