Barriers on projective convex sets

Pages: 672 - 683, Issue Special, September 2011

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Roland Hildebrand - LJK, Université Grenoble 1/CNRS, 51 rue des Mathématiques, BP 53, 38041 Grenoble cedex 09, France (email)

Abstract: 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, projective convex sets, Codazzi structure
Mathematics Subject Classification:  Primary: 52A99; Secondary: 53B05

Received: August 2010;      Revised: January 2011;      Published: October 2011.