= Self-concordant barrier
{title2=$F,\nu$}
A convex three-times differentiable barrier $F$ is self-concordant when
$$
|D^3F(x)[h,h,h]|\leq2\bigl(D^2F(x)[h,h]\bigr)^{3/2}.
$$
A barrier of parameter $\nu$ additionally satisfies $|DF(x)[h]|\leq\sqrt{\nu D^2F(x)[h,h]}$ 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 $-\log(t^2-\|z\|^2)$ on $t>\|z\|$ has parameter two. Their Hessians define <Dikin ellipsoids> and control interior Newton steps.
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