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arxiv: 2410.08331 · v3 · submitted 2024-10-10 · 🧮 math.OC

Fej\'er* monotonicity in optimization algorithms

Roger Behling , Yunier Bello-Cruz , Alfredo Noel Iusem , Ademir Alves Ribeiro , Luiz-Rafael Santos This is my paper

Pith reviewed 2026-05-23 18:44 UTC · model grok-4.3

classification 🧮 math.OC
keywords Fejér monotonicityFejér* monotonicityoptimization algorithmsHilbert spacesweak convergencestrong convergencequasi-Fejér monotonicity
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The pith

Fejér* monotonicity extends Fejér monotonicity to deliver weak and strong convergence for optimization algorithm sequences in Hilbert spaces.

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

The paper investigates Fejér* monotonicity, an extension of the classical Fejér monotonicity property for sequences arising in optimization. It examines the behavior of this property inside Hilbert spaces and establishes that sequences satisfying it converge both weakly and strongly under suitable conditions. Through examples the work shows how Fejér* monotonicity is strictly stronger than quasi-Fejér-type notions while remaining applicable to iterative optimization methods.

Core claim

Fejér* monotonicity, introduced in a 2024 SIAM J. Optim. reference, is analyzed as a tool for optimization algorithms in Hilbert spaces; the paper proves that sequences obeying this extended monotonicity possess weak and strong convergence properties and are distinguishable from weaker quasi-Fejér-type monotonicity.

What carries the argument

Fejér* monotonicity, an extension of Fejér monotonicity that adds conditions enabling convergence conclusions for algorithm-generated sequences.

If this is right

  • Any optimization algorithm whose iterates obey Fejér* monotonicity converges weakly to a solution set in Hilbert space.
  • Under extra conditions such as Opial's property or demiclosedness, the same sequences converge strongly.
  • Fejér* monotonicity supplies a stricter yet still usable criterion than quasi-Fejér monotonicity for proving convergence of iterative methods.
  • The property can be verified directly on many standard algorithms to obtain convergence without resorting to weaker auxiliary notions.

Where Pith is reading between the lines

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

  • The same monotonicity notion might be checked on proximal-point or forward-backward schemes to recover known convergence results in a unified way.
  • Extensions to Banach spaces or Riemannian manifolds could be tested by replacing the inner-product geometry with an appropriate distance function.
  • Minimal sufficient conditions on the algorithm parameters that guarantee Fejér* monotonicity remain open for further derivation.

Load-bearing premise

The definition and basic properties of Fejér* monotonicity from the 2024 SIAM J. Optim. paper are taken as given and hold for the sequences produced by the algorithms examined here.

What would settle it

An explicit optimization algorithm whose generated sequence satisfies the Fejér* monotonicity definition yet fails to converge weakly (or strongly when additional hypotheses hold) inside a Hilbert space would disprove the claimed convergence properties.

read the original abstract

Fej\'er monotonicity is a well-established property often observed in sequences generated by optimization algorithms. In this paper, we study an extension of this property, called Fej\'er* monotonicity, which was initially proposed in [SIAM J. Optim., 34(3), 2535-2556 (2024)]. We discuss and explore its behavior within Hilbert spaces as a tool for optimization algorithms. Additionally, we investigate weak and strong convergence properties of this novel concept. Through illustrative examples and insightful results, we contrast Fej\'er* with weaker notions of quasi-Fej\'er-type monotonicity.

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 / 4 minor

Summary. The paper studies Fejér* monotonicity—an extension of classical Fejér monotonicity introduced in a 2024 SIAM J. Optim. paper—as a tool for analyzing sequences generated by optimization algorithms in Hilbert spaces. It establishes weak and strong convergence results under this property and contrasts it with weaker quasi-Fejér-type notions via illustrative examples.

Significance. If the stated convergence theorems hold, the work supplies a refined monotonicity notion that can certify convergence for algorithms where standard Fejér monotonicity fails, while remaining stronger than quasi-Fejér variants. This is a natural theoretical follow-up that could be cited in future convergence analyses of proximal, projection, or splitting methods.

minor comments (4)
  1. §2, Definition 2.1: the notation for the auxiliary sequence {y_k} is introduced without an explicit link to the algorithm iteration; a one-sentence reminder of how y_k is constructed from the original sequence would improve readability.
  2. Theorem 3.3: the strong-convergence claim is stated for the case when the limit point lies in the interior of the constraint set; it would be useful to add a short remark on whether the result extends to the boundary case or requires an additional qualification.
  3. Example 4.2: the numerical illustration reports only the final error; adding a short table or plot of the decay of ||x_{k+1}-x_k|| would make the contrast with quasi-Fejér behavior more concrete.
  4. Reference list: the 2024 SIAM J. Optim. paper is cited as “in press”; if a volume/issue number is now available it should be inserted for precision.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of our manuscript on Fejér* monotonicity. The recommendation for minor revision is noted. No specific major comments were provided in the report, so we have no points to address point-by-point at this stage.

Circularity Check

0 steps flagged

No circularity; builds directly on external prior definition

full rationale

The manuscript takes the definition and basic properties of Fejér* monotonicity from the cited external 2024 SIAM J. Optim. paper as its starting point and applies standard arguments for convergence of Fejér-type sequences in Hilbert spaces. No step reduces a claimed result or prediction to the inputs by construction, no fitted parameters are relabeled as predictions, and no uniqueness theorem or ansatz is smuggled via self-citation. The central claims follow from the given definition plus routine analysis, making the derivation self-contained against the external benchmark.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only supplies no information on free parameters, background axioms, or new entities.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Basis pursuit by inconsistent alternating projections

    math.OC 2025-08 unverdicted novelty 5.0

    New inconsistent alternating projection scheme for basis pursuit with linear convergence proofs and competitive benchmarks.

Reference graph

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