On the Self-Concordance of the Universal Barrier Function

Osman Guler

Let $K$ be a regular convex cone in ${\Bbb R^n}$ and $F(x)$ its universal barrier function. Let $D^kF(x)[h,\ldots,h]$ be $k$th order directional derivative at the point $x\in K^0$ and direction $h\in{\Bbb R^n}$. 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.

Technical Report GU 95-2, Department of Mathematics and Statistics, University of Maryland Baltimore County, Baltimore, MD 21228-5398.