{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:7GP5TX34CI7ZDKNJFQ3GN3PTMO","short_pith_number":"pith:7GP5TX34","canonical_record":{"source":{"id":"2403.02530","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2024-03-04T22:51:11Z","cross_cats_sorted":["cs.NA","math.NA"],"title_canon_sha256":"cf74f75819065bf5dfc836418da0b685ce5e8262012fb1baa62e17566a7cc823","abstract_canon_sha256":"25f2dde7ea1fe83fd756999098eac872b97255e81941133d0cb2667bde4db120"},"schema_version":"1.0"},"canonical_sha256":"f99fd9df7c123f91a9a92c3666edf363b76404975b835ed3f92934cc681b3f20","source":{"kind":"arxiv","id":"2403.02530","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2403.02530","created_at":"2026-07-05T11:28:29Z"},{"alias_kind":"arxiv_version","alias_value":"2403.02530v3","created_at":"2026-07-05T11:28:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2403.02530","created_at":"2026-07-05T11:28:29Z"},{"alias_kind":"pith_short_12","alias_value":"7GP5TX34CI7Z","created_at":"2026-07-05T11:28:29Z"},{"alias_kind":"pith_short_16","alias_value":"7GP5TX34CI7ZDKNJ","created_at":"2026-07-05T11:28:29Z"},{"alias_kind":"pith_short_8","alias_value":"7GP5TX34","created_at":"2026-07-05T11:28:29Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:7GP5TX34CI7ZDKNJFQ3GN3PTMO","target":"record","payload":{"canonical_record":{"source":{"id":"2403.02530","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2024-03-04T22:51:11Z","cross_cats_sorted":["cs.NA","math.NA"],"title_canon_sha256":"cf74f75819065bf5dfc836418da0b685ce5e8262012fb1baa62e17566a7cc823","abstract_canon_sha256":"25f2dde7ea1fe83fd756999098eac872b97255e81941133d0cb2667bde4db120"},"schema_version":"1.0"},"canonical_sha256":"f99fd9df7c123f91a9a92c3666edf363b76404975b835ed3f92934cc681b3f20","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:28:29.508573Z","signature_b64":"tZkVZW4gT6Jrtzu/IcjEZkZKvF+o3XvvxCok2GHwsPQ8pOQeYmK4O7GeOBAFjQchdcIZDmPFaZU5kmHGOmqTAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f99fd9df7c123f91a9a92c3666edf363b76404975b835ed3f92934cc681b3f20","last_reissued_at":"2026-07-05T11:28:29.508098Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:28:29.508098Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2403.02530","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T11:28:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HtmlyC4pNnIm5hadfLCA5kZsHPcKVWTHzn/eukDtT/eoKTqPgfLia9GX4gaAurLoKGQw2zDc/Z6Xwa7eSyTjDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-10T19:27:08.911037Z"},"content_sha256":"491d4ec77b4b96d4cb821b473c070cd65d7896ead4138e7627e479d2c1ba3fac","schema_version":"1.0","event_id":"sha256:491d4ec77b4b96d4cb821b473c070cd65d7896ead4138e7627e479d2c1ba3fac"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:7GP5TX34CI7ZDKNJFQ3GN3PTMO","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Projected gradient descent accumulates at Bouligand stationary points","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NA","math.NA"],"primary_cat":"math.OC","authors_text":"Guillaume Olikier, Ir\\`ene Waldspurger","submitted_at":"2024-03-04T22:51:11Z","abstract_excerpt":"This paper considers the projected gradient descent (PGD) algorithm for the problem of minimizing a continuously differentiable function on a nonempty closed subset of a Euclidean vector space. Without further assumptions, this problem is intractable and algorithms are only expected to find a stationary point. PGD generates a sequence in the set whose accumulation points are known to be Mordukhovich stationary. In this paper, these accumulation points are proven to be Bouligand stationary, and even proximally stationary if the gradient is locally Lipschitz continuous. These are the strongest s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2403.02530","kind":"arxiv","version":3},"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/2403.02530/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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T11:28:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"286G4wWd0u12nLWqaqA10UIt73k7V+u7wJbwVX2COEpp3H9rtkSLcOvh4Jo34U4jfhrizQ/UqGcinUfN9mhoDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-10T19:27:08.911918Z"},"content_sha256":"b051f05685bfdd36399dd42a24d3afa9f944605cfe54261331318f155e80363b","schema_version":"1.0","event_id":"sha256:b051f05685bfdd36399dd42a24d3afa9f944605cfe54261331318f155e80363b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7GP5TX34CI7ZDKNJFQ3GN3PTMO/bundle.json","state_url":"https://pith.science/pith/7GP5TX34CI7ZDKNJFQ3GN3PTMO/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7GP5TX34CI7ZDKNJFQ3GN3PTMO/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-08-10T19:27:08Z","links":{"resolver":"https://pith.science/pith/7GP5TX34CI7ZDKNJFQ3GN3PTMO","bundle":"https://pith.science/pith/7GP5TX34CI7ZDKNJFQ3GN3PTMO/bundle.json","state":"https://pith.science/pith/7GP5TX34CI7ZDKNJFQ3GN3PTMO/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7GP5TX34CI7ZDKNJFQ3GN3PTMO/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:7GP5TX34CI7ZDKNJFQ3GN3PTMO","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"25f2dde7ea1fe83fd756999098eac872b97255e81941133d0cb2667bde4db120","cross_cats_sorted":["cs.NA","math.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2024-03-04T22:51:11Z","title_canon_sha256":"cf74f75819065bf5dfc836418da0b685ce5e8262012fb1baa62e17566a7cc823"},"schema_version":"1.0","source":{"id":"2403.02530","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2403.02530","created_at":"2026-07-05T11:28:29Z"},{"alias_kind":"arxiv_version","alias_value":"2403.02530v3","created_at":"2026-07-05T11:28:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2403.02530","created_at":"2026-07-05T11:28:29Z"},{"alias_kind":"pith_short_12","alias_value":"7GP5TX34CI7Z","created_at":"2026-07-05T11:28:29Z"},{"alias_kind":"pith_short_16","alias_value":"7GP5TX34CI7ZDKNJ","created_at":"2026-07-05T11:28:29Z"},{"alias_kind":"pith_short_8","alias_value":"7GP5TX34","created_at":"2026-07-05T11:28:29Z"}],"graph_snapshots":[{"event_id":"sha256:b051f05685bfdd36399dd42a24d3afa9f944605cfe54261331318f155e80363b","target":"graph","created_at":"2026-07-05T11:28:29Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2403.02530/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"This paper considers the projected gradient descent (PGD) algorithm for the problem of minimizing a continuously differentiable function on a nonempty closed subset of a Euclidean vector space. Without further assumptions, this problem is intractable and algorithms are only expected to find a stationary point. PGD generates a sequence in the set whose accumulation points are known to be Mordukhovich stationary. In this paper, these accumulation points are proven to be Bouligand stationary, and even proximally stationary if the gradient is locally Lipschitz continuous. These are the strongest s","authors_text":"Guillaume Olikier, Ir\\`ene Waldspurger","cross_cats":["cs.NA","math.NA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2024-03-04T22:51:11Z","title":"Projected gradient descent accumulates at Bouligand stationary points"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2403.02530","kind":"arxiv","version":3},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:491d4ec77b4b96d4cb821b473c070cd65d7896ead4138e7627e479d2c1ba3fac","target":"record","created_at":"2026-07-05T11:28:29Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"25f2dde7ea1fe83fd756999098eac872b97255e81941133d0cb2667bde4db120","cross_cats_sorted":["cs.NA","math.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2024-03-04T22:51:11Z","title_canon_sha256":"cf74f75819065bf5dfc836418da0b685ce5e8262012fb1baa62e17566a7cc823"},"schema_version":"1.0","source":{"id":"2403.02530","kind":"arxiv","version":3}},"canonical_sha256":"f99fd9df7c123f91a9a92c3666edf363b76404975b835ed3f92934cc681b3f20","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f99fd9df7c123f91a9a92c3666edf363b76404975b835ed3f92934cc681b3f20","first_computed_at":"2026-07-05T11:28:29.508098Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:28:29.508098Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"tZkVZW4gT6Jrtzu/IcjEZkZKvF+o3XvvxCok2GHwsPQ8pOQeYmK4O7GeOBAFjQchdcIZDmPFaZU5kmHGOmqTAg==","signature_status":"signed_v1","signed_at":"2026-07-05T11:28:29.508573Z","signed_message":"canonical_sha256_bytes"},"source_id":"2403.02530","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:491d4ec77b4b96d4cb821b473c070cd65d7896ead4138e7627e479d2c1ba3fac","sha256:b051f05685bfdd36399dd42a24d3afa9f944605cfe54261331318f155e80363b"],"state_sha256":"1ddd8f708c143dc8f657b3175814651d752631b8edb035e53b30f47af6eb90b8"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"An64Oen7Wxf/7KV3YIN5yjYPZ4/3UCvmgDYAxLzbuP2BxkKBAoS3zVEjl7Yt9R7yHs6On5XFRl0i3+3UfOBWAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-10T19:27:08.917433Z","bundle_sha256":"24e0079a1e8eef3250a033f29cf0997f95ca40f1a198d5d70c08a936cf0255c9"}}