{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2020:3EVOXSCXDR5N4JXBJR4IGG4C3A","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":"ca84ec83ddef804543293e73a14670d45e014a3a4773144db0341a7014b80e64","cross_cats_sorted":["math.IT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2020-09-24T02:04:03Z","title_canon_sha256":"279dc08b2f17cac09052148861bbc3f554a0f61f584eec60f54a8b6ba77feba6"},"schema_version":"1.0","source":{"id":"2009.12215","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2009.12215","created_at":"2026-07-05T02:17:34Z"},{"alias_kind":"arxiv_version","alias_value":"2009.12215v1","created_at":"2026-07-05T02:17:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2009.12215","created_at":"2026-07-05T02:17:34Z"},{"alias_kind":"pith_short_12","alias_value":"3EVOXSCXDR5N","created_at":"2026-07-05T02:17:34Z"},{"alias_kind":"pith_short_16","alias_value":"3EVOXSCXDR5N4JXB","created_at":"2026-07-05T02:17:34Z"},{"alias_kind":"pith_short_8","alias_value":"3EVOXSCX","created_at":"2026-07-05T02:17:34Z"}],"graph_snapshots":[{"event_id":"sha256:ed0e3b55be8ffa5e64d9438e77d1fe86d90d50859b45a9aab5f183a7d0e651cd","target":"graph","created_at":"2026-07-05T02:17:34Z","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/2009.12215/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In contrast to Part I of this treatise [1] that focuses on the optimization problems associated with single matrix variables, in this paper, we investigate the application of the matrix-monotonic optimization framework in the optimization problems associated with multiple matrix variables. It is revealed that matrix-monotonic optimization still works even for multiple matrix-variate based optimization problems, provided that certain conditions are satisfied. Using this framework, the optimal structures of the matrix variables can be derived and the associated multiple matrix-variate optimizati","authors_text":"Chengwen Xing, H. Vincent Poor, Lajos Hanzo, Shaodan Ma, Sheng Chen, Shuai Wang","cross_cats":["math.IT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2020-09-24T02:04:03Z","title":"Matrix-Monotonic Optimization Part II: Multi-Variable Optimization"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2009.12215","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:cab083cf9cbb90f0795a3b6cebf1a8772d279befa27b0bfd597350dd6903b7f6","target":"record","created_at":"2026-07-05T02:17:34Z","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":"ca84ec83ddef804543293e73a14670d45e014a3a4773144db0341a7014b80e64","cross_cats_sorted":["math.IT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2020-09-24T02:04:03Z","title_canon_sha256":"279dc08b2f17cac09052148861bbc3f554a0f61f584eec60f54a8b6ba77feba6"},"schema_version":"1.0","source":{"id":"2009.12215","kind":"arxiv","version":1}},"canonical_sha256":"d92aebc8571c7ade26e14c78831b82d82902038c3a14d13862609005a64cbb2b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d92aebc8571c7ade26e14c78831b82d82902038c3a14d13862609005a64cbb2b","first_computed_at":"2026-07-05T02:17:34.721960Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T02:17:34.721960Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"4qRKbgOWwe4/MAz2cZCeTslrGdMA8NyEuIWyNMrkSSNBUJvK3dgHPwhiRHxLLQynM56OoOx6W5gEretT0oEHBg==","signature_status":"signed_v1","signed_at":"2026-07-05T02:17:34.722448Z","signed_message":"canonical_sha256_bytes"},"source_id":"2009.12215","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:cab083cf9cbb90f0795a3b6cebf1a8772d279befa27b0bfd597350dd6903b7f6","sha256:ed0e3b55be8ffa5e64d9438e77d1fe86d90d50859b45a9aab5f183a7d0e651cd"],"state_sha256":"fb5aa9858917f5be1df3f9c6978a546adc148cd4228d8da1c4625315551ec77d"}