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The Volumetric Barrier for Convex Quadratic Constraints

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Kurt M. Anstreicher

Let ${\cal Q}=\{x | s_i(x)\ge 0, i=1,\ldots,k\}$, where
$s_i(x)=a_i^T x-x^T Q_i x-c_i$, and $Q_i$ is an $n\times n$
positive semidefinite matrix. We prove that the volumetric and combined
volumetric-logarithmic barriers for ${\cal Q}$ are $O(\sqrt{k}n)$ and
$O(\sqrt{kn})$ self-concordant, respectively. Our analysis uses the
semidefinite programming (SDP) representation for the convex quadratic
constraints defining ${\cal Q}$, and our earlier results on the volumetric
barrier for SDP. The self-concordancy results actually hold for a class
of SDP problems more general than those corresponding to the SDP
representation of ${\cal Q}$.
Dept. of Management Sciences, University of Iowa, Iowa City,
October 1998.

Contact: kurt-anstreicher@uiowa.edu