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Completely positive factorization by a Riemannian smoothing method

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  • Zhijian Lai

    (University of Tsukuba)

  • Akiko Yoshise

    (University of Tsukuba)

Abstract

Copositive optimization is a special case of convex conic programming, and it consists of optimizing a linear function over the cone of all completely positive matrices under linear constraints. Copositive optimization provides powerful relaxations of NP-hard quadratic problems or combinatorial problems, but there are still many open problems regarding copositive or completely positive matrices. In this paper, we focus on one such problem; finding a completely positive (CP) factorization for a given completely positive matrix. We treat it as a nonsmooth Riemannian optimization problem, i.e., a minimization problem of a nonsmooth function over a Riemannian manifold. To solve this problem, we present a general smoothing framework for solving nonsmooth Riemannian optimization problems and show convergence to a stationary point of the original problem. An advantage is that we can implement it quickly with minimal effort by directly using the existing standard smooth Riemannian solvers, such as Manopt. Numerical experiments show the efficiency of our method especially for large-scale CP factorizations.

Suggested Citation

  • Zhijian Lai & Akiko Yoshise, 2022. "Completely positive factorization by a Riemannian smoothing method," Computational Optimization and Applications, Springer, vol. 83(3), pages 933-966, December.
    • Handle: RePEc:spr:coopap:v:83:y:2022:i:3:d:10.1007_s10589-022-00417-4
    DOI: 10.1007/s10589-022-00417-4
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    References listed on IDEAS

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    1. Immanuel Bomze & Werner Schachinger & Gabriele Uchida, 2012. "Think co(mpletely)positive ! Matrix properties, examples and a clustered bibliography on copositive optimization," Journal of Global Optimization, Springer, vol. 52(3), pages 423-445, March.
    2. Jasmina Hasanhodzic & Andrew Lo & Emanuele Viola, 2011. "A computational view of market efficiency," Quantitative Finance, Taylor & Francis Journals, vol. 11(7), pages 1043-1050.
    3. Chen Chen & Ting Kei Pong & Lulin Tan & Liaoyuan Zeng, 2020. "A difference-of-convex approach for split feasibility with applications to matrix factorizations and outlier detection," Journal of Global Optimization, Springer, vol. 78(1), pages 107-136, September.
    4. Garms, Marco A. & Andrade, Marco T.C. & Caldas, Iberê L., 2009. "Fuzzy computational control for real Chua circuit," Chaos, Solitons & Fractals, Elsevier, vol. 39(5), pages 2169-2178.
    5. Bomze, Immanuel M., 2012. "Copositive optimization – Recent developments and applications," European Journal of Operational Research, Elsevier, vol. 216(3), pages 509-520.
    6. Glaydston Carvalho Bento & João Xavier Cruz Neto & Paulo Roberto Oliveira, 2016. "A New Approach to the Proximal Point Method: Convergence on General Riemannian Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 168(3), pages 743-755, March.
    7. Peter Dickinson & Luuk Gijben, 2014. "On the computational complexity of membership problems for the completely positive cone and its dual," Computational Optimization and Applications, Springer, vol. 57(2), pages 403-415, March.
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