{"paper":{"title":"Optimal Acceleration for Proximal Minimization of the Sum of Convex and Strongly Convex Functions","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"Fast Douglas-Rachford Splitting achieves optimal O(1/N²) convergence for sums of convex and strongly convex functions.","cross_cats":[],"primary_cat":"math.OC","authors_text":"Beh\\c{c}et A\\c{c}{\\i}kme\\c{s}e, Ernest K. Ryu, Govind M. Chari, Uijeong Jang","submitted_at":"2026-05-09T01:17:24Z","abstract_excerpt":"When minimizing the sum of a convex and a strongly convex function, or when finding the zero of the sum of a monotone operator and a strongly monotone operator, Chambolle and Pock (2010) and Davis and Yin (2015) proposed accelerated mechanisms that achieve an $\\mathcal{O}(1/N^2)$ convergence rate for the squared distance to the solution, but the optimality of this rate was not established. In this work, we present Fast Douglas--Rachford Splitting (FDR), an accelerated method that improves the constants established in the prior works, and provide a complexity lower bound establishing that both "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"we present Fast Douglas--Rachford Splitting (FDR), an accelerated method that improves the constants established in the prior works, and provide a complexity lower bound establishing that both the O(1/N^2) convergence rate and the leading-order constant of FDR's rate are optimal.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The lower bound holds for the general class of convex-plus-strongly-convex problems (or monotone-plus-strongly-monotone operators) without extra structure or restrictions on the proximal operators.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"FDR achieves the optimal O(1/N²) convergence rate with the best leading constant for proximal minimization of convex plus strongly convex functions.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Fast Douglas-Rachford Splitting achieves optimal O(1/N²) convergence for sums of convex and strongly convex functions.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"44bd5081a84a60745e538d2e324740c111ac80de631b411628e0611d1999991e"},"source":{"id":"2605.08593","kind":"arxiv","version":2},"verdict":{"id":"d508e283-8f96-4034-b064-fa6ea4207002","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-12T01:40:31.915340Z","strongest_claim":"we present Fast Douglas--Rachford Splitting (FDR), an accelerated method that improves the constants established in the prior works, and provide a complexity lower bound establishing that both the O(1/N^2) convergence rate and the leading-order constant of FDR's rate are optimal.","one_line_summary":"FDR achieves the optimal O(1/N²) convergence rate with the best leading constant for proximal minimization of convex plus strongly convex functions.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The lower bound holds for the general class of convex-plus-strongly-convex problems (or monotone-plus-strongly-monotone operators) without extra structure or restrictions on the proximal operators.","pith_extraction_headline":"Fast Douglas-Rachford Splitting achieves optimal O(1/N²) convergence for sums of convex and strongly convex functions."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.08593/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"claim_evidence","ran_at":"2026-05-20T09:22:01.952391Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T22:38:10.085488Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-19T14:31:18.233612Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T10:59:19.992276Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"a91f20907de8745d18110d33a0546f922e0874f4a8c6f5d58b5052145aab64b6"},"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"}