{"ID":6497633,"CreatedAt":"2026-07-13T01:19:40.13847098Z","UpdatedAt":"2026-07-14T01:36:59.12045529Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2607.09517","arxiv_id":"2607.09517","title":"Riemannian Multilevel Optimization with Application to Constrained Energy Minimization Problems","abstract":"Multilevel optimization methods are highly effective for discretized energy minimization problems, but their Euclidean formulation does not directly apply to manifold constraints. We introduce a Riemannian extension of multilevel optimization based on a coarse model that is first-order coherent with the fine-level objective and yields descent directions under mild retraction-convexity assumptions. The framework includes metric-compatible vector transfer operators for passing first-order information between levels, covering both intrinsic and extrinsic constructions. We formulate two-level and multilevel algorithms and prove global convergence using a Riemannian Zoutendijk-type argument. Applications to Kohn--Sham density functional theory, Gross--Pitaevskii ground-state computation, and binary continuous cuts demonstrate the method on Stiefel, ellipsoid and Bernoulli manifolds. The experiments show significant reductions in computational time compared with single-level Riemannian optimization.","short_abstract":"Multilevel optimization methods are highly effective for discretized energy minimization problems, but their Euclidean formulation does not directly apply to manifold constraints. We introduce a Riemannian extension of multilevel optimization based on a coarse model that is first-order coherent with the fine-level obje...","url_abs":"https://arxiv.org/abs/2607.09517","url_pdf":"https://arxiv.org/pdf/2607.09517v1","authors":"[\"Yara Elshiaty\",\"Stefania Petra\",\"Jonas Püschel\",\"Tatjana Stykel\",\"Ferdinand-Joseph Vanmaele\"]","published":"2026-07-10T15:28:05Z","proceeding":"math.OC","tasks":"[\"math.OC\"]","methods":"[]","has_code":false}