This paper numerically quantifies the degradation of the optimal barrier parameter under convex relaxation and shows the loss is small when the optimal parameter is near 2.
Towards non-symmetric conic optimization
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Numerical analysis of the convex relaxation of the barrier parameter functional of self-concordant barriers
This paper numerically quantifies the degradation of the optimal barrier parameter under convex relaxation and shows the loss is small when the optimal parameter is near 2.