Barriers on projective convex sets

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2011, 2011(Special): 672-683. doi: 10.3934/proc.2011.2011.672

Barriers on projective convex sets

Roland Hildebrand ^1,

1.	LJK, Université Grenoble 1/CNRS, 51 rue des Mathématiques, BP 53, 38041 Grenoble cedex 09, France

Received August 2010 Revised January 2011 Published October 2011

Abstract
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Under a Creative Commons license

Modern interior-point methods used for optimization on convex sets in ane space are based on the notion of a barrier function. Projective space lacks crucial properties inherent to ane space, and the concept of a barrier function cannot be directly carried over. We present a self-contained theory of barriers on convex sets in projective space which is build upon the projective cross-ratio. Such a projective barrier equips the set with a Codazzi structure, which is a generalization of the Hessian structure induced by a barrier in the ane case. The results provide a new interpretation of the ane theory and serve as a base for constructing a theory of interior-point methods for projective convex optimization.

Keywords: self-concordant barriers, Codazzi structure, projective convex sets.

Mathematics Subject Classification: Primary: 52A99; Secondary: 53B05.

Citation: Roland Hildebrand. Barriers on projective convex sets. Conference Publications, 2011, 2011 (Special) : 672-683. doi: 10.3934/proc.2011.2011.672

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