{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:MDIJS5QAN44Y4L74G3BV4SSW7K","short_pith_number":"pith:MDIJS5QA","canonical_record":{"source":{"id":"1204.0166","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2012-04-01T05:08:08Z","cross_cats_sorted":["math.IT"],"title_canon_sha256":"e87349609601cf0d8a1860ab04c93e8770f988bc4b2945b746d924003dbbdef0","abstract_canon_sha256":"68033894aee76907941b0543074fd5ff18b92dbaf331be99d5797741296dfb5a"},"schema_version":"1.0"},"canonical_sha256":"60d09976006f398e2ffc36c35e4a56fa98dad2fbc371f269f4518a6aea549919","source":{"kind":"arxiv","id":"1204.0166","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1204.0166","created_at":"2026-05-18T00:58:43Z"},{"alias_kind":"arxiv_version","alias_value":"1204.0166v1","created_at":"2026-05-18T00:58:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1204.0166","created_at":"2026-05-18T00:58:43Z"},{"alias_kind":"pith_short_12","alias_value":"MDIJS5QAN44Y","created_at":"2026-05-18T12:27:14Z"},{"alias_kind":"pith_short_16","alias_value":"MDIJS5QAN44Y4L74","created_at":"2026-05-18T12:27:14Z"},{"alias_kind":"pith_short_8","alias_value":"MDIJS5QA","created_at":"2026-05-18T12:27:14Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:MDIJS5QAN44Y4L74G3BV4SSW7K","target":"record","payload":{"canonical_record":{"source":{"id":"1204.0166","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2012-04-01T05:08:08Z","cross_cats_sorted":["math.IT"],"title_canon_sha256":"e87349609601cf0d8a1860ab04c93e8770f988bc4b2945b746d924003dbbdef0","abstract_canon_sha256":"68033894aee76907941b0543074fd5ff18b92dbaf331be99d5797741296dfb5a"},"schema_version":"1.0"},"canonical_sha256":"60d09976006f398e2ffc36c35e4a56fa98dad2fbc371f269f4518a6aea549919","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:58:43.126287Z","signature_b64":"2YCJmiKFVkeC36JU/+yc7mEQm5maSYxVZkVAgj3I5ZwOLBJabDcPd0rOBzXJ18vqCRN4/EZW83mF2mCQUc1xBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"60d09976006f398e2ffc36c35e4a56fa98dad2fbc371f269f4518a6aea549919","last_reissued_at":"2026-05-18T00:58:43.125528Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:58:43.125528Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1204.0166","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":"2026-05-18T00:58:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CpT6WRx2Fn2qyUN54rMN23unU4GoG0ncM21+tWZQx22iaSVQ8BiyOvC6ZaHt+wCY/7UTZzvHJLo2TG7qrpjnCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T11:20:10.356870Z"},"content_sha256":"fcdf6613dc5d3b35fe87f1583d63485b03deabcc3ab36ac9bac7bd5d339a6fb7","schema_version":"1.0","event_id":"sha256:fcdf6613dc5d3b35fe87f1583d63485b03deabcc3ab36ac9bac7bd5d339a6fb7"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:MDIJS5QAN44Y4L74G3BV4SSW7K","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Worst-Case Robust Multiuser Transmit Beamforming Using Semidefinite Relaxation: Duality and Implications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.IT"],"primary_cat":"cs.IT","authors_text":"Chong-Yung Chi, Tsung-Hui Chang, Wing-Kin Ma","submitted_at":"2012-04-01T05:08:08Z","abstract_excerpt":"This paper studies a downlink multiuser transmit beamforming design under spherical channel uncertainties, using a worst-case robust formulation. This robust design problem is nonconvex. Recently, a convex approximation formulation based on semidefinite relaxation (SDR) has been proposed to handle the problem. Curiously, simulation results have consistently indicated that SDR can attain the global optimum of the robust design problem. This paper intends to provide some theoretical insights into this important empirical finding. Our main result is a dual representation of the SDR formulation, w"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1204.0166","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":""},"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":"2026-05-18T00:58:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uWJlk7u5/EIb02BoVeygdUbEwVmjAmOZLMnqDQJJ8WlobIrUDU33ddhHBErT9ic04TvKsyxib3ihdstQ0r/9DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T11:20:10.357606Z"},"content_sha256":"4073bf754f83a6d857acdc8a1d4bd5a19b1bcd2ba65102a394027b29f8ec7851","schema_version":"1.0","event_id":"sha256:4073bf754f83a6d857acdc8a1d4bd5a19b1bcd2ba65102a394027b29f8ec7851"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/MDIJS5QAN44Y4L74G3BV4SSW7K/bundle.json","state_url":"https://pith.science/pith/MDIJS5QAN44Y4L74G3BV4SSW7K/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/MDIJS5QAN44Y4L74G3BV4SSW7K/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":"2026-05-27T11:20:10Z","links":{"resolver":"https://pith.science/pith/MDIJS5QAN44Y4L74G3BV4SSW7K","bundle":"https://pith.science/pith/MDIJS5QAN44Y4L74G3BV4SSW7K/bundle.json","state":"https://pith.science/pith/MDIJS5QAN44Y4L74G3BV4SSW7K/state.json","well_known_bundle":"https://pith.science/.well-known/pith/MDIJS5QAN44Y4L74G3BV4SSW7K/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:MDIJS5QAN44Y4L74G3BV4SSW7K","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":"68033894aee76907941b0543074fd5ff18b92dbaf331be99d5797741296dfb5a","cross_cats_sorted":["math.IT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2012-04-01T05:08:08Z","title_canon_sha256":"e87349609601cf0d8a1860ab04c93e8770f988bc4b2945b746d924003dbbdef0"},"schema_version":"1.0","source":{"id":"1204.0166","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1204.0166","created_at":"2026-05-18T00:58:43Z"},{"alias_kind":"arxiv_version","alias_value":"1204.0166v1","created_at":"2026-05-18T00:58:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1204.0166","created_at":"2026-05-18T00:58:43Z"},{"alias_kind":"pith_short_12","alias_value":"MDIJS5QAN44Y","created_at":"2026-05-18T12:27:14Z"},{"alias_kind":"pith_short_16","alias_value":"MDIJS5QAN44Y4L74","created_at":"2026-05-18T12:27:14Z"},{"alias_kind":"pith_short_8","alias_value":"MDIJS5QA","created_at":"2026-05-18T12:27:14Z"}],"graph_snapshots":[{"event_id":"sha256:4073bf754f83a6d857acdc8a1d4bd5a19b1bcd2ba65102a394027b29f8ec7851","target":"graph","created_at":"2026-05-18T00:58:43Z","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":"This paper studies a downlink multiuser transmit beamforming design under spherical channel uncertainties, using a worst-case robust formulation. This robust design problem is nonconvex. Recently, a convex approximation formulation based on semidefinite relaxation (SDR) has been proposed to handle the problem. Curiously, simulation results have consistently indicated that SDR can attain the global optimum of the robust design problem. This paper intends to provide some theoretical insights into this important empirical finding. Our main result is a dual representation of the SDR formulation, w","authors_text":"Chong-Yung Chi, Tsung-Hui Chang, Wing-Kin Ma","cross_cats":["math.IT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2012-04-01T05:08:08Z","title":"Worst-Case Robust Multiuser Transmit Beamforming Using Semidefinite Relaxation: Duality and Implications"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1204.0166","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:fcdf6613dc5d3b35fe87f1583d63485b03deabcc3ab36ac9bac7bd5d339a6fb7","target":"record","created_at":"2026-05-18T00:58:43Z","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":"68033894aee76907941b0543074fd5ff18b92dbaf331be99d5797741296dfb5a","cross_cats_sorted":["math.IT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2012-04-01T05:08:08Z","title_canon_sha256":"e87349609601cf0d8a1860ab04c93e8770f988bc4b2945b746d924003dbbdef0"},"schema_version":"1.0","source":{"id":"1204.0166","kind":"arxiv","version":1}},"canonical_sha256":"60d09976006f398e2ffc36c35e4a56fa98dad2fbc371f269f4518a6aea549919","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"60d09976006f398e2ffc36c35e4a56fa98dad2fbc371f269f4518a6aea549919","first_computed_at":"2026-05-18T00:58:43.125528Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:58:43.125528Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"2YCJmiKFVkeC36JU/+yc7mEQm5maSYxVZkVAgj3I5ZwOLBJabDcPd0rOBzXJ18vqCRN4/EZW83mF2mCQUc1xBw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:58:43.126287Z","signed_message":"canonical_sha256_bytes"},"source_id":"1204.0166","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fcdf6613dc5d3b35fe87f1583d63485b03deabcc3ab36ac9bac7bd5d339a6fb7","sha256:4073bf754f83a6d857acdc8a1d4bd5a19b1bcd2ba65102a394027b29f8ec7851"],"state_sha256":"5ddb420a75af3327380b446456c8c1ecb02b62f12a79f1d178e6502c55be0021"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"oU5zMgJDDcpAnVYDKh5D8K16BrHuARDU4a0SjAig9xpD8uE9Y7psSBOOwWmZYuLF7YAbqHaUVIqJJQGmrq4HDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T11:20:10.361407Z","bundle_sha256":"661b3ac8264c20c8010caa3c1ce06404a7a21a6486dc1f64de436ed24728dcbf"}}