{"paper":{"title":"Convergence Rates for $\\ell_p$ Norm Minimization in Convex Vector Optimization","license":"http://creativecommons.org/licenses/by/4.0/","headline":"The Hausdorff approximation error converges at rate O(k^{2/(1-q)}) for every ℓ_p norm in convex vector optimization.","cross_cats":["cs.NA","math.NA"],"primary_cat":"math.OC","authors_text":"Mohammed Alshahrani","submitted_at":"2026-05-14T03:39:06Z","abstract_excerpt":"We analyze convergence rates of norm-minimization-based outer approximation algorithms for convex vector optimization when the scalarization uses an $\\ell_p$ norm with $p \\in (1,\\infty)$. While the Euclidean case ($p=2$) achieves the optimal rate $O(k^{2/(1-q)})$, the behavior under general $\\ell_p$ norms has remained open. A direct approach via the modulus of smoothness yields only the weaker exponent $\\min(p,2)$, which degrades for $1 < p < 2$. We prove that the Hausdorff approximation error satisfies $\\delta_H(P_k, A) = O(k^{2/(1-q)})$ for \\emph{every} $p \\in (1,\\infty)$, where $q$ is the n"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We prove that the Hausdorff approximation error satisfies δ_H(P_k, A) = O(k^{2/(1-q)}) for every p ∈ (1,∞)","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The technique assumes the ambient space is R^q with its standard inner product structure, which enables the quadratic bound on hyperplane distance; this may not extend directly to non-Euclidean settings or infinite dimensions.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"The Hausdorff error for ℓ_p-norm based outer approximations in convex vector optimization converges at the optimal rate O(k^{2/(1-q)}) independently of p.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The Hausdorff approximation error converges at rate O(k^{2/(1-q)}) for every ℓ_p norm in convex vector optimization.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"9801f7ba844be2fded2bf6206c34636fd78689b724b006ae543948207c4e4e3a"},"source":{"id":"2605.14324","kind":"arxiv","version":1},"verdict":{"id":"29928b0c-8c2c-4561-87af-e2f5167777e6","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T02:34:52.741938Z","strongest_claim":"We prove that the Hausdorff approximation error satisfies δ_H(P_k, A) = O(k^{2/(1-q)}) for every p ∈ (1,∞)","one_line_summary":"The Hausdorff error for ℓ_p-norm based outer approximations in convex vector optimization converges at the optimal rate O(k^{2/(1-q)}) independently of p.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The technique assumes the ambient space is R^q with its standard inner product structure, which enables the quadratic bound on hyperplane distance; this may not extend directly to non-Euclidean settings or infinite dimensions.","pith_extraction_headline":"The Hausdorff approximation error converges at rate O(k^{2/(1-q)}) for every ℓ_p norm in convex vector optimization."},"references":{"count":30,"sample":[{"doi":"10.1007/s10957-022-02045-8","year":2022,"title":"A Norm Minimization-Based Convex Vector Optimization Algorithm,","work_id":"650744a5-28c3-49a5-bf19-832c03b90861","ref_index":1,"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":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1515/form.1993.5.521","year":1993,"title":"Asymptotic estimates for best and stepwise approximation of convex bodies II,","work_id":"06fc47b2-612f-4daa-a63a-6c01c757fc46","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1515/form.1993.5.281","year":1993,"title":"Asymptotic estimates for best and stepwise approximation of convex bodies I,","work_id":"dced9a03-6e52-45e5-b03d-2b64b6eb513b","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1515/form.1997.9","year":1997,"title":"Asymptotic estimates for best and stepwise approximation of convex bodies III,","work_id":"ebd95fb1-0523-40b4-9f7c-a3f53071b0cd","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":30,"snapshot_sha256":"5abbddd83481452b65fc10423e2b1dddb480bcd027ea2f36c8cfd47c9452bf34","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"da77f47229a7e73292a8191f55978e4f4a9027171667a4a1df97438fea24d396"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}