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Proximal Point Algorithm with Euclidean Distance on the Stiefel Manifold

Author

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  • Harry Oviedo

    (Facultad de Ingeniería y Ciencias, Universidad Adolfo Ibáñez, Av. Diag. Las Torres 2640, Santiago de Chile 7941169, Chile)

Abstract

In this paper, we consider the problem of minimizing a continuously differentiable function on the Stiefel manifold. To solve this problem, we develop a geodesic-free proximal point algorithm equipped with Euclidean distance that does not require use of the Riemannian metric. The proposed method can be regarded as an iterative fixed-point method that repeatedly applies a proximal operator to an initial point. In addition, we establish the global convergence of the new approach without any restrictive assumption. Numerical experiments on linear eigenvalue problems and the minimization of sums of heterogeneous quadratic functions show that the developed algorithm is competitive with some procedures existing in the literature.

Suggested Citation

  • Harry Oviedo, 2023. "Proximal Point Algorithm with Euclidean Distance on the Stiefel Manifold," Mathematics, MDPI, vol. 11(11), pages 1-17, May.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:11:p:2414-:d:1153538
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    References listed on IDEAS

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    1. Glaydston C. Bento & Orizon P. Ferreira & Jefferson G. Melo, 2017. "Iteration-Complexity of Gradient, Subgradient and Proximal Point Methods on Riemannian Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 173(2), pages 548-562, May.
    2. J. Souza & P. Oliveira, 2015. "A proximal point algorithm for DC fuctions on Hadamard manifolds," Journal of Global Optimization, Springer, vol. 63(4), pages 797-810, December.
    3. Xiaojing Zhu, 2017. "A Riemannian conjugate gradient method for optimization on the Stiefel manifold," Computational Optimization and Applications, Springer, vol. 67(1), pages 73-110, May.
    4. Raoul Pietersz & Patrick Groenen, 2004. "Rank reduction of correlation matrices by majorization," Quantitative Finance, Taylor & Francis Journals, vol. 4(6), pages 649-662.
    5. Yldenilson Torres Almeida & João Xavier Cruz Neto & Paulo Roberto Oliveira & João Carlos de Oliveira Souza, 2020. "A modified proximal point method for DC functions on Hadamard manifolds," Computational Optimization and Applications, Springer, vol. 76(3), pages 649-673, July.
    6. Balendu Bhooshan Upadhyay & Arnav Ghosh, 2023. "On Constraint Qualifications for Mathematical Programming Problems with Vanishing Constraints on Hadamard Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 199(1), pages 1-35, October.
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