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Convergence Rates for $\ell_p$ Norm Minimization in Convex Vector Optimization

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Pith's one-line read Any ℓ_p norm in norm-minimization algorithms for convex vector optimization achieves the optimal convergence rate O(k^{2/(1-q)}).

desk verdict 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. read the letter →

arxiv 2605.14324 v2 pith:5MGIFMV4 submitted 2026-05-14 math.OC cs.NAmath.NA

Mohammed Alshahrani This is my paper · ORCID
classification math.OCcs.NAmath.NA
keywords convexvectoroptimizationouterapproximationconvergenceratesHausdorffdistanceℓ_pnormsnormminimizationmultiobjective
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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).

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 3 minor

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.

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.

minor comments (3)
  1. §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.
  2. 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.
  3. 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.

Simulated Author's Rebuttal

0 responses · 0 unresolved

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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; derivation is self-contained

full rationale

The paper derives the claimed Hausdorff rate O(k^{2/(1-q)}) via an explicit Euclidean intermediary that produces a quadratic hyperplane-distance bound using only the ambient inner-product structure on R^q, followed by standard norm equivalence to transfer the bound to any ℓ_p metric. Norm equivalence affects only the multiplicative constant (depending on q and p) and leaves the exponent unchanged. The manuscript supplies the explicit construction of the intermediary bound, confirms the resulting rate analytically, and validates it numerically. No step reduces the target rate to a fitted parameter, a self-referential definition, or a load-bearing self-citation whose content is unverified outside the present work. The argument is therefore independent of the inputs it seeks to bound.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The proof rests on standard finite-dimensional geometry and norm equivalence; no free parameters or new entities are introduced.

assumptions (1)
  • standard math All norms on R^q are equivalent, with constants depending only on dimension q and the choice of norms.
    Invoked to transfer the Euclidean quadratic bound to any ℓ_p metric without changing the exponent.

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Cite this review

Pith. "Pith review of Convergence Rates for $\ell_p$ Norm Minimization in Convex Vector Optimization." pith.science (2026). https://pith.science/paper/5MGIFMV4

@misc{pith2026260514324,
  author       = {Pith},
  title        = {Pith review of: Convergence Rates for $\ell_p$ Norm Minimization in Convex Vector Optimization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5MGIFMV4}},
  note         = {Machine review of arXiv:2605.14324}
}
abstract

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 number of objectives and $k$ is the iteration count. 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 $\ell_p$ smoothness limitation; norm equivalence then converts this to any $\ell_p$ metric at the cost of only a dimension-dependent constant, not a loss of exponent. Numerical experiments confirm the $p$-independent rate predicted by the theory.

Figures

Figures reproduced from arXiv: 2605.14324 by the authors.

Figure 1
Figure 1. Convergence of the Hausdorff error δH(Pk, A; ∥ · ∥p) for Example 1 with q = 3 and p ∈ {1.25, 1.5, 2, 3, 4, 8}. Solid lines show the monotone envelope; dashed lines show fitted power-law models in log-log coordinates. All curves decay at approximately the same rate, confirming c(p) = 2. For Example 1 with q = 2, [PITH_FULL_IMAGE:figures/full_fig_p016_1.png] view at source ↗
Figure 2
Figure 2. Convergence of the Hausdorff error δH(Pk, A; ∥ · ∥p) for Example 1 with q = 2 and p ∈ {1.25, 1.5, 2, 3, 4, 8}. The convergence tolerance is ε = 10−4 . For the rotated ellipse with q = 2, [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗
Figure 3
Figure 3. Convergence of the Hausdorff error δH(Pk, A; ∥ · ∥p) for Example 2 with q = 3, p ∈ {1.25, 1.5, 2, 3, 4, 8}, and ε = 0.05. 6.4 Discussion The experiments support the main theoretical result. Across all four experimental configurations (Example 1 with q = 2 and q = 3, the rotated ellipse with q = 2, and Example 2 with q = 3) and all six values of p, the fitted exponent ˆc(p) shows no systematic dependence on p. Within… view at source ↗

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Cited by 1 Pith paper

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  1. On Parallel and Batch-Cutting Strategies for Norm-Minimization-Based Convex Vector Optimization

    math.OC 2026-06 unverdicted novelty 4.0 of 10

    Introduces parallel subproblem evaluation and batch addition of up to K cuts per iteration for a convex vector optimization algorithm, proves the batch variant preserves the O(k^{2/(1-q)}) convergence rate, and report...

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