On the optimal parameter of a self-concordant barrier over a symmetric cone

The properties of the barrier F(x) = −log(det(x)), defined over the cone of squares of a Euclidean Jordan algebra, are analyzed using pure algebraic techniques. Furthermore, relating the Carathéodory number of a symmetric cone with the rank of an underlying Euclidean Jordan algebra, conclusions about the optimal parameter of F are suitably obtained. Namely, in a more direct and suitable way than the one presented by Güler and Tunçel (Characterization of the barrier parameter of homogeneous convex cones, Mathematical Programming 81 (1998) 55–76), it is proved that the Carathéodory number of the cone of squares of a Euclidean Jordan algebra is equal to the rank of the algebra. Then, taking into account the result obtained in the same paper where it is stated that the Carathéodory number of a symmetric cone Q is the optimal parameter of a self-concordant barrier defined over Q, we may conclude that the rank of every underlying Euclidean Jordan algebra is also the self-concordant barrier optimal parameter.