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Sequential optimality conditions for nonlinear optimization on Riemannian manifolds and a globally convergent augmented Lagrangian method

Author

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  • Yuya Yamakawa

    (Kyoto University)

  • Hiroyuki Sato

    (Kyoto University)

Abstract

Recently, the approximate Karush–Kuhn–Tucker (AKKT) conditions, also called the sequential optimality conditions, have been proposed for nonlinear optimization in Euclidean spaces, and several methods to find points satisfying such conditions have been developed by researchers. These conditions are known as genuine necessary optimality conditions because all local optima satisfy them with no constraint qualification (CQ). In this paper, we extend the AKKT conditions to nonlinear optimization on Riemannian manifolds and propose an augmented Lagrangian (AL) method that globally converges to points satisfying such conditions. In addition, we prove that the AKKT and KKT conditions are indeed equivalent under a certain CQ. Finally, we examine the effectiveness of the proposed AL method via several numerical experiments.

Suggested Citation

  • Yuya Yamakawa & Hiroyuki Sato, 2022. "Sequential optimality conditions for nonlinear optimization on Riemannian manifolds and a globally convergent augmented Lagrangian method," Computational Optimization and Applications, Springer, vol. 81(2), pages 397-421, March.
    • Handle: RePEc:spr:coopap:v:81:y:2022:i:2:d:10.1007_s10589-021-00336-w
    DOI: 10.1007/s10589-021-00336-w
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    References listed on IDEAS

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    1. Huixian Wu & Hezhi Luo & Xiaodong Ding & Guanting Chen, 2013. "Global convergence of modified augmented Lagrangian methods for nonlinear semidefinite programming," Computational Optimization and Applications, Springer, vol. 56(3), pages 531-558, December.
    2. Hiroyuki Sato & Kensuke Aihara, 2019. "Cholesky QR-based retraction on the generalized Stiefel manifold," Computational Optimization and Applications, Springer, vol. 72(2), pages 293-308, March.
    3. Xiaojing Zhu & Hiroyuki Sato, 2020. "Riemannian conjugate gradient methods with inverse retraction," Computational Optimization and Applications, Springer, vol. 77(3), pages 779-810, December.
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    Citations

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    Cited by:

    1. Caroline Geiersbach & Tim Suchan & Kathrin Welker, 2024. "Stochastic Augmented Lagrangian Method in Riemannian Shape Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 203(1), pages 165-195, October.
    2. Brennan McCann & Morad Nazari & Christopher Petersen, 2024. "Numerical Approaches for Constrained and Unconstrained, Static Optimization on the Special Euclidean Group SE(3)," Journal of Optimization Theory and Applications, Springer, vol. 201(3), pages 1116-1150, June.
    3. Yasushi Narushima & Shummin Nakayama & Masashi Takemura & Hiroshi Yabe, 2023. "Memoryless Quasi-Newton Methods Based on the Spectral-Scaling Broyden Family for Riemannian Optimization," Journal of Optimization Theory and Applications, Springer, vol. 197(2), pages 639-664, May.

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