{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:GPB2XZDTILMC66U2WZSNNOV4YH","short_pith_number":"pith:GPB2XZDT","canonical_record":{"source":{"id":"2607.13383","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.OC","submitted_at":"2026-07-15T02:22:55Z","cross_cats_sorted":[],"title_canon_sha256":"e4b9c9184b2ec6c456f13b95effc659519dd84ec0ae214424771dc443f0d529b","abstract_canon_sha256":"ccf3fab7c2cd91b4cc7860223afadb6fafe0170156333bb5581018898544827b"},"schema_version":"1.0"},"canonical_sha256":"33c3abe47342d82f7a9ab664d6babcc1e4eb8fd80c64161c5a9f3c47721df898","source":{"kind":"arxiv","id":"2607.13383","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.13383","created_at":"2026-07-16T00:22:15Z"},{"alias_kind":"arxiv_version","alias_value":"2607.13383v1","created_at":"2026-07-16T00:22:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.13383","created_at":"2026-07-16T00:22:15Z"},{"alias_kind":"pith_short_12","alias_value":"GPB2XZDTILMC","created_at":"2026-07-16T00:22:15Z"},{"alias_kind":"pith_short_16","alias_value":"GPB2XZDTILMC66U2","created_at":"2026-07-16T00:22:15Z"},{"alias_kind":"pith_short_8","alias_value":"GPB2XZDT","created_at":"2026-07-16T00:22:15Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:GPB2XZDTILMC66U2WZSNNOV4YH","target":"record","payload":{"canonical_record":{"source":{"id":"2607.13383","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.OC","submitted_at":"2026-07-15T02:22:55Z","cross_cats_sorted":[],"title_canon_sha256":"e4b9c9184b2ec6c456f13b95effc659519dd84ec0ae214424771dc443f0d529b","abstract_canon_sha256":"ccf3fab7c2cd91b4cc7860223afadb6fafe0170156333bb5581018898544827b"},"schema_version":"1.0"},"canonical_sha256":"33c3abe47342d82f7a9ab664d6babcc1e4eb8fd80c64161c5a9f3c47721df898","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-16T00:22:15.020794Z","signature_b64":"cihCebuypPXDLIMXe+pueWV8vY1B+4dp9dHuCVuV6mLg9rKOPe76NaKnJkSoLMr8sTz2Zac0dF4a6U/oZJ0bDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"33c3abe47342d82f7a9ab664d6babcc1e4eb8fd80c64161c5a9f3c47721df898","last_reissued_at":"2026-07-16T00:22:15.019924Z","signature_status":"signed_v1","first_computed_at":"2026-07-16T00:22:15.019924Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2607.13383","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-16T00:22:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VDtTZJnRlWw77PSw2Nl5HajopBRPbXNmA9h9VNYBlzQ5cV4fv0QqHThWwb9kVVTHlXn30KPWIrHMOBGgwZ7TBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T11:52:46.106707Z"},"content_sha256":"beceb92b9ae422cd9f312569c337d78c39355ba0a4297413ccd63f4fc689d662","schema_version":"1.0","event_id":"sha256:beceb92b9ae422cd9f312569c337d78c39355ba0a4297413ccd63f4fc689d662"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:GPB2XZDTILMC66U2WZSNNOV4YH","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Hager-Zhang Conjugate Gradient Method for Set Optimization with Set-Valued Objective Map of Finite Cardinality","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Debdas Ghosh, Ravi Raushan, Zai-Yun Peng","submitted_at":"2026-07-15T02:22:55Z","abstract_excerpt":"This work introduces a nonlinear Hager-Zhang conjugate gradient method for solving set optimization problems. The objective function under consideration is defined by a finite collection of continuously differentiable functions. Notably, the proposed approach imposes restrictions neither on the existence of a finite generator of the ordering cone nor on any regularity condition at the optimal solution. As a result, the proposed method holds considerable significance for both set optimization and vector optimization problems, with the latter serving as a special case of the former. The study be"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.13383","kind":"arxiv","version":1},"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/2607.13383/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":"2026-07-16T00:22:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nG43Cm9hBp1hFl3PnGq6/BX9azE0o5AkqbJGO+ZXeevRuASGesMOlF9xijR8N5Tt1+Z8SS3P7003NDTFLBkkAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T11:52:46.107621Z"},"content_sha256":"f6f60ed3d9a0021a0eafdcf00590d3c54149afe0b1ea100227a7a6f39f823df4","schema_version":"1.0","event_id":"sha256:f6f60ed3d9a0021a0eafdcf00590d3c54149afe0b1ea100227a7a6f39f823df4"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/GPB2XZDTILMC66U2WZSNNOV4YH/bundle.json","state_url":"https://pith.science/pith/GPB2XZDTILMC66U2WZSNNOV4YH/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/GPB2XZDTILMC66U2WZSNNOV4YH/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":"2026-08-13T11:52:46Z","links":{"resolver":"https://pith.science/pith/GPB2XZDTILMC66U2WZSNNOV4YH","bundle":"https://pith.science/pith/GPB2XZDTILMC66U2WZSNNOV4YH/bundle.json","state":"https://pith.science/pith/GPB2XZDTILMC66U2WZSNNOV4YH/state.json","well_known_bundle":"https://pith.science/.well-known/pith/GPB2XZDTILMC66U2WZSNNOV4YH/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:GPB2XZDTILMC66U2WZSNNOV4YH","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":"ccf3fab7c2cd91b4cc7860223afadb6fafe0170156333bb5581018898544827b","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.OC","submitted_at":"2026-07-15T02:22:55Z","title_canon_sha256":"e4b9c9184b2ec6c456f13b95effc659519dd84ec0ae214424771dc443f0d529b"},"schema_version":"1.0","source":{"id":"2607.13383","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.13383","created_at":"2026-07-16T00:22:15Z"},{"alias_kind":"arxiv_version","alias_value":"2607.13383v1","created_at":"2026-07-16T00:22:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.13383","created_at":"2026-07-16T00:22:15Z"},{"alias_kind":"pith_short_12","alias_value":"GPB2XZDTILMC","created_at":"2026-07-16T00:22:15Z"},{"alias_kind":"pith_short_16","alias_value":"GPB2XZDTILMC66U2","created_at":"2026-07-16T00:22:15Z"},{"alias_kind":"pith_short_8","alias_value":"GPB2XZDT","created_at":"2026-07-16T00:22:15Z"}],"graph_snapshots":[{"event_id":"sha256:f6f60ed3d9a0021a0eafdcf00590d3c54149afe0b1ea100227a7a6f39f823df4","target":"graph","created_at":"2026-07-16T00:22:15Z","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/2607.13383/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"This work introduces a nonlinear Hager-Zhang conjugate gradient method for solving set optimization problems. The objective function under consideration is defined by a finite collection of continuously differentiable functions. Notably, the proposed approach imposes restrictions neither on the existence of a finite generator of the ordering cone nor on any regularity condition at the optimal solution. As a result, the proposed method holds considerable significance for both set optimization and vector optimization problems, with the latter serving as a special case of the former. The study be","authors_text":"Debdas Ghosh, Ravi Raushan, Zai-Yun Peng","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.OC","submitted_at":"2026-07-15T02:22:55Z","title":"Hager-Zhang Conjugate Gradient Method for Set Optimization with Set-Valued Objective Map of Finite Cardinality"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.13383","kind":"arxiv","version":1},"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:beceb92b9ae422cd9f312569c337d78c39355ba0a4297413ccd63f4fc689d662","target":"record","created_at":"2026-07-16T00:22:15Z","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":"ccf3fab7c2cd91b4cc7860223afadb6fafe0170156333bb5581018898544827b","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.OC","submitted_at":"2026-07-15T02:22:55Z","title_canon_sha256":"e4b9c9184b2ec6c456f13b95effc659519dd84ec0ae214424771dc443f0d529b"},"schema_version":"1.0","source":{"id":"2607.13383","kind":"arxiv","version":1}},"canonical_sha256":"33c3abe47342d82f7a9ab664d6babcc1e4eb8fd80c64161c5a9f3c47721df898","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"33c3abe47342d82f7a9ab664d6babcc1e4eb8fd80c64161c5a9f3c47721df898","first_computed_at":"2026-07-16T00:22:15.019924Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-16T00:22:15.019924Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"cihCebuypPXDLIMXe+pueWV8vY1B+4dp9dHuCVuV6mLg9rKOPe76NaKnJkSoLMr8sTz2Zac0dF4a6U/oZJ0bDg==","signature_status":"signed_v1","signed_at":"2026-07-16T00:22:15.020794Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.13383","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:beceb92b9ae422cd9f312569c337d78c39355ba0a4297413ccd63f4fc689d662","sha256:f6f60ed3d9a0021a0eafdcf00590d3c54149afe0b1ea100227a7a6f39f823df4"],"state_sha256":"9ea0307436a4517c9bc53bb03a15dad1e23016211da72a34df3c0a9f95bdab20"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fxaii1yDf3/SRKS4C3zpzKvzT0sqVIKZIegXTe8TwosU5gRlc1Ox/o+Gl4AYpfTlDz/FbcqcV59Hl0GeJ00OBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-13T11:52:46.114473Z","bundle_sha256":"d39390b2c01222a2a7c6db31b754db0b3f46a068e82657f3d225be8c46d600df"}}