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We define the standard dual function $\\phi(\\lambda) = \\inf_x \\{f(x) + \\langle \\lambda, A x - b \\rangle\\}$, the augmented Lagrangian $\\mathcal{L}_{\\rho}(x, \\lambda) = f(x) + \\langle \\lambda, Ax - b \\rangle + \\frac{\\rho}{2}\\|Ax - b\\|^2$ ($\\rho > 0$), and the augmented Lagrangian dual function $\\phi_{\\rho}(\\l"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2505.01824","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2025-05-03T14:04:04Z","cross_cats_sorted":["cs.SY","eess.SY"],"title_canon_sha256":"ec87fd105555a55100b456698e1a115b15bca93a32efd97897a070a183a0bacc","abstract_canon_sha256":"76e429e5bfbfeec1c2aea84303a96c2ec92b4fd37b41bc68fb0acddc590f4048"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-20T02:05:33.409978Z","signature_b64":"wPYtD8pkOaSHasSKmUxdc1w9x9Ai71iAdCAJOsR+byLr65jxONs40e8Y+6H7/tuFv+SP5vciWRokLDhpZQuWDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5f44e3f68d640b8315aadb6ce0e9fcd14c3e8df09d89d478efdeee02945d18cd","last_reissued_at":"2026-05-20T02:05:33.409250Z","signature_status":"signed_v1","first_computed_at":"2026-05-20T02:05:33.409250Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Smoothness of the Augmented Lagrangian Dual in Convex Optimization","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.SY","eess.SY"],"primary_cat":"math.OC","authors_text":"Jingwang Li, Vincent Lau","submitted_at":"2025-05-03T14:04:04Z","abstract_excerpt":"This paper focuses on the general linearly constrained optimization problem: $\\min_{x \\in \\mathbb{R}^d} f(x) \\ \\text{s.t.} \\ Ax = b$, where $f: \\mathbb{R}^d \\rightarrow \\mathbb{R} \\cup \\{+\\infty\\}$ is a closed proper convex function, $A \\in \\mathbb{R}^{p \\times d}$, and $b \\in \\mathbb{R}^p$. We define the standard dual function $\\phi(\\lambda) = \\inf_x \\{f(x) + \\langle \\lambda, A x - b \\rangle\\}$, the augmented Lagrangian $\\mathcal{L}_{\\rho}(x, \\lambda) = f(x) + \\langle \\lambda, Ax - b \\rangle + \\frac{\\rho}{2}\\|Ax - b\\|^2$ ($\\rho > 0$), and the augmented Lagrangian dual function $\\phi_{\\rho}(\\l"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2505.01824","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2505.01824/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"2505.01824","created_at":"2026-05-20T02:05:33.409346+00:00"},{"alias_kind":"arxiv_version","alias_value":"2505.01824v2","created_at":"2026-05-20T02:05:33.409346+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2505.01824","created_at":"2026-05-20T02:05:33.409346+00:00"},{"alias_kind":"pith_short_12","alias_value":"L5COH5UNMQFY","created_at":"2026-05-20T02:05:33.409346+00:00"},{"alias_kind":"pith_short_16","alias_value":"L5COH5UNMQFYGFNK","created_at":"2026-05-20T02:05:33.409346+00:00"},{"alias_kind":"pith_short_8","alias_value":"L5COH5UN","created_at":"2026-05-20T02:05:33.409346+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":2,"internal_anchor_count":2,"sample":[{"citing_arxiv_id":"2505.03719","citing_title":"Accelerated Decentralized Constraint-Coupled Optimization: A Dual$^2$ Approach","ref_index":29,"is_internal_anchor":true},{"citing_arxiv_id":"2605.11694","citing_title":"Augmented Lagrangian Method for Last-Iterate Convergence for Constrained MDPs","ref_index":8,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/L5COH5UNMQFYGFNK3NWOB2P42F","json":"https://pith.science/pith/L5COH5UNMQFYGFNK3NWOB2P42F.json","graph_json":"https://pith.science/api/pith-number/L5COH5UNMQFYGFNK3NWOB2P42F/graph.json","events_json":"https://pith.science/api/pith-number/L5COH5UNMQFYGFNK3NWOB2P42F/events.json","paper":"https://pith.science/paper/L5COH5UN"},"agent_actions":{"view_html":"https://pith.science/pith/L5COH5UNMQFYGFNK3NWOB2P42F","download_json":"https://pith.science/pith/L5COH5UNMQFYGFNK3NWOB2P42F.json","view_paper":"https://pith.science/paper/L5COH5UN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2505.01824&json=true","fetch_graph":"https://pith.science/api/pith-number/L5COH5UNMQFYGFNK3NWOB2P42F/graph.json","fetch_events":"https://pith.science/api/pith-number/L5COH5UNMQFYGFNK3NWOB2P42F/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/L5COH5UNMQFYGFNK3NWOB2P42F/action/timestamp_anchor","attest_storage":"https://pith.science/pith/L5COH5UNMQFYGFNK3NWOB2P42F/action/storage_attestation","attest_author":"https://pith.science/pith/L5COH5UNMQFYGFNK3NWOB2P42F/action/author_attestation","sign_citation":"https://pith.science/pith/L5COH5UNMQFYGFNK3NWOB2P42F/action/citation_signature","submit_replication":"https://pith.science/pith/L5COH5UNMQFYGFNK3NWOB2P42F/action/replication_record"}},"created_at":"2026-05-20T02:05:33.409346+00:00","updated_at":"2026-05-20T02:05:33.409346+00:00"}