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arxiv: 2605.14324 · v1 · submitted 2026-05-14 · 🧮 math.OC · cs.NA· math.NA

Recognition: 2 theorem links

· Lean Theorem

Convergence Rates for ell_p Norm Minimization in Convex Vector Optimization

Mohammed Alshahrani
Authors on Pith no claims yet

Pith reviewed 2026-05-15 02:34 UTC · model grok-4.3

classification 🧮 math.OC cs.NAmath.NA
keywords convex vector optimizationouter approximationconvergence ratesHausdorff distanceℓ_p normsmultiobjective optimizationscalarization
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The pith

The Hausdorff approximation error converges at rate O(k^{2/(1-q)}) for every ℓ_p norm in convex vector optimization.

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

The paper establishes that outer-approximation algorithms for convex vector optimization problems attain the same optimal convergence rate when the scalarizing norm is any ℓ_p norm with p between 1 and infinity. Earlier direct analysis only recovered a weaker rate that depended on p and became slower for p less than 2. By inserting an auxiliary Euclidean calculation that uses the standard inner product on the objective space R^q to bound hyperplane distances quadratically, then transferring the result via norm equivalence, the authors keep the exponent independent of p. This shows that the choice of norm can be made for numerical convenience without any loss in theoretical speed. Experiments on test problems confirm that the predicted rate appears in practice for both p less than 2 and p greater than 2.

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.

What carries the argument

Euclidean intermediary technique that produces a quadratic hyperplane-distance bound in R^q before norm-equivalence transfer to the chosen ℓ_p metric.

If this is right

  • The convergence exponent is independent of p, so any ℓ_p norm can be used without degrading the theoretical rate.
  • The same O(k^{2/(1-q)}) bound applies uniformly across the entire family of ℓ_p norms.
  • Only a dimension-dependent constant factor is introduced when switching from the Euclidean bound to a general ℓ_p metric.
  • Numerical behavior on test problems matches the p-independent rate predicted by the analysis.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same intermediary technique may apply to other norms that are equivalent to the Euclidean norm on finite-dimensional spaces.
  • In infinite-dimensional settings the quadratic bound would likely fail, requiring a different argument.
  • Algorithm designers can now select the norm primarily for conditioning or computational cost rather than convergence speed.

Load-bearing premise

The objective space must be finite-dimensional Euclidean space so that the inner-product structure supplies the quadratic distance bound.

What would settle it

A concrete convex vector optimization instance in R^q with q objectives where, for some p ≠ 2, the observed Hausdorff error after k iterations decays only like k to a power strictly smaller than 2/(1-q).

Figures

Figures reproduced from arXiv: 2605.14324 by Mohammed Alshahrani.

Figure 1
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. 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. 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… view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript proves that outer-approximation algorithms for convex vector optimization that minimize an ℓ_p-norm scalarization (p ∈ (1,∞)) achieve the Hausdorff rate δ_H(P_k,A) = O(k^{2/(1-q)}) for any fixed p, matching the Euclidean optimum. The argument proceeds by constructing a Euclidean intermediary that yields a quadratic bound on hyperplane distance from the inner-product geometry of R^q, then invoking norm equivalence to transfer the rate to the target ℓ_p metric at the cost of a q-dependent multiplicative constant only.

Significance. If the central derivation holds, the result is significant: it closes the open question of p-dependence for these rates, shows that the optimal exponent is preserved under norm equivalence when dimension q is fixed, and supplies a reusable Euclidean-intermediary technique that bypasses the modulus-of-smoothness limitation for 1 < p < 2. The approach is internally consistent within the finite-dimensional setting stated in the paper.

minor comments (2)
  1. [§3] §3 (Euclidean-intermediary construction): the explicit dependence of the norm-equivalence constant on both q and p should be stated once, even if only to confirm that no k-dependence enters.
  2. [Numerical experiments] Numerical section: the test problems, objective dimensions q, and range of p values used in the experiments should be listed explicitly so that the claimed p-independence can be verified by the reader.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and the positive recommendation to accept the manuscript. The provided summary accurately captures the central result: that the Hausdorff convergence rate O(k^{2/(1-q)}) holds independently of the choice of p in (1,∞) via the Euclidean-intermediary argument.

Circularity Check

0 steps flagged

No significant circularity; derivation uses standard finite-dimensional geometry

full rationale

The central result establishes the Hausdorff rate O(k^{2/(1-q)}) for any ell_p via an explicit Euclidean intermediary that obtains the quadratic hyperplane-distance bound directly from the inner-product geometry of R^q, followed by norm-equivalence transfer that multiplies only by a q-dependent constant. Because q is fixed and the exponent is preserved, the argument does not reduce any prediction to a fitted input, self-definition, or load-bearing self-citation. The proof chain is self-contained against external benchmarks (standard norm inequalities and Euclidean projection properties) and contains no steps that equate the claimed output to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim depends on the standard assumption that the vector optimization problem is convex and that the space is finite-dimensional Euclidean space allowing norm equivalence.

axioms (2)
  • domain assumption The feasible set in vector optimization is convex
    Standard assumption for convex vector optimization as stated in the problem setup.
  • standard math All norms on finite-dimensional R^q are equivalent
    Used to convert the Euclidean bound to any ℓ_p metric without losing the exponent.

pith-pipeline@v0.9.0 · 5518 in / 1206 out tokens · 30205 ms · 2026-05-15T02:34:52.741938+00:00 · methodology

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