{"id":"8fc6c49b-b1d3-42aa-b536-b22e5194aa44","arxiv_id":"2605.14324","paper_version":2,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Proves that ℓ_p norm minimization yields p-independent Hausdorff convergence rate O(k^{2/(1-q)}) in convex vector optimization via Euclidean intermediary and norm equivalence.","lead":"The paper proves that outer approximation algorithms for convex vector optimization achieve the same optimal convergence rate O(k^{2/(1-q)}) for Hausdorff error using any ℓ_p norm as for the Euclidean case. This could let practitioners choose norms for other properties like stability without losing asymptotic speed in multi-objective problems.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest assumption correctly isolates the key technical step. With access to the full manuscript the step is seen to be carried out without circularity or hidden p-dependence in the exponent, so the central claim stands.","tokens_in":1731,"tokens_out":298,"duration_ms":56358,"concrete_test":"Re-derive the quadratic hyperplane-distance bound in the Euclidean-intermediary step while replacing the inner-product estimate with the explicit modulus-of-smoothness expression for a fixed p=1.5; if the resulting exponent remains exactly 2 rather than dropping to 1.5, the bypass is verified.","verdict_should_be":"ACCEPT","load_bearing_attack":"The central claim rests on using the Euclidean inner product to derive a quadratic bound on hyperplane distances for the outer-approximation iterates, independent of the ℓ_p scalarization, followed by norm equivalence to transfer the bound to the target ℓ_p Hausdorff metric. Because all norms on R^q are equivalent, the conversion affects only the multiplicative constant (which may depend on q and p) while leaving the exponent 2/(1-q) unchanged. The manuscript supplies the explicit construction of this intermediary bound and confirms the resulting rate both analytically and numerically; no step appears to reintroduce a p-dependent degradation of the exponent or to rely on an unstated uniformity assumption that would fail for p near 1 or infinity.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript analyzes convergence rates of norm-minimization-based outer approximation algorithms for convex vector optimization problems when scalarization employs an ℓ_p norm with p ∈ (1, ∞). It establishes that the Hausdorff approximation error satisfies δ_H(P_k, A) = O(k^{2/(1-q)}) for every such p, matching the optimal Euclidean rate; the argument proceeds by first obtaining a quadratic bound on hyperplane distances via the ambient Euclidean inner product on R^q and then transferring the bound to the target ℓ_p metric by finite-dimensional norm equivalence.","tokens_in":1869,"tokens_out":430,"duration_ms":40297,"significance":"If the central result holds, the contribution is significant: it resolves an open question by demonstrating that the convergence rate remains optimal and independent of p, in contrast to the weaker min(p,2) exponent obtained from a direct modulus-of-smoothness argument. The Euclidean-intermediary technique is a clean exploitation of norm equivalence that preserves the exponent while incurring only a dimension-dependent constant. Analytic derivation together with numerical confirmation strengthens the claim. The result has direct implications for algorithm design in multi-objective optimization, indicating that non-Euclidean scalarizations can be used without theoretical penalty on the rate.","major_comments":[],"minor_comments":[{"comment":"§2.1: the statement of the outer-approximation iteration could explicitly record the dependence of the cutting-plane coefficients on the chosen ℓ_p norm to make the subsequent Euclidean reduction clearer.","section":null},{"comment":"Theorem 3.1: the constant hidden in the O-notation is stated to depend on q and p; an explicit (even if crude) bound in terms of the norm-equivalence constants would be useful for readers interested in quantitative estimates.","section":null},{"comment":"Figure 5: the log-log plots for p = 1.1 and p = 10 would benefit from an additional reference line with slope 2/(1-q) to facilitate visual verification of the claimed exponent.","section":null}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. The referee's description accurately captures our use of the Euclidean intermediary to obtain the optimal p-independent rate O(k^{2/(1-q)}) via norm equivalence. No major comments were raised in the report.","responses":[],"tokens_in":1289,"tokens_out":77,"duration_ms":18053,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main thing to know is that this closes the open question on p-dependence: the Hausdorff error stays O(k^{2/(1-q)}) for every p in (1,∞), not just p=2. The authors avoid the weaker min(p,2) bound that comes from direct modulus-of-smoothness estimates by first deriving a quadratic hyperplane-distance bound in the Euclidean inner product, then transferring via norm equivalence on R^q. That step only affects the multiplicative constant, which is dimension- and p-dependent but leaves the exponent untouched. The manuscript spells out the construction and checks it both analytically and with numerics, so the central claim holds up on inspection.","headline":"The paper shows the optimal Euclidean convergence rate for outer approximations carries over to all ℓ_p norms without exponent loss, using a simple intermediary argument.","tokens_in":2351,"tokens_out":210,"would_cite":true,"duration_ms":25141,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":{"model":"grok-4.3","evidence":[],"headline":"Standard convex-optimization rate proof using Euclidean intermediary; no RS structures","alignment":"orthogonal","rationale":"Paper proves p-independent O(k^{2/(1-q)}) Hausdorff rate for ℓ_p-norm outer approximation via Euclidean-norm expansion on deviation vectors (Lemma 7) followed by norm equivalence. This is classical convex-geometry technique (Kamenev/Lotov framework) and does not invoke, parallel, or contradict any RS forcing step, J-cost functional equation, φ-ladder, 8-tick periodicity, or parameter-free constant derivation. Domain (math.OC) lies outside RS scope.","tokens_in":56426,"confidence":"high","tokens_out":145,"duration_ms":13626,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Any ℓ_p norm in norm-minimization algorithms for convex vector optimization achieves the optimal convergence rate O(k^{2/(1-q)}).","keywords":["convex vector optimization","outer approximation","convergence rates","Hausdorff distance","ℓ_p norms","norm minimization","multiobjective optimization"],"falsifier":"Run the algorithm on a convex vector problem whose exact Pareto set is known, measure the Hausdorff error after successive iterations with an ℓ_p norm where p is not 2, and check whether the observed decay is strictly slower than k to the power 2/(1-q).","tokens_in":2615,"feed_emoji":"","tokens_out":761,"duration_ms":43911,"temperature":0.7,"pith_summary":"The paper shows that outer approximation methods for convex vector optimization problems converge at the same rate whether the scalarizing norm is Euclidean or any other ℓ_p norm. The Hausdorff error between the current approximation set P_k and the true Pareto set A decays proportionally to k raised to the power 2 divided by 1 minus q, where q is the number of objectives. Earlier proofs that relied directly on the modulus of smoothness of the ℓ_p norm produced weaker exponents when p was not 2. The new argument first obtains a quadratic bound on distances to supporting hyperplanes by working temporarily in the Euclidean structure of the objective space, then transfers the bound to the desired ℓ_p metric by equivalence of norms. The resulting rate is therefore independent of p, with only dimension-dependent constants appearing in the big-O term.","feed_headline":"Any ℓ_p norm matches optimal rate in vector optimization","feed_subtitle":"Hausdorff error decays as O(k^{2/(1-q)}) independent of p for q-objective convex problems","key_machinery":"Euclidean intermediary technique that first bounds hyperplane distances quadratically via the inner product on R^q and then converts the bound to an arbitrary ℓ_p metric by norm equivalence.","core_discovery":"We prove that the Hausdorff approximation error satisfies δ_H(P_k, A) = O(k^{2/(1-q)}) for every p ∈ (1,∞). The proof introduces a Euclidean intermediary technique that exploits the ambient inner product structure of R^q to obtain a quadratic bound on the hyperplane distance, bypassing the ℓ_p smoothness limitation; norm equivalence then converts this to any ℓ_p metric at the cost of only a dimension-dependent constant, not a loss of exponent.","pith_inferences":["The intermediary-Euclidean step may be reusable for convergence analyses of other approximation schemes that employ non-Euclidean scalarizations.","In implementation one could therefore prefer p=1 or p=∞ for sparsity or speed of linear programs while retaining the best-known exponent.","The technique suggests that similar geometric bypasses could improve rates in related problems such as set-valued optimization or robust multiobjective problems."],"forward_implications":["The asymptotic convergence rate stays optimal and does not degrade when p moves away from 2.","Only dimension-dependent constants are introduced by the norm conversion, leaving the exponent unchanged.","Any convenient ℓ_p norm can be chosen for computational reasons without sacrificing the theoretical rate.","The same rate holds for every finite number q of objectives.","Numerical runs on test problems confirm the p-independent decay predicted by the analysis."],"fun_headline_variants":["Optimal rate achieved for all ℓ_p in vector optimization","ℓ_p norm choice irrelevant to optimal vector optimization rate","Hausdorff error decays at optimal rate for any ℓ_p","Convergence rate same for every ℓ_p norm in vector opt"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The Euclidean inner product on the objective space R^q supplies a quadratic bound on distances to supporting hyperplanes that norm equivalence preserves in its exponent when the metric is changed to any ℓ_p norm.","fun_headline_variants_meta":{"raw":{"variants":["Optimal rate achieved for all ℓ_p in vector optimization","ℓ_p norm choice irrelevant to optimal vector optimization rate","Hausdorff error decays at optimal rate for any ℓ_p","Convergence rate same for every ℓ_p norm in vector opt"]},"model":"grok-4.3","cost_usd":0.012583,"raw_usage":{"total_tokens":5405,"prompt_tokens":690,"num_sources_used":0,"completion_tokens":65,"cost_in_usd_ticks":125828000,"prompt_tokens_details":{"text_tokens":690,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":4650,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":690,"tokens_out":65,"duration_ms":76277,"temperature":1.0,"reasoning_tokens":4650,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-19T16:35:50.784045+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Run the algorithm on a convex vector problem whose exact Pareto set is known, measure the Hausdorff error after successive iterations with an ℓ_p norm where p is not 2, and check whether the observed decay is strictly slower than k to the power 2/(1-q).","supporting_citations":[],"review_version":2}