{"paper":{"title":"Adaptive Metrics for Norm-Minimization-Based Outer Approximation in Convex Vector Optimization","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Adaptive metrics extend improved convergence rates to all inner-product norms in convex vector optimization outer approximation.","cross_cats":["cs.NA","math.NA"],"primary_cat":"math.OC","authors_text":"Mohammed Alshahrani","submitted_at":"2026-05-14T03:32:59Z","abstract_excerpt":"We develop an adaptive-metric framework for norm-minimization-based outer approximation algorithms in bounded convex vector optimization. The key idea is to let the scalarization metric vary across iterations while measuring approximation error in a fixed Euclidean norm. This enables the algorithm to exploit problem geometry dynamically. Our approach rests on two theoretical foundations. First, we prove that the improved Euclidean convergence rate $O(k^{2/(1-q)})$ -- previously known only for the standard $\\ell_2$ norm -- extends to all fixed inner-product norms. Second, we establish a dispers"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We prove that the improved Euclidean convergence rate O(k^{2/(1-q)}) extends to all fixed inner-product norms and establish a dispersion theorem showing that the cut normals generated by the algorithm naturally spread across all directions when the upper image has a strictly convex boundary with bounded curvature.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The upper image has a strictly convex boundary with bounded curvature, which is required to guarantee that the adaptive metric remains well-conditioned throughout execution.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"An adaptive metric framework for outer approximation in convex vector optimization extends convergence rates to inner-product norms, proves a dispersion theorem under strict convexity, and achieves 31-33% fewer iterations than fixed Euclidean norm on curved Pareto fronts.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Adaptive metrics extend improved convergence rates to all inner-product norms in convex vector optimization outer approximation.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"c9acb063fa38b68cd019f232ae9c16ab6fe5aa8c792638009203916c583b2ed3"},"source":{"id":"2605.14320","kind":"arxiv","version":1},"verdict":{"id":"34a90024-2ed3-4fd7-ad8b-4805c18d5e19","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T02:42:26.871740Z","strongest_claim":"We prove that the improved Euclidean convergence rate O(k^{2/(1-q)}) extends to all fixed inner-product norms and establish a dispersion theorem showing that the cut normals generated by the algorithm naturally spread across all directions when the upper image has a strictly convex boundary with bounded curvature.","one_line_summary":"An adaptive metric framework for outer approximation in convex vector optimization extends convergence rates to inner-product norms, proves a dispersion theorem under strict convexity, and achieves 31-33% fewer iterations than fixed Euclidean norm on curved Pareto fronts.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The upper image has a strictly convex boundary with bounded curvature, which is required to guarantee that the adaptive metric remains well-conditioned throughout execution.","pith_extraction_headline":"Adaptive metrics extend improved convergence rates to all inner-product norms in convex vector optimization outer approximation."},"references":{"count":25,"sample":[{"doi":"10.1007/978-3-319-63049-6","year":2018,"title":"Q. H. Ansari, E. K¨ obis, and J.-C. Yao,Vector Variational Inequalities and Vector Optimization(Vector Optimization), en. Cham: Springer International Publishing, 2018.doi:10.1007/978-3-319-63049-6","work_id":"f2cde9a9-0b19-4d23-8ffe-2012e2fe4bf0","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2022,"title":"A Norm Minimization-Based Convex Vector Optimization Algorithm,","work_id":"104bfda2-ea9e-422b-bc55-e2f711eda42e","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2023,"title":"Outer Approximation Algorithms for Convex Vector Optimization Problems,","work_id":"71309eac-3772-461e-abf5-014427db40e6","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1137/23m1574580","year":2024,"title":"Convergence Analysis of a Norm Minimization-Based Convex Vector Optimization Algorithm,","work_id":"f84b4e15-7f19-43c9-be85-1193fecf1f0d","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1992,"title":"A Class of Adaptive Algorithms for Approximating Convex Bodies by Polyhedra,","work_id":"2437bde4-aaae-4c02-ad80-32ba31631e94","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":25,"snapshot_sha256":"f17a053871e22ba7395c11655d31c1bbfdd6ed7d4ee6c98beca19eaccc8ef22e","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"ed9dcfc4f4065758de2778fe135cdb3645e9aabf74b104d0aba68bca6c0bbeb4"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}