The Volumetric Barrier for Semidefinite Programming

Kurt M. Anstreicher

We consider the volumetric barrier for semidefinite programming, or "generalized" volumetric barrier, as introduced by Nesterov and Nemirovskii. We extend several fundamental properties of the volumetric barrier for a polyhedral set to the semidefinite case. Our analysis facilitates a simplified proof of self-concordancy for the semidefinite volumetric barrier, as well as for the combined volumetric-logarithmic barrier for semidefinite programming. For both of these barriers we obtain self-concordancy parameters equal to those previously shown to hold in the polyhedral case.