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Projectively Self-Concordant Barriers

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  • Roland Hildebrand

    (Laboratoire Jean Kuntzmann, University Grenoble Alpes, French National Centre for Scientific Research, 38000 Grenoble, France)

Abstract

Self-concordance is the most important property required for barriers in convex programming. It is intrinsically linked to the affine structure of the underlying space. Here we introduce an alternative notion of self-concordance that is linked to the projective structure. A function on a set X in an affine space is projectively self-concordant if and only if it can be extended to an affinely self-concordant logarithmically homogeneous function on the conic extension of X . The feasible sets in conic programs, notably linear and semidefinite programs, are naturally equipped with projectively self-concordant barriers. However, the interior-point methods used to solve these programs use only affine self-concordance. We show that estimates used in the analysis of interior-point methods are tighter for projective self-concordance, in particular inner and outer approximations of the set. This opens the way to a better tuning of parameters in interior-points algorithms to allow larger steps and hence faster convergence. Projective self-concordance is also a useful tool in the theoretical analysis of logarithmically homogeneous barriers on cones.

Suggested Citation

  • Roland Hildebrand, 2022. "Projectively Self-Concordant Barriers," Mathematics of Operations Research, INFORMS, vol. 47(3), pages 2444-2463, August.
    • Handle: RePEc:inm:ormoor:v:47:y:2022:i:3:p:2444-2463
    DOI: 10.1287/moor.2021.1215
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    References listed on IDEAS

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    1. Yu. E. Nesterov & M. J. Todd, 1997. "Self-Scaled Barriers and Interior-Point Methods for Convex Programming," Mathematics of Operations Research, INFORMS, vol. 22(1), pages 1-42, February.
    2. Sébastien Bubeck & Ronen Eldan, 2019. "The Entropic Barrier: Exponential Families, Log-Concave Geometry, and Self-Concordance," Mathematics of Operations Research, INFORMS, vol. 44(1), pages 264-276, February.
    3. Roland Hildebrand, 2014. "Canonical Barriers on Convex Cones," Mathematics of Operations Research, INFORMS, vol. 39(3), pages 841-850, August.
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