{"paper":{"title":"A Proximal Gradient Framework for Composite Multiobjective Optimization on Riemannian Manifolds","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"A proximal gradient method converges composite multiobjective optimization problems on Riemannian manifolds to Pareto stationary points at an O(1/k) rate.","cross_cats":[],"primary_cat":"math.OC","authors_text":"Kangming Chen","submitted_at":"2026-05-16T00:47:45Z","abstract_excerpt":"This paper proposes a Riemannian Multiobjective Proximal Gradient Method (RMPGM) for composite optimization problems on manifolds. Unlike scalarization-based approaches, the proposed framework directly handles vector-valued objectives and establishes global convergence to Pareto stationary points, together with an $\\mathcal{O}(1/k)$ convergence rate. We further develop two variants to enhance practicality and performance: an inexact RMPGM that allows controlled inexactness in solving subproblems, and a trust-region RMPGM that adaptively adjusts the penalty parameter and achieves an $\\mathcal{O"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"establishes global convergence to Pareto stationary points, together with an O(1/k) convergence rate","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The composite structure of the vector-valued objective functions and the Riemannian manifold admit well-defined proximal mappings and retraction operations that preserve the necessary descent properties (implicit in the framework description for composite optimization problems on manifolds).","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"The Riemannian Multiobjective Proximal Gradient Method (RMPGM) directly optimizes vector-valued composite objectives on Riemannian manifolds and converges globally to Pareto stationary points with an O(1/k) rate.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A proximal gradient method converges composite multiobjective optimization problems on Riemannian manifolds to Pareto stationary points at an O(1/k) rate.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"8dec5983df02c036e594b4783aa37bfa7c41ff0b4e3c83b3f821f2f7740fd6d0"},"source":{"id":"2605.16731","kind":"arxiv","version":1},"verdict":{"id":"62480791-9750-4774-9dd1-64d8cc8b554b","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T21:34:48.109882Z","strongest_claim":"establishes global convergence to Pareto stationary points, together with an O(1/k) convergence rate","one_line_summary":"The Riemannian Multiobjective Proximal Gradient Method (RMPGM) directly optimizes vector-valued composite objectives on Riemannian manifolds and converges globally to Pareto stationary points with an O(1/k) rate.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The composite structure of the vector-valued objective functions and the Riemannian manifold admit well-defined proximal mappings and retraction operations that preserve the necessary descent properties (implicit in the framework description for composite optimization problems on manifolds).","pith_extraction_headline":"A proximal gradient method converges composite multiobjective optimization problems on Riemannian manifolds to Pareto stationary points at an O(1/k) rate."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.16731/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_title_agreement","ran_at":"2026-05-19T22:01:23.068936Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T21:40:57.053372Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T19:01:56.343140Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T18:33:26.469390Z","status":"skipped","version":"1.0.0","findings_count":0}],"snapshot_sha256":"9f7f5c2d3287eb7a4a655ecbb1387f566d561a32ebc8d3509eb149c9ce374af3"},"references":{"count":31,"sample":[{"doi":"","year":2008,"title":"Absil, P.-A., Mahony, R., Sepulchre, R.: Optimization Algorithms on Matrix Manifolds, p. 224. 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