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Primal Central Paths and Riemannian Distances for Convex Sets

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Abstract

In this paper, we study the Riemannian length of the primal central path in a convex set computed with respect to the local metric defined by a self-concordant function. Despite some negative examples, in many important situations, the length of this path is quite close to the length of a geodesic curve. We show that in the case of a bounded convex set endowed with a ν-self-concordant barrier, the length of the central path is within a factor O(ν 1/4) of the length of the shortest geodesic curve.

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Authors and Affiliations

  1. CORE, Catholic University of Louvain, 34 voie du Roman Pays, 1348, Louvain-la-Neuve, Belgium

    Y. Nesterov

  2. Technion-Israel Institute of Technology, Haifa, Israel

    A. Nemirovski

Authors
  1. Y. Nesterov
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  2. A. Nemirovski
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Corresponding author

Correspondence to Y. Nesterov.

Additional information

Communicated by Mike Todd.

This paper presents research results of the Belgian Program on Interuniversity Poles of Attraction initiated by the Belgian State, Prime Minister’s Office, Science Policy Programming.

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Nesterov, Y., Nemirovski, A. Primal Central Paths and Riemannian Distances for Convex Sets. Found Comput Math 8, 533–560 (2008). https://doi.org/10.1007/s10208-007-9019-4

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  • DOI: https://doi.org/10.1007/s10208-007-9019-4

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