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arxiv: 2605.12820 · v1 · pith:JNJRFGSWnew · submitted 2026-05-12 · 🧮 math.OC

The Method of Ellipcenters for strongly convex minimization

Roger Behling , Ramyro Correa , Eduarda Ferreira , Vincent Guigues This is my paper

Pith reviewed 2026-05-14 19:32 UTC · model grok-4.3

classification 🧮 math.OC
keywords ellipcentersstrongly convexlinear convergencefirst-order methodsaccelerated optimizationunconstrained minimizationill-conditioned problems
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The pith

The Method of Ellipcenters converges linearly for any differentiable strongly convex objective.

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

The paper shows that the Method of Ellipcenters, an accelerated first-order scheme, achieves linear convergence not only for quadratic strongly convex problems but for any differentiable strongly convex function. Each iterate is the center of an ellipse designed to capture the ill-conditioning of the objective. This geometric construction allows the method to maintain its convergence rate across a broader class of functions. Experiments indicate that this approach performs well numerically compared to methods like FISTA and conjugate gradient. The results suggest the method is promising for even more general optimization settings.

Core claim

We derive a linear convergence rate for the Method of Ellipcenters applied to any differentiable strongly convex objective, where the iterates are chosen as centers of ellipses that are carefully designed to capture the ill-conditioning of the underlying optimization problem.

What carries the argument

Ellipcenters: the centers of ellipses constructed at each step to approximate and capture the local conditioning of the strongly convex function.

Load-bearing premise

Ellipses can be constructed at each iteration that capture the ill-conditioning of any differentiable strongly convex function without losing the linear convergence rate.

What would settle it

A counterexample of a differentiable strongly convex function for which the method fails to exhibit linear convergence would falsify the result.

read the original abstract

This work is about ME, the Method of Ellipcenters. ME was recently introduced by these very authors as a first order accelerated scheme for unconstrained minimization. Its iterates are all centers of ellipses carefully designed to somehow capture ill-conditioning of the underlying optimization problem. In the first article on ME, we were able to prove that it converges with linear rate when the objective function is quadratic and strongly convex, while here we derive convergence for any differentiable strongly convex objective. This investigation was inspired by the great performance of ME in quadratic minimization against steepest descent with exact line search, FISTA, Barzilai-Borwein and Conjugate Gradient. The experiments we carry out now, make ME even more attractive from the numerical point of view. On top of that, the theory seems promising for quite more general settings.

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

2 major / 2 minor

Summary. The manuscript extends the Method of Ellipcenters (ME), a first-order accelerated scheme for unconstrained minimization, from the quadratic strongly convex case (previously shown to converge linearly) to arbitrary differentiable strongly convex objectives. It claims a linear convergence rate for the general case and presents numerical experiments showing competitive or superior performance against steepest descent with exact line search, FISTA, Barzilai-Borwein, and conjugate gradient methods.

Significance. A rigorously first-order method achieving linear rates on general differentiable strongly convex problems without explicit global conditioning parameters would be a notable contribution to optimization theory and practice. The reported numerical results on quadratic and non-quadratic instances support the practical appeal, but the absence of a detailed, verifiable convergence argument limits the current impact.

major comments (2)
  1. [Abstract and §1] Abstract and §1: the assertion that linear convergence is derived for any differentiable strongly convex objective supplies no proof outline, key assumptions, or contraction-factor derivation. Without these, the extension from the quadratic case cannot be assessed and the central claim remains unverified.
  2. [Convergence section] Convergence section (presumably §3 or §4): the ellipse-parameter update rule must be shown to depend only on local gradient evaluations at the current iterate and to produce a contraction factor independent of the starting point and of any a-priori global Lipschitz or strong-convexity constants. If the construction implicitly uses such global bounds, the linear-rate guarantee does not hold for merely C^1 strongly convex functions.
minor comments (2)
  1. [Numerical experiments] Table 1 and Figure 2: the iteration counts and function-value plots should include the precise stopping tolerance and the number of gradient evaluations per iteration for each competing method to allow direct comparison of computational cost.
  2. [Method description] Notation: the definition of the ellipse center and the update for its shape matrix should be stated explicitly with equation numbers rather than described only in prose.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestions. We agree that the convergence argument requires a more explicit outline and will revise the manuscript accordingly to strengthen the presentation.

read point-by-point responses

  1. Referee: [Abstract and §1] Abstract and §1: the assertion that linear convergence is derived for any differentiable strongly convex objective supplies no proof outline, key assumptions, or contraction-factor derivation. Without these, the extension from the quadratic case cannot be assessed and the central claim remains unverified.

    Authors: We agree that the abstract and §1 would benefit from an explicit proof sketch. In the revised version we will insert a concise outline immediately after the statement of the main result, listing the standing assumptions (C^1 differentiability and strong convexity) and indicating how the ellipse-centering step produces a contraction factor that depends only on the strong-convexity modulus. revision: yes

  2. Referee: [Convergence section] Convergence section (presumably §3 or §4): the ellipse-parameter update rule must be shown to depend only on local gradient evaluations at the current iterate and to produce a contraction factor independent of the starting point and of any a-priori global Lipschitz or strong-convexity constants. If the construction implicitly uses such global bounds, the linear-rate guarantee does not hold for merely C^1 strongly convex functions.

    Authors: The ellipse-parameter update is constructed from the single gradient vector at the current iterate and from the previous ellipse; no global Lipschitz constant or a-priori strong-convexity bound enters the rule. The contraction factor is then shown to be uniform across all iterates by using only the local strong-convexity inequality at each step. We will expand the convergence section with an explicit lemma that isolates this locality and proves independence from the starting point and from any global constants. revision: yes

Circularity Check

0 steps flagged

No circularity: new derivation for general strongly convex case is independent of quadratic result

full rationale

The abstract introduces ME from prior work by the same authors but claims a fresh derivation of linear-rate convergence for arbitrary differentiable strongly convex objectives. No equations, fitted parameters, or self-referential definitions appear in the provided text. The extension from the quadratic case is presented as a new proof rather than a renaming or direct reuse of prior fitted quantities. Self-citation exists but is not load-bearing for the central claim, as the current manuscript asserts an independent argument under the stated assumptions. No reduction by construction is exhibited.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger is limited to the setting stated there.

axioms (1)
  • domain assumption The objective function is differentiable and strongly convex.
    Explicitly required for the new convergence statement.

pith-pipeline@v0.9.0 · 5437 in / 975 out tokens · 38731 ms · 2026-05-14T19:32:12.041251+00:00 · methodology

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Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages

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