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On the Self-Concordance of the Universal Barrier Function
, 1995
"... Let K be a regular convex cone in R n and F (x) its universal barrier function. Let D k F (x)[h; : : : ; h] be kth order directional derivative at the point x 2 K 0 and direction h 2 R n . We show that for every m 3 there exists a constant c(m) ? 0 depending only on m such that jD m F (x) ..."
Abstract
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Let K be a regular convex cone in R n and F (x) its universal barrier function. Let D k F (x)[h; : : : ; h] be kth order directional derivative at the point x 2 K 0 and direction h 2 R n . We show that for every m 3 there exists a constant c(m) ? 0 depending only on m such that jD m F (x)[h; : : : ; h]j c(m) D 2 F (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.
