Adaptive Riemannian ADMM for Nonsmooth Optimization: Optimal Complexity without Smoothing
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866912663031775232 |
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| author | Deng, Kangkang Jin, Jiachen Hu, Jiang Wang, Hongxia |
| author_facet | Deng, Kangkang Jin, Jiachen Hu, Jiang Wang, Hongxia |
| contents | We study the problem of minimizing the sum of a smooth function and a nonsmooth convex regularizer over a compact Riemannian submanifold embedded in Euclidean space. By introducing an auxiliary splitting variable, we propose an adaptive Riemannian alternating direction method of multipliers (ARADMM), which, for the first time, achieves convergence without requiring smoothing of the nonsmooth term. Our approach involves only one Riemannian gradient evaluation and one proximal update per iteration. Through careful and adaptive coordination of the stepsizes and penalty parameters, we establish an optimal iteration complexity of order $\mathcal{O}(ε^{-3})$ for finding an $ε$-approximate KKT point, matching the complexity of existing smoothing technique-based Riemannian ADMM methods. Extensive numerical experiments on sparse PCA and robust subspace recovery demonstrate that our ARADMM consistently outperforms state-of-the-art Riemannian ADMM variants in convergence speed and solution quality. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_18617 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Adaptive Riemannian ADMM for Nonsmooth Optimization: Optimal Complexity without Smoothing Deng, Kangkang Jin, Jiachen Hu, Jiang Wang, Hongxia Optimization and Control We study the problem of minimizing the sum of a smooth function and a nonsmooth convex regularizer over a compact Riemannian submanifold embedded in Euclidean space. By introducing an auxiliary splitting variable, we propose an adaptive Riemannian alternating direction method of multipliers (ARADMM), which, for the first time, achieves convergence without requiring smoothing of the nonsmooth term. Our approach involves only one Riemannian gradient evaluation and one proximal update per iteration. Through careful and adaptive coordination of the stepsizes and penalty parameters, we establish an optimal iteration complexity of order $\mathcal{O}(ε^{-3})$ for finding an $ε$-approximate KKT point, matching the complexity of existing smoothing technique-based Riemannian ADMM methods. Extensive numerical experiments on sparse PCA and robust subspace recovery demonstrate that our ARADMM consistently outperforms state-of-the-art Riemannian ADMM variants in convergence speed and solution quality. |
| title | Adaptive Riemannian ADMM for Nonsmooth Optimization: Optimal Complexity without Smoothing |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2510.18617 |