An inexact variable metric proximal linearization method for composite optimization on manifolds
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arXiv
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| Format: | Preprint |
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2025
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| author | He, Hao Liu, Ruyu Qian, Yitian Pan, Shaohua |
| author_facet | He, Hao Liu, Ruyu Qian, Yitian Pan, Shaohua |
| contents | This paper concerns the minimization of the composition of a nonsmooth convex function and a $\mathcal{C}^{1,1}$ mapping $F$ over a $\mathcal{C}^2$-smooth embedded closed submanifold $\mathcal{M}$. For this class of nonconvex and nonsmooth problems, we propose an inexact variable metric proximal linearization method by leveraging its composite structure and the retraction and first-order information of $\mathcal{M}$, which at each iteration seeks an inexact solution to a subspace constrained strongly convex problem by a practical inexactness criterion. Under the boundedness assumption on the iterate sequence, we establish the $O(ε^{-3})$ oracle complexity with a dual fast gradient method as the inner solver, and prove that any cluster point of the iterate sequence is a stationary point. If in addition the constructed potential function has the Kurdyka-Lojasiewicz (KL) property on the set of cluster points, the iterate sequence converges to a stationary point, and if the potential function has the KL property of exponent $q\in[\frac{1}{2},1)$, the local convergence rate is characterized. We also provide a condition only involving the original data to identify the KL property of the potential function with an exponent $q\in[0,1)$. Numerical comparisons with the existing methods validate the efficiency of the proposed method. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_12003 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An inexact variable metric proximal linearization method for composite optimization on manifolds He, Hao Liu, Ruyu Qian, Yitian Pan, Shaohua Optimization and Control This paper concerns the minimization of the composition of a nonsmooth convex function and a $\mathcal{C}^{1,1}$ mapping $F$ over a $\mathcal{C}^2$-smooth embedded closed submanifold $\mathcal{M}$. For this class of nonconvex and nonsmooth problems, we propose an inexact variable metric proximal linearization method by leveraging its composite structure and the retraction and first-order information of $\mathcal{M}$, which at each iteration seeks an inexact solution to a subspace constrained strongly convex problem by a practical inexactness criterion. Under the boundedness assumption on the iterate sequence, we establish the $O(ε^{-3})$ oracle complexity with a dual fast gradient method as the inner solver, and prove that any cluster point of the iterate sequence is a stationary point. If in addition the constructed potential function has the Kurdyka-Lojasiewicz (KL) property on the set of cluster points, the iterate sequence converges to a stationary point, and if the potential function has the KL property of exponent $q\in[\frac{1}{2},1)$, the local convergence rate is characterized. We also provide a condition only involving the original data to identify the KL property of the potential function with an exponent $q\in[0,1)$. Numerical comparisons with the existing methods validate the efficiency of the proposed method. |
| title | An inexact variable metric proximal linearization method for composite optimization on manifolds |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2508.12003 |