{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:ZG6C4XWZZ3XJR7E2ODTZXSRT2A","short_pith_number":"pith:ZG6C4XWZ","canonical_record":{"source":{"id":"1507.00887","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2015-07-03T12:27:09Z","cross_cats_sorted":["cs.NA","math.NA"],"title_canon_sha256":"a43c1c8bc92990dccf3de16a3ca3ea454aa7295e223d271b88d115e9eaead0c2","abstract_canon_sha256":"1447932ede1241ec9c850e88a12db2077576c6eb86f6a09c2926e057c7cf1fde"},"schema_version":"1.0"},"canonical_sha256":"c9bc2e5ed9ceee98fc9a70e79bca33d039b76d468f8e092981fcdb5753788b4c","source":{"kind":"arxiv","id":"1507.00887","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1507.00887","created_at":"2026-05-18T00:53:12Z"},{"alias_kind":"arxiv_version","alias_value":"1507.00887v3","created_at":"2026-05-18T00:53:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.00887","created_at":"2026-05-18T00:53:12Z"},{"alias_kind":"pith_short_12","alias_value":"ZG6C4XWZZ3XJ","created_at":"2026-05-18T12:29:52Z"},{"alias_kind":"pith_short_16","alias_value":"ZG6C4XWZZ3XJR7E2","created_at":"2026-05-18T12:29:52Z"},{"alias_kind":"pith_short_8","alias_value":"ZG6C4XWZ","created_at":"2026-05-18T12:29:52Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:ZG6C4XWZZ3XJR7E2ODTZXSRT2A","target":"record","payload":{"canonical_record":{"source":{"id":"1507.00887","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2015-07-03T12:27:09Z","cross_cats_sorted":["cs.NA","math.NA"],"title_canon_sha256":"a43c1c8bc92990dccf3de16a3ca3ea454aa7295e223d271b88d115e9eaead0c2","abstract_canon_sha256":"1447932ede1241ec9c850e88a12db2077576c6eb86f6a09c2926e057c7cf1fde"},"schema_version":"1.0"},"canonical_sha256":"c9bc2e5ed9ceee98fc9a70e79bca33d039b76d468f8e092981fcdb5753788b4c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:53:12.593008Z","signature_b64":"8QFy9ENJxoYYvEP3jYLx+tQXmCrZjxupzpWUn0pXtY2ufk+z9XcBzwQw8IXHwN6Ww43LrXuiYVjzXlfwY+q/Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c9bc2e5ed9ceee98fc9a70e79bca33d039b76d468f8e092981fcdb5753788b4c","last_reissued_at":"2026-05-18T00:53:12.592502Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:53:12.592502Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1507.00887","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":"2026-05-18T00:53:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"OOAZDVSwNqZhzm/x5sncnvRg7UjjpRmf15Tefw2QBlcj/0esjtZKxvHyEKNhVcEk8sEn1r7byD3PcIIV155sAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T11:00:16.440507Z"},"content_sha256":"cd95cb73d6c6d165983dbe1f5e8e8cb260eb3470baaa7189fad088ebcc1b1d66","schema_version":"1.0","event_id":"sha256:cd95cb73d6c6d165983dbe1f5e8e8cb260eb3470baaa7189fad088ebcc1b1d66"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:ZG6C4XWZZ3XJR7E2ODTZXSRT2A","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Peaceman-Rachford splitting for a class of nonconvex optimization problems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NA","math.NA"],"primary_cat":"math.OC","authors_text":"Guoyin Li, Tianxiang Liu, Ting Kei Pong","submitted_at":"2015-07-03T12:27:09Z","abstract_excerpt":"We study the applicability of the Peaceman-Rachford (PR) splitting method for solving nonconvex optimization problems. When applied to minimizing the sum of a strongly convex Lipschitz differentiable function and a proper closed function, we show that if the strongly convex function has a large enough strong convexity modulus and the step-size parameter is chosen below a threshold that is computable, then any cluster point of the sequence generated, if exists, will give a stationary point of the optimization problem. We also give sufficient conditions guaranteeing boundedness of the sequence g"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.00887","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":""},"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":"2026-05-18T00:53:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"KnE5KgfdoVKAQLTvEhCSAHKVNvUMFW/Xk/A7Dk2NhMPyyPt5G7rc5F1fsFQx34l5QbBPm0oIhEbmKYJMnQ3MAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T11:00:16.440885Z"},"content_sha256":"a8614d25989a21206b1265f461c3d7fd0f782487addc0f442fd3720feac98e5e","schema_version":"1.0","event_id":"sha256:a8614d25989a21206b1265f461c3d7fd0f782487addc0f442fd3720feac98e5e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ZG6C4XWZZ3XJR7E2ODTZXSRT2A/bundle.json","state_url":"https://pith.science/pith/ZG6C4XWZZ3XJR7E2ODTZXSRT2A/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ZG6C4XWZZ3XJR7E2ODTZXSRT2A/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":"2026-05-30T11:00:16Z","links":{"resolver":"https://pith.science/pith/ZG6C4XWZZ3XJR7E2ODTZXSRT2A","bundle":"https://pith.science/pith/ZG6C4XWZZ3XJR7E2ODTZXSRT2A/bundle.json","state":"https://pith.science/pith/ZG6C4XWZZ3XJR7E2ODTZXSRT2A/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ZG6C4XWZZ3XJR7E2ODTZXSRT2A/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:ZG6C4XWZZ3XJR7E2ODTZXSRT2A","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":"1447932ede1241ec9c850e88a12db2077576c6eb86f6a09c2926e057c7cf1fde","cross_cats_sorted":["cs.NA","math.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2015-07-03T12:27:09Z","title_canon_sha256":"a43c1c8bc92990dccf3de16a3ca3ea454aa7295e223d271b88d115e9eaead0c2"},"schema_version":"1.0","source":{"id":"1507.00887","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1507.00887","created_at":"2026-05-18T00:53:12Z"},{"alias_kind":"arxiv_version","alias_value":"1507.00887v3","created_at":"2026-05-18T00:53:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.00887","created_at":"2026-05-18T00:53:12Z"},{"alias_kind":"pith_short_12","alias_value":"ZG6C4XWZZ3XJ","created_at":"2026-05-18T12:29:52Z"},{"alias_kind":"pith_short_16","alias_value":"ZG6C4XWZZ3XJR7E2","created_at":"2026-05-18T12:29:52Z"},{"alias_kind":"pith_short_8","alias_value":"ZG6C4XWZ","created_at":"2026-05-18T12:29:52Z"}],"graph_snapshots":[{"event_id":"sha256:a8614d25989a21206b1265f461c3d7fd0f782487addc0f442fd3720feac98e5e","target":"graph","created_at":"2026-05-18T00:53:12Z","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"},"paper":{"abstract_excerpt":"We study the applicability of the Peaceman-Rachford (PR) splitting method for solving nonconvex optimization problems. When applied to minimizing the sum of a strongly convex Lipschitz differentiable function and a proper closed function, we show that if the strongly convex function has a large enough strong convexity modulus and the step-size parameter is chosen below a threshold that is computable, then any cluster point of the sequence generated, if exists, will give a stationary point of the optimization problem. We also give sufficient conditions guaranteeing boundedness of the sequence g","authors_text":"Guoyin Li, Tianxiang Liu, Ting Kei Pong","cross_cats":["cs.NA","math.NA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2015-07-03T12:27:09Z","title":"Peaceman-Rachford splitting for a class of nonconvex optimization problems"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.00887","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:cd95cb73d6c6d165983dbe1f5e8e8cb260eb3470baaa7189fad088ebcc1b1d66","target":"record","created_at":"2026-05-18T00:53:12Z","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":"1447932ede1241ec9c850e88a12db2077576c6eb86f6a09c2926e057c7cf1fde","cross_cats_sorted":["cs.NA","math.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2015-07-03T12:27:09Z","title_canon_sha256":"a43c1c8bc92990dccf3de16a3ca3ea454aa7295e223d271b88d115e9eaead0c2"},"schema_version":"1.0","source":{"id":"1507.00887","kind":"arxiv","version":3}},"canonical_sha256":"c9bc2e5ed9ceee98fc9a70e79bca33d039b76d468f8e092981fcdb5753788b4c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c9bc2e5ed9ceee98fc9a70e79bca33d039b76d468f8e092981fcdb5753788b4c","first_computed_at":"2026-05-18T00:53:12.592502Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:53:12.592502Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"8QFy9ENJxoYYvEP3jYLx+tQXmCrZjxupzpWUn0pXtY2ufk+z9XcBzwQw8IXHwN6Ww43LrXuiYVjzXlfwY+q/Aw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:53:12.593008Z","signed_message":"canonical_sha256_bytes"},"source_id":"1507.00887","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:cd95cb73d6c6d165983dbe1f5e8e8cb260eb3470baaa7189fad088ebcc1b1d66","sha256:a8614d25989a21206b1265f461c3d7fd0f782487addc0f442fd3720feac98e5e"],"state_sha256":"869d2b8bd7147866bd777ca483c5333bd232acf26ca49b0868ff0579e6d0bcdb"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"g9hhzzII2LqIa6FB/BG5IIQJG6j/AySesF1wQUeQ6UVTJ5k9h5N/+OHdUBSvNSonyDnHCh0T4Jef2NCSQbUWDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-30T11:00:16.443777Z","bundle_sha256":"df9bf53a080695ecc6a82a305975f34dad888c3761d9a9f787fbb99fa94aff1a"}}