 Self-concordant barrier

A convex three-times differentiable barrier $F$ is self-concordant whenA barrier of parameter $\nu$ additionally satisfies $|DF(x)[h]|\le \sqrt{\nu D^{2}F(x)[h,h]}$ and diverges at the domain boundary. Logarithmically homogeneous cone barriers satisfy $F(tx)=F(x)-\nu \log t$. The orthant barrier $-{\sum }_i\log x_i$ has parameter equal to the dimension, while the Lorentz-cone barrier $-\log (t^2-\Vert z{\Vert }^2)$ on $t>\Vert z\Vert$ has parameter two. Their Hessians define Dikin ellipsoids and control interior Newton steps.