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Barrier functions in interior point methods. (English) Zbl 0867.90090
Summary: We show that the universal barrier function of a convex cone introduced by Nesterov and Nemirovskii is the logarithm of the characteristic function of the cone. This interpretation demonstrates the invariance of the universal barrier under the automorphism group of the underlying cone. This provides a simple method for calculating the universal barrier for homogeneous cones. We identify some known barriers as the universal barrier scaled by an appropriate constant. We also calculate some new universal barrier functions. Our results connect the field of interior point methods to several branches of mathematics such as Lie groups, Jordan algebras, Siegel domains, differential geometry, complex analysis of several variables, etc.

90C25 Convex programming
90C60 Abstract computational complexity for mathematical programming problems
52A41 Convex functions and convex programs in convex geometry
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