{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:255WVFSUMKFFSS4SGJ3K4IEPGI","short_pith_number":"pith:255WVFSU","canonical_record":{"source":{"id":"2605.13468","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2026-05-13T12:55:22Z","cross_cats_sorted":["cs.NA","cs.NE","math.NA"],"title_canon_sha256":"d86bdcab9cf1ba92cfa7ed846c70b53e43408df3ea71e4890c0a0f0fb51256aa","abstract_canon_sha256":"d9043667ea4dcb8d09300620ea86d03693ff07b5511e568e8c290cf01437ed50"},"schema_version":"1.0"},"canonical_sha256":"d77b6a9654628a594b923276ae208f321fd88e1fc2a9f554a69affaad6d654c5","source":{"kind":"arxiv","id":"2605.13468","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.13468","created_at":"2026-05-18T02:44:41Z"},{"alias_kind":"arxiv_version","alias_value":"2605.13468v1","created_at":"2026-05-18T02:44:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.13468","created_at":"2026-05-18T02:44:41Z"},{"alias_kind":"pith_short_12","alias_value":"255WVFSUMKFF","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_16","alias_value":"255WVFSUMKFFSS4S","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_8","alias_value":"255WVFSU","created_at":"2026-05-18T12:33:37Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:255WVFSUMKFFSS4SGJ3K4IEPGI","target":"record","payload":{"canonical_record":{"source":{"id":"2605.13468","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2026-05-13T12:55:22Z","cross_cats_sorted":["cs.NA","cs.NE","math.NA"],"title_canon_sha256":"d86bdcab9cf1ba92cfa7ed846c70b53e43408df3ea71e4890c0a0f0fb51256aa","abstract_canon_sha256":"d9043667ea4dcb8d09300620ea86d03693ff07b5511e568e8c290cf01437ed50"},"schema_version":"1.0"},"canonical_sha256":"d77b6a9654628a594b923276ae208f321fd88e1fc2a9f554a69affaad6d654c5","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:44:41.589532Z","signature_b64":"aNntR5vkWey+ceHgZImllXmigJsVHtNFCGoy24aKis07xf+CIbc6WhZqdO+03keusOLhzviHD3aGdndogM8ZDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d77b6a9654628a594b923276ae208f321fd88e1fc2a9f554a69affaad6d654c5","last_reissued_at":"2026-05-18T02:44:41.589087Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:44:41.589087Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2605.13468","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":"2026-05-18T02:44:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"M1EPi0Yoa4xHAbFUxYeFUIEQQbxfWg8GR9STQN8cz6ITszZF9Ao0U61TQAVttl/Y/Rs37yCSDRLUr/sS5vJQAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-24T13:53:27.109377Z"},"content_sha256":"97eb75289be2f268c2becb9a51192b6b552928e1c6bb9ffa325c23a04635dfd0","schema_version":"1.0","event_id":"sha256:97eb75289be2f268c2becb9a51192b6b552928e1c6bb9ffa325c23a04635dfd0"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:255WVFSUMKFFSS4SGJ3K4IEPGI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Nonsmooth Set-Gradient Ascent to the Pareto Front via Layered Hypervolume and Magnitude Indicators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Layered weighting of nondomination layers gives set-gradient ascent directions that improve the first Pareto front without compensation from deeper layers.","cross_cats":["cs.NA","cs.NE","math.NA"],"primary_cat":"math.OC","authors_text":"Michael T.M. Emmerich","submitted_at":"2026-05-13T12:55:22Z","abstract_excerpt":"A nonsmooth set-gradient ascent method is developed for moving finite approximation sets toward the Pareto front in multiobjective optimization. The method optimizes layered set indicators: a base indicator is evaluated on successive nondomination layers, and the layer values are combined with rapidly decreasing weights. This gives ascent directions to nondominated and dominated points while preventing deeper layers from compensating for deterioration of the first front. Two base indicators are treated: the hypervolume indicator and the magnitude indicator of the dominated set, whose expansion"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"The method gives ascent directions to nondominated and dominated points while preventing deeper layers from compensating for deterioration of the first front, with an exact gradient formula for the magnitude indicator derived as a linear combination of hypervolume gradients of projected shadow sets.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That rapidly decreasing weights on successive nondomination layers suffice to isolate improvement of the first front and that chamberwise Lipschitz continuity on bounded sets holds for the finite-epsilon surrogates used in practice.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Nonsmooth gradient ascent on layered hypervolume and magnitude indicators moves sets to the Pareto front.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Layered weighting of nondomination layers gives set-gradient ascent directions that improve the first Pareto front without compensation from deeper layers.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"82d0290806dc558513f12a6876d893ddd69fd326b64c3af72f87dfd71df7f30c"},"source":{"id":"2605.13468","kind":"arxiv","version":1},"verdict":{"id":"37a36112-7b61-474c-8794-2282a53d9526","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T17:42:55.848980Z","strongest_claim":"The method gives ascent directions to nondominated and dominated points while preventing deeper layers from compensating for deterioration of the first front, with an exact gradient formula for the magnitude indicator derived as a linear combination of hypervolume gradients of projected shadow sets.","one_line_summary":"Nonsmooth gradient ascent on layered hypervolume and magnitude indicators moves sets to the Pareto front.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That rapidly decreasing weights on successive nondomination layers suffice to isolate improvement of the first front and that chamberwise Lipschitz continuity on bounded sets holds for the finite-epsilon surrogates used in practice.","pith_extraction_headline":"Layered weighting of nondomination layers gives set-gradient ascent directions that improve the first Pareto front without compensation from deeper layers."},"references":{"count":21,"sample":[{"doi":"","year":1990,"title":"F. H. Clarke,Optimization and Nonsmooth Analysis, Classics in Applied Mathematics 5, SIAM, Philadelphia, 1990","work_id":"5cf19cdd-e319-4f7c-9631-45deb735c75e","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1016/j.tcs.2011.03.012","year":2012,"title":"Hypervolume-Based Multiobjective Optimization: Theoretical Foundations and Practical Implications,","work_id":"bf46f82f-4f71-4a3d-be9c-bd470fa4be41","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1137/10079731x","year":2011,"title":"Direct Multisearch for Multiobjective Opti- mization,","work_id":"3ef1fba3-8c03-4789-82f9-b06671128add","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2001,"title":"Deb,Multi-Objective Optimization Using Evolutionary Algorithms, Wiley, Chichester, 2001","work_id":"f92a6530-270a-4038-83a4-9000d7248d04","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1007/978-3-030-58115-2_13","year":2020,"title":"Multi-objective Optimization by Uncrowded Hypervolume Gradient Ascent,","work_id":"fcbcf5f8-e8c1-4643-9920-071f76e91685","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":21,"snapshot_sha256":"cc3f88420b10a460468ff8d882403eee563260eccfd7290326d74617d5945765","internal_anchors":1},"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":"37a36112-7b61-474c-8794-2282a53d9526"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:44:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"//PxsFl4aySe43pH+hjI5nTN2F35+0P54ZaqD+txghflW/Brtm7pr/GNqlCtYdfW1uDeD8ERCu+A2kZeCtsYCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-24T13:53:27.110268Z"},"content_sha256":"0da9c96ae4c1b394015791dd2185361f3233da16f000f930a354dc89510997d1","schema_version":"1.0","event_id":"sha256:0da9c96ae4c1b394015791dd2185361f3233da16f000f930a354dc89510997d1"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/255WVFSUMKFFSS4SGJ3K4IEPGI/bundle.json","state_url":"https://pith.science/pith/255WVFSUMKFFSS4SGJ3K4IEPGI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/255WVFSUMKFFSS4SGJ3K4IEPGI/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":"2026-05-24T13:53:27Z","links":{"resolver":"https://pith.science/pith/255WVFSUMKFFSS4SGJ3K4IEPGI","bundle":"https://pith.science/pith/255WVFSUMKFFSS4SGJ3K4IEPGI/bundle.json","state":"https://pith.science/pith/255WVFSUMKFFSS4SGJ3K4IEPGI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/255WVFSUMKFFSS4SGJ3K4IEPGI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:255WVFSUMKFFSS4SGJ3K4IEPGI","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":"d9043667ea4dcb8d09300620ea86d03693ff07b5511e568e8c290cf01437ed50","cross_cats_sorted":["cs.NA","cs.NE","math.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2026-05-13T12:55:22Z","title_canon_sha256":"d86bdcab9cf1ba92cfa7ed846c70b53e43408df3ea71e4890c0a0f0fb51256aa"},"schema_version":"1.0","source":{"id":"2605.13468","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.13468","created_at":"2026-05-18T02:44:41Z"},{"alias_kind":"arxiv_version","alias_value":"2605.13468v1","created_at":"2026-05-18T02:44:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.13468","created_at":"2026-05-18T02:44:41Z"},{"alias_kind":"pith_short_12","alias_value":"255WVFSUMKFF","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_16","alias_value":"255WVFSUMKFFSS4S","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_8","alias_value":"255WVFSU","created_at":"2026-05-18T12:33:37Z"}],"graph_snapshots":[{"event_id":"sha256:0da9c96ae4c1b394015791dd2185361f3233da16f000f930a354dc89510997d1","target":"graph","created_at":"2026-05-18T02:44:41Z","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":4,"items":[{"attestation":"unclaimed","claim_id":"C1","kind":"strongest_claim","source":"verdict.strongest_claim","status":"machine_extracted","text":"The method gives ascent directions to nondominated and dominated points while preventing deeper layers from compensating for deterioration of the first front, with an exact gradient formula for the magnitude indicator derived as a linear combination of hypervolume gradients of projected shadow sets."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","text":"That rapidly decreasing weights on successive nondomination layers suffice to isolate improvement of the first front and that chamberwise Lipschitz continuity on bounded sets holds for the finite-epsilon surrogates used in practice."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"Nonsmooth gradient ascent on layered hypervolume and magnitude indicators moves sets to the Pareto front."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"Layered weighting of nondomination layers gives set-gradient ascent directions that improve the first Pareto front without compensation from deeper layers."}],"snapshot_sha256":"82d0290806dc558513f12a6876d893ddd69fd326b64c3af72f87dfd71df7f30c"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"A nonsmooth set-gradient ascent method is developed for moving finite approximation sets toward the Pareto front in multiobjective optimization. The method optimizes layered set indicators: a base indicator is evaluated on successive nondomination layers, and the layer values are combined with rapidly decreasing weights. This gives ascent directions to nondominated and dominated points while preventing deeper layers from compensating for deterioration of the first front. Two base indicators are treated: the hypervolume indicator and the magnitude indicator of the dominated set, whose expansion","authors_text":"Michael T.M. Emmerich","cross_cats":["cs.NA","cs.NE","math.NA"],"headline":"Layered weighting of nondomination layers gives set-gradient ascent directions that improve the first Pareto front without compensation from deeper layers.","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2026-05-13T12:55:22Z","title":"Nonsmooth Set-Gradient Ascent to the Pareto Front via Layered Hypervolume and Magnitude Indicators"},"references":{"count":21,"internal_anchors":1,"resolved_work":21,"sample":[{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":1,"title":"F. H. Clarke,Optimization and Nonsmooth Analysis, Classics in Applied Mathematics 5, SIAM, Philadelphia, 1990","work_id":"5cf19cdd-e319-4f7c-9631-45deb735c75e","year":1990},{"cited_arxiv_id":"","doi":"10.1016/j.tcs.2011.03.012","is_internal_anchor":false,"ref_index":2,"title":"Hypervolume-Based Multiobjective Optimization: Theoretical Foundations and Practical Implications,","work_id":"bf46f82f-4f71-4a3d-be9c-bd470fa4be41","year":2012},{"cited_arxiv_id":"","doi":"10.1137/10079731x","is_internal_anchor":false,"ref_index":3,"title":"Direct Multisearch for Multiobjective Opti- mization,","work_id":"3ef1fba3-8c03-4789-82f9-b06671128add","year":2011},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":4,"title":"Deb,Multi-Objective Optimization Using Evolutionary Algorithms, Wiley, Chichester, 2001","work_id":"f92a6530-270a-4038-83a4-9000d7248d04","year":2001},{"cited_arxiv_id":"","doi":"10.1007/978-3-030-58115-2_13","is_internal_anchor":false,"ref_index":5,"title":"Multi-objective Optimization by Uncrowded Hypervolume Gradient Ascent,","work_id":"fcbcf5f8-e8c1-4643-9920-071f76e91685","year":2020}],"snapshot_sha256":"cc3f88420b10a460468ff8d882403eee563260eccfd7290326d74617d5945765"},"source":{"id":"2605.13468","kind":"arxiv","version":1},"verdict":{"created_at":"2026-05-14T17:42:55.848980Z","id":"37a36112-7b61-474c-8794-2282a53d9526","model_set":{"reader":"grok-4.3"},"one_line_summary":"Nonsmooth gradient ascent on layered hypervolume and magnitude indicators moves sets to the Pareto front.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"Layered weighting of nondomination layers gives set-gradient ascent directions that improve the first Pareto front without compensation from deeper layers.","strongest_claim":"The method gives ascent directions to nondominated and dominated points while preventing deeper layers from compensating for deterioration of the first front, with an exact gradient formula for the magnitude indicator derived as a linear combination of hypervolume gradients of projected shadow sets.","weakest_assumption":"That rapidly decreasing weights on successive nondomination layers suffice to isolate improvement of the first front and that chamberwise Lipschitz continuity on bounded sets holds for the finite-epsilon surrogates used in practice."}},"verdict_id":"37a36112-7b61-474c-8794-2282a53d9526"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:97eb75289be2f268c2becb9a51192b6b552928e1c6bb9ffa325c23a04635dfd0","target":"record","created_at":"2026-05-18T02:44:41Z","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":"d9043667ea4dcb8d09300620ea86d03693ff07b5511e568e8c290cf01437ed50","cross_cats_sorted":["cs.NA","cs.NE","math.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2026-05-13T12:55:22Z","title_canon_sha256":"d86bdcab9cf1ba92cfa7ed846c70b53e43408df3ea71e4890c0a0f0fb51256aa"},"schema_version":"1.0","source":{"id":"2605.13468","kind":"arxiv","version":1}},"canonical_sha256":"d77b6a9654628a594b923276ae208f321fd88e1fc2a9f554a69affaad6d654c5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d77b6a9654628a594b923276ae208f321fd88e1fc2a9f554a69affaad6d654c5","first_computed_at":"2026-05-18T02:44:41.589087Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:44:41.589087Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"aNntR5vkWey+ceHgZImllXmigJsVHtNFCGoy24aKis07xf+CIbc6WhZqdO+03keusOLhzviHD3aGdndogM8ZDA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:44:41.589532Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.13468","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:97eb75289be2f268c2becb9a51192b6b552928e1c6bb9ffa325c23a04635dfd0","sha256:0da9c96ae4c1b394015791dd2185361f3233da16f000f930a354dc89510997d1"],"state_sha256":"a4c9f34e5dab116dde076527c3fb4279b415916e4d340e919c1aef0e9beaf579"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WSInwpFJXeR7ONH+UVK2LGPnEvjH3dAMsXZAclgo6Mi1rQyVoQSC3GhI6joKRVT3rjUTDRADMhEAlVYOADU8Bw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-24T13:53:27.114790Z","bundle_sha256":"c3e2fc7ed997caf5e2493ed1d232012a42e8b6c2c43753a15dce176f30441bb2"}}