{"ID":2881648,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2508.12003","arxiv_id":"2508.12003","title":"An inexact variable metric proximal linearization method for composite optimization on manifolds","abstract":"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.","short_abstract":"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...","url_abs":"https://arxiv.org/abs/2508.12003","url_pdf":"https://arxiv.org/pdf/2508.12003v2","authors":"[\"Hao He\",\"Ruyu Liu\",\"Yitian Qian\",\"Shaohua Pan\"]","published":"2025-08-16T10:08:17Z","proceeding":"math.OC","tasks":"[\"math.OC\"]","methods":"[]","has_code":false}