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The Riemannian approach has become a popular technique for addressing these problems due to the advantageous computational and analytical properties of the Stiefel manifold. Nonetheless, the interplay of nonsmoothness alongside orthogonality constraints introduces substantial challenges to current Riemannian methods, including scalability, parallelizability, complicated subproblems, and cumulative numerical errors that threaten feasibility. In this paper, we take a retraction-free primal-dual approach and propose a linearized smoothing augmented Lagrangian method specifically designed for nonsmooth and nonconvex optimization with orthogonality constraints. Our proposed method is single-loop and free of subproblem solving. 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In this paper, we take a retraction-free primal-dual approach and propose a linearized smoothing augmented Lagrangian method specifically designed for nonsmooth and nonconvex optimization with orthogonality constraints. Our proposed method is single-loop and free of subproblem solving. We establish its iteration complexity of $O(ε^{-3})$ for finding $ε$-KKT points, matching the best-known results in the Riemannian optimiz","title":"Primal-Dual Methods for Nonsmooth Nonconvex Optimization with Orthogonality Constraints","vendor":"arxiv"},"confidence":{"method":"deterministic"},"decision":"POSITIVE","issued_at":"2026-05-03T16:20:10Z","notes":"Spider arxiv_retraction (research) research.arxiv_retraction.v1","object":{"captured_by":"crovia.spider.arxiv_retraction","primary_source_url":"https://arxiv.org/pdf/2604.04130v1"},"predecessors":[],"schema":"crovia.axiom.v1","signature":"ed25519:a155d1cb405162ce6509ba3f81e0b140ac37255a39a44c66b53432d8d190ffa82dd0a0844aaf724c197c480fc1cbe2834a84f9f2b3bf0e5c2df742e354be0105","signer":"crovia.substrate","subject":{"observed_at":"2026-05-03T16:20:10Z","source_collector":"spider:arxiv_retraction","target_id":"https://arxiv.org/pdf/2604.04130v1"},"tsa":{"authority":"crovia.substrate.bootstrap","rfc3161_token":"{\"kind\":\"crovia.bootstrap.tsa\",\"source_jsonl\":\"/opt/crovia/spider/data/research/arxiv_retraction_v1.jsonl\",\"source_seal_merkle_root\":\"spider_arxiv_retraction_v1\",\"upgrade_path\":\"Sessione H \\u2014 OpenTimestamps weekly anchor\"}"},"zk_mode":"clear","zk_proof":null},"ledger":{"leaf_hash":"2df9e9764b1ccf58b354cfd1e803a09b539ec83f1a7367325f6be90218ac3718","leaf_index":111938,"ledger_path":"/opt/crovia/substrate/axiom_ledger.jsonl"},"merkle_proof":{"hash_alg":"sha256","leaf_prefix":"0x00","node_prefix":"0x01","odd_leaf_rule":"duplicate_last","path":[{"sibling":"d1ccfca96fb983bd3c5615d187b8f5455e431cf91b3a55e4da0b5046faba60b3","side":"right"},{"sibling":"d7018d2fecda2bf3fb5c095e3b8bf5aacac2a14bf379b4fbb8df4055ffecdaff","side":"left"},{"sibling":"0569b533cd8159c1c294589c57b105cf8f036169bad17416d72a0056fd8c0e43","side":"right"},{"sibling":"e34d335cae85ba6a34caa45f35a0cc16a495442d839437de3a1820d7491a905a","side":"right"},{"sibling":"753774e59ee1a795f876ec420e258ae77d369aa892917f6778e4fda7d080fe34","side":"right"},{"sibling":"04f08a05f60539e5d320a1d658ae9463ad3d1673f88287d19b483af959df3046","side":"right"},{"sibling":"7cc6ec0331a098790dc04ced8e8abc2b910e0fcce6830c8442fff364ca231498","side":"left"},{"sibling":"0994d51d2178913b0e3ac7346ceed16c5ea436d841a8e7f28a99362fa269bad0","side":"right"},{"sibling":"015d2ce3c58c7a790ad078159267c63cd54f9ae7e20530115a6a423419cefe22","side":"left"},{"sibling":"5bcdb02d4310fc8bfd791789d990714b0a10f44358ffa98c446f1755352c49b8","side":"right"},{"sibling":"cf8897e1feec248f5d05e2cafec4f8c2739e2005227d28b5ccc8cfa11ed66ee4","side":"left"},{"sibling":"a6d6e5bc888edc394757fbba02b675bac6ad0831e36a6f5ebbaa4e5548f25603","side":"right"},{"sibling":"76d7c4385d71ae8104107105be66eddb792e01ed6e493db5a9b4eec5441abb76","side":"left"},{"sibling":"b03f005860bf95148ea89c703a32ca19ab7a2ca71b58bb628ff94a9edb8703b0","side":"left"},{"sibling":"21d1e30b556f553dc939fddc23e6367f0a7755ebe3dd489dc0001cef017beb7f","side":"right"},{"sibling":"f2817ab288b5324fe49770372c7a10f33f7cd11005f8d4c0a730316f5229dc98","side":"left"},{"sibling":"725fac972e772ca0dc598810ea1abc70df472f72d2d6ab8a0baee2b80e5d2f4c","side":"left"},{"sibling":"98fc57dfef8873b512edc8340f7181df57302bb96777625e072235c62d7c5895","side":"right"}]},"schema":"crovia.axiom_proof.v1","seal":{"first_collector_run_id":"","first_receipt_hash":"","jsonl_path":"/opt/crovia/substrate/axiom_ledger.jsonl","key_id":"430895f101d38164","last_collector_run_id":"","last_receipt_hash":"","leaf_count":134292,"merkle_root":"77fc9c28fae777b81da5b495b3115474df6592dfac590333213d3bdf8b94a9b3","public_key_hex":"cf742e26f75669dc673cb5c0786a1ae23ae8ca19c347317192ce40c28a7ff25c","run_id":"hourly_json_retrofit_20260515T023701Z","schema":"crovia.seal.v1","seal_family_version":"crovia-seal-family/1","seal_kind":"substrate_batch","sealed_at":"2026-05-15T02:37:25Z","sig_algorithm":"ed25519","signature":"68107a834b00b24f5d4501e5ec727445311f132a486567ecc4c72a4e6dff24c8c21f2de3105293353ba5fdbe370d032819af6aa70f694e2e39b6af6737507009","signer_version":"1.1.0"},"trust_root":{"key_id":"430895f101d38164","public_key_hex":"cf742e26f75669dc673cb5c0786a1ae23ae8ca19c347317192ce40c28a7ff25c","signature_algorithm":"ed25519","url":"/registry/canon/TRUST_ROOT.md"},"verifier":{"spec":"/registry/canon/AXIOM_RECEIPT_v1.md","url":"/v/axm_e06791a6c51d4388a5c0c16ad965eae9bcf4cd066634be522e525085632def2c"}}