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Barriers on projective convex sets

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


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