Anstreicher, K. Math. Program., Ser. A (2004) 100: 613. doi:10.1007/s10107-003-0513-4
Let where and i is an n×n positive semidefinite matrix. We prove that the volumetric and combined volumetric-logarithmic barriers for are and self-concordant, respectively. Our analysis uses the semidefinite programming (SDP) representation for the convex quadratic constraints defining , and our earlier results on the volumetric barrier for SDP. The self-concordance results actually hold for a class of SDP problems more general than those corresponding to the SDP representation of .