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Friday, October 12, 2007 |
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On the Tunçel Conjecture: A New Class of Self-Concordant Barriers on Sets of Symmetric Matrices |
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Given a separable strongly self-concordant function $f:\R^n
\rightarrow \R$, we show the associated spectral function $F(X)= (f \circ
\lambda)(X)$ is also strongly self-concordant function. In addition, there
is a universal constant $\mathcal{O}$ such that, if $f(x)$ is separable
self-concordant barrier then $\mathcal{O}^2F(X)$ is a self-concordant
barrier. We estimate that for the universal constant we have $\mathcal{O}
\le 22$. This generalizes the relationship between the standard
logarithmic barriers $-\sum_{i=1}^n\log x_i$ and $-\log \det X$ and gives
a partial solution to a conjecture of L. Tunçel.
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