On the Self-Concordance of the Universal Barrier Function

Abstract

Let K be a regular convex cone in $\hbox{{\bbb R}}^n$ and let $F(x)$ be its universal barrier function. Let $D^kF(x)[h,\ldots,h]$ be the kth order directional derivative at the point $x\in K^0$ and direction $h\in\hbox{{\bbb R}}^n$. We show that for every $m\ge3$ there exists a constant $c(m)>0$ depending only on m such that $|D^mF(x)[h,\ldots,h]|\le c(m)\,D^2F(x)[h,h]^{m/2}$. For $m=3$, this is the self-concordance inequality of Nesterov and Nemirovskii. Our proof uses a powerful recent result of Bourgain.