On the optimal parameter of a self-concordant barrier over a symmetric cone

On the optimal parameter of a self-concordant barrier over a symmetric cone

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Article ID: iaor2007422
Country: Netherlands
Volume: 169
Issue: 3
Start Page Number: 1148
End Page Number: 1157
Publication Date: Mar 2006
Journal: European Journal of Operational Research
Authors: ,
Keywords: barrier function
Abstract:

The properties of the barrier F(x) = -log(det(x)), defined over the cone of squares of a Euclidean Jordan algebra, are analyzed using pure algebraic techniques. Furthermore, relating the Carathéodory number of a symmetric cone with the rank of an underlying Euclidean Jordan algebra, conclusions about the optimal parameter of F are suitably obtained. Namely, in a more direct and suitable way than the one presented by Güler and Tunçel, it is proved that the Carathéodory number of the cone of squares of a Euclidean Jordan algebra is equal to the rank of the algebra. Then, taking into account the result obtained in the same paper where it is stated that the Carathéodory number of a symmetric cone Q is the optimal parameter of a self-concordant barrier defined over Q, we may conclude that the rank of every underlying Euclidean Jordan algebra is also the self-concordant barrier optimal parameter.

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