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Convexity of Sets and Quadratic Functions on the Hyperbolic Space

Author

Listed:
  • Orizon P. Ferreira

    (IME, Universidade Federal de Goiás)

  • Sándor Z. Németh

    (University of Birmingham)

  • Jinzhen Zhu

    (University of Birmingham)

Abstract

In this paper, some concepts of convex analysis on hyperbolic spaces are studied. We first study properties of the intrinsic distance, for instance, we present the spectral decomposition of its Hessian. Next, we study the concept of convex sets and the intrinsic projection onto these sets. We also study the concept of convex functions and present first- and second-order characterizations of these functions, as well as some optimization concepts related to them. An extensive study of the hyperbolically convex quadratic functions is also presented.

Suggested Citation

  • Orizon P. Ferreira & Sándor Z. Németh & Jinzhen Zhu, 2024. "Convexity of Sets and Quadratic Functions on the Hyperbolic Space," Journal of Optimization Theory and Applications, Springer, vol. 202(1), pages 421-455, July.
    • Handle: RePEc:spr:joptap:v:202:y:2024:i:1:d:10.1007_s10957-022-02073-4
    DOI: 10.1007/s10957-022-02073-4
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    References listed on IDEAS

    as
    1. Li-wen Zhou & Yi-bin Xiao & Nan-jing Huang, 2017. "New Characterization of Geodesic Convexity on Hadamard Manifolds with Applications," Journal of Optimization Theory and Applications, Springer, vol. 172(3), pages 824-844, March.
    2. O. P. Ferreira & S. Z. Németh, 2019. "On the spherical convexity of quadratic functions," Journal of Global Optimization, Springer, vol. 73(3), pages 537-545, March.
    3. Alessandro Muscoloni & Josephine Maria Thomas & Sara Ciucci & Ginestra Bianconi & Carlo Vittorio Cannistraci, 2017. "Machine learning meets complex networks via coalescent embedding in the hyperbolic space," Nature Communications, Nature, vol. 8(1), pages 1-19, December.
    Full references (including those not matched with items on IDEAS)

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