|
|
Pure Mathematics 2023
Hadamard流形上的多目标邻近梯度算法
|
Abstract:
邻近梯度算法是求解非光滑优化问题的经典算法。本文将多目标优化问题的邻近梯度算法推广到Hadammard流形上。在一定条件下，证明了算法产生序列的聚点是Pareto稳定点。在目标函数满足Polyak-Loiasiewicz不等式时，得到算法的收敛速度是线性的。所得结果在Hadamard流形上是新的。
Proximal gradient algorithm is a classical algorithm for solving nonsmooth optimiza-
tion problems. In this paper, the multiobjective proximal gradient algorithm is ex-
tended to Hadamard manifold. Under certain conditions, it is proved that the cluster
point of the sequence generated by the algorithm is Pareto stationary. In the case,
when the objective function satisfies the Polyak-Lojasiewicz inequality, the conver-
gence rate of the algorithm is linear.
| [1] | Adler, R.L., Dedieu, J.P., Margulies, J.Y., et al. (2002) Newton's Method on Riemannian
Manifolds and a Geometric Model for the Human Spine. IMA Journal of Numerical Analysis,
22, 359-390. https://doi.org/10.1093/imanum/22.3.359 |
| [2] | Li, C. and Wang, J. (2006) Newton's Method on Riemannian Manifolds: Smale's Point Estimate
Theory under the
-Condition. IMA Journal of Numerical Analysis, 26, 228-251.
https://doi.org/10.1093/imanum/dri039 |
| [3] | Gabay, D. (1982) Minimizing a Differentiable Function over a Differential Manifold. Journal
of Optimization Theory and Applications, 37, 177-219. https://doi.org/10.1007/BF00934767 |
| [4] | Wang, J.H., Li, C., Lopez, G. and Yao, J.C. (2015) Convergence Analysis of Inexact Proximal
Point Algorithms on Hadamard Manifolds. Journal of Global Optimization, 61, 553-573.
https://doi.org/10.1007/s10898-014-0182-2 |
| [5] | Wang, X.M. (2018) Subgradient Algorithms on Riemannian Manifolds of Lower Bounded Curvatures.
Optimization, 67, 179-194. https://doi.org/10.1080/02331934.2017.1387548 |
| [6] | Absil, P.A., Baker, C.G. and Gallivan, K.A. (2007) Trust-Region Methods on Riemannian
Manifolds. Foundations of Computational Mathematics, 7, 303-330.
https://doi.org/10.1007/s10208-005-0179-9 |
| [7] | Huang, W. and Wei, K. (2022) Riemannian Proximal Gradient Methods. Mathematical Pro-
gramming, 194, 371-413. https://doi.org/10.1007/s10107-021-01632-3 |
| [8] | Jolliffe, I.T., Trendafilov, N.T. and Uddin, M. (2003) A Modified Principal Component Technique
Based on the LASSO. Journal of Computational and Graphical Statistics, 12, 531-547.
https://doi.org/10.1198/1061860032148 |
| [9] | Genicot, M., Huang, W. and Trendafilov, N.T. (2015) Weakly Correlated Sparse Components
with Nearly Orthonormal Loadings. Geometric Science of Information: Second International
Conference, Palaiseau, 28-30 October 2015, 484-490.
https://doi.org/10.1007/978-3-319-25040-3 52 |
| [10] | Zhang, Y., Lau, Y., Kuo, H., et al. (2017) On the Global Geometry of Sphere-Constrained
Sparse Blind Deconvolution. 2017 IEEE Conference on Computer Vision and Pattern Recog-
nition, Honolulu, HI, 21-26 July 2017, 4894-4902. |
| [11] | Tang, J. and Liu, H. (2012) Unsupervised Feature Selection for Linked Social Media Data.
Proceedings of the 18th ACM SIGKDD international conference on Knowledge discovery and
data mining, Beijing, 12-16 August 2012, 904-912. |
| [12] | Beck, A. and Teboulle, M. (2009) A Fast Iterative Shrinkage-Thresholding Algorithm for Linear
Inverse Problems. SIAM Journal on Imaging Sciences, 2, 183-202.
https://doi.org/10.1137/080716542 |
| [13] | Huang, W. (2013) Optimization Algorithms on Riemannian Manifolds with Applications. The
Florida State University, Tallahassee, FL. |
| [14] | Hosseini, S. and Uschmajew, A. (2017) A Riemannian Gradient Sampling Algorithm for Nonsmooth
Optimization on Manifolds. SIAM Journal on Optimization, 27, 173-189.
https://doi.org/10.1137/16M1069298 |
| [15] | Hosseini, S., Huang, W. and Yousefpour, R. (2018) Line Search Algorithms for Locally Lipschitz
Functions on Riemannian Manifolds. SIAM Journal on Optimization, 28, 596-619.
https://doi.org/10.1137/16M1108145 |
| [16] | Liu, Y., Shang, F., Cheng, J., et al. (2017) Accelerated First-Order Methods for Geodesically
Convex Optimization on Riemannian Manifolds. Advances in Neural Information Processing
Systems, 30. |
| [17] | Chen, S., Ma, S., So, A.M.-C., et al. (2020) Proximal Gradient Method for Nonsmooth Optimization
over the Stiefel Manifold. SIAM Journal on Optimization, 30, 210-239.
https://doi.org/10.1137/18M122457X |
| [18] | Huang, W. and Wei, K. (2019) An Extension of Fast Iterative Shrinkage-thresholding to Riemannian
Optimization for Sparse Principal Component Analysis.
https://doi.org/10.48550/arXiv.1909.05485 |
| [19] | Huang, W. and Wei, K. (2022) Riemannian Proximal Gradient Methods. Mathematical Pro-
gramming, 194, 371-413. https://doi.org/10.1007/s10107-021-01632-3 |
| [20] | Tanabe, H., Fukuda, E.H. and Yamashita, N. (2019) Proximal Gradient Methods for Multiobjective
Optimization and Their Applications. Computational Optimization and Applications,
72, 339-361. https://doi.org/10.1007/s10589-018-0043-x |
| [21] | Tanabe, H., Fukuda, E.H. and Yamashita, N. (2023) Convergence Rates Analysis of a Multiobjective
Proximal Gradient Method. Optimization Letters, 17, 333-350.
https://doi.org/10.1007/s11590-022-01877-7 |
| [22] | Do Carmo, M.P. and Flaherty Francis, J. (1992) Riemannian Geometry. Birkhauser. |
| [23] | 陈维桓,李兴校.黎曼几何引论[M].北京: 北京大学出版社，2002: 12. |
| [24] | Boothby, W.M. (1986) An Introduction to Differentiable Manifolds and Riemannian Geometry.
Academic Press, Cambridge, MA. |
| [25] | Padlewska, B. and Darmochwal, A. (1990) Topological Spaces and Continuous Functions.
Formalized Mathematics, 1, 223-230. |
| [26] | Hogan, W.W. (1973) Point-to-Set Maps in Mathematical Programming. SIAM Review, 15,
591-603. https://doi.org/10.1137/1015073 |
| [27] | Tanabe, H., Fukuda, E.H. and Yamashita, N. (2023) New Merit Functions for Multiobjective
Optimization and Their Properties. Optimization, 1-38.
https://doi.org/10.1080/02331934.2023.2232794 |
