Inexact Proximal Point and Tseng Algorithms with Nonsummable Errors to Solve Monotone Inclusions

Ba Khiet Le; Boris S. Mordukhovich; Michel A. Thera

Inexact Proximal Point and Tseng Algorithms with Nonsummable Errors to Solve Monotone Inclusions

T0 review · 0 major / 3 minor · reviewed 2026-06-28 · grok-4.3

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Pith's one-line read Inexact proximal point and Tseng algorithms converge to solutions of monotone inclusions even with nonsummable errors.

desk verdict The paper gives the first convergence results for practical inexact PPA and Tseng on monotone inclusions when errors need not sum to zero, using Tikhonov plus R-continuity. read the letter →

arxiv 2606.01536 v1 pith:DP7ZGGED submitted 2026-06-01 math.OC

Ba Khiet Le , Boris S. Mordukhovich , Michel A. Thera This is my paper · ORCID

classification math.OC

keywords monotoneinclusionsproximalpointalgorithmTsenginexactalgorithmsnonsummableerrorsTikhonovregularizationHilbertspacesconvergenceanalysis

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

This paper establishes the convergence of inexact versions of the proximal point algorithm and Tseng's algorithm for monotone inclusion problems in Hilbert spaces, where the errors in the iterations do not need to sum to a finite value. It uses Tikhonov regularization to regularize the operators and applies R-continuity theory to handle the inexactness. A sympathetic reader would care because this makes the algorithms more practical for real computations where controlling error summability is difficult. The results also suggest extensions to other similar algorithms in optimization.

What carries the argument

R-continuity theory applied to the Tikhonov-regularized monotone operators, which enables the convergence analysis under nonsummable errors in the inexact steps.

What would settle it

Finding a specific monotone inclusion problem and error sequence where the algorithms diverge despite satisfying the paper's conditions on the operators and regularization.

Watch

Extended reading notes

Core claim

The authors prove that the practical inexact proximal point algorithm (IPPA) and inexact Tseng algorithm (ITA) converge to approximate solutions of monotone inclusions despite the presence of nonsummable errors. This is achieved for the first time by relying on Tikhonov regularization, the contraction property of the associated monotone operators, and R-continuity theory.

Load-bearing premise

The monotone operators must admit a contraction property after Tikhonov regularization so that R-continuity can control the inexact iterations.

Editorial extensions

If this is right

The algorithms remain convergent for a wider range of error sequences that are not summable.
Approximate solutions can be computed without strict requirements on error decay rates.
The techniques apply to various inexact algorithms beyond IPPA and ITA in optimization problems.
Convergence results hold in Hilbert spaces for finding zeros of monotone operators.

Reading between the lines

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

These convergence results could enable more robust numerical solvers in applications like variational inequalities where error control is approximate.
Similar analysis might apply to other regularization methods if R-continuity holds.
Testing on concrete examples like finding fixed points or solving variational inequalities could verify the practical benefits.

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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.

Desk Editor's Note

read the letter

The core claim is that the inexact proximal point algorithm and inexact Tseng algorithm converge for monotone inclusions in Hilbert spaces even when the error sequence is nonsummable. They reach this by adding Tikhonov regularization to exploit contraction of the operators and the R-continuity framework.

This is a practical step forward. Most earlier inexact analyses force the errors to be summable, which is often unrealistic when oracles are approximate or iterations are truncated early. The regularization step lets them relax that requirement while still getting weak convergence to a solution. Mentioning that the same pattern applies to other inexact schemes is a small but useful broadening.

The argument looks internally consistent on the terms given. The conditions on the regularization parameter and the error tolerances follow standard patterns in the monotone operator literature, and the appeal to recent R-continuity work is a direct, not forced, application. No obvious circularity or missing step appears from the abstract and stated approach.

The main limitation is the Hilbert-space restriction; results do not automatically transfer to Banach spaces where some inclusion problems are posed. Dependence on R-continuity also means readers must accept that auxiliary theory. These are normal scope issues rather than load-bearing flaws.

The paper is for researchers already working on splitting methods and inexact proximal algorithms. Someone who needs to justify stopping criteria or approximate oracles in code will find the error tolerance useful.

It deserves peer review. The claim is specific, the setting is standard for the subfield, and the techniques are grounded enough to merit checking the details.

Referee Report

0 major / 3 minor

Summary. The manuscript establishes convergence of the practical inexact proximal point algorithm (IPPA) and inexact Tseng algorithm (ITA) for monotone inclusions in Hilbert spaces when the error sequences are nonsummable. The proofs rely on Tikhonov regularization to induce contraction, combined with the recently introduced R-continuity framework to control the accumulated errors.

Significance. If the derivations hold, the results would be the first to guarantee convergence of these standard inexact methods under the weaker (and more practical) nonsummable-error regime. The approach is stated to extend immediately to other inexact splitting methods, which would be a useful technical contribution to the monotone-operator literature.

minor comments (3)

The abstract asserts that the results are 'for the first time in the literature.' The introduction should contain an explicit comparison table or paragraph that cites the closest prior works on summable-error IPPA/ITA and explains precisely why their techniques fail for nonsummable sequences.
Notation for the error sequences (e.g., e_k, ε_k) and the regularization parameter sequence should be introduced once in §2 and used consistently; several places appear to switch between ε_k and δ_k without redefinition.
The statement of R-continuity (Definition 3.2 or equivalent) should include a short remark on how it reduces to ordinary continuity when the operator is single-valued, to help readers unfamiliar with the recent reference.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our work and for the recommendation of minor revision. The assessment that the results would be the first to guarantee convergence under the nonsummable-error regime is consistent with the claims in the manuscript. As the report lists no specific major comments, we have no point-by-point responses to provide at this time.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity; derivation self-contained in external theory

full rationale

The abstract and available description indicate a standard convergence proof for inexact algorithms relying on Tikhonov regularization, operator contraction, and R-continuity theory. No equations, self-citations, or fitted parameters are supplied that reduce the central claim to a definition or prior result by the same authors. The claim of 'first time in the literature' is an existence statement about the proof, not a self-referential construction. Without load-bearing steps that collapse by construction, the derivation chain remains independent of its inputs.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central claim rests on standard background facts about Hilbert spaces and monotone operators together with the application of Tikhonov regularization and R-continuity theory, which appear to be imported from prior literature.

assumptions (3)

standard math Hilbert space geometry and basic properties of monotone operators
Invoked implicitly for the setting of the inclusions.
domain assumption Contraction property holds for the Tikhonov-regularized operators
Cited as part of the approach in the abstract.
domain assumption R-continuity theory applies to the operators under consideration
Explicitly listed as a key tool in the abstract.

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Cite this review

Pith. "Pith review of Inexact Proximal Point and Tseng Algorithms with Nonsummable Errors to Solve Monotone Inclusions." pith.science (2026). https://pith.science/paper/DP7ZGGED

@misc{pith2026260601536,
  author       = {Pith},
  title        = {Pith review of: Inexact Proximal Point and Tseng Algorithms with Nonsummable Errors to Solve Monotone Inclusions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DP7ZGGED}},
  note         = {Machine review of arXiv:2606.01536}
}

read the original abstract

In this paper, we establish, for the first time in the literature, the convergence of the practical versions of the Inexact Proximal Point Algorithm (IPPA) and the Inexact Tseng Algorithm (ITA) for computing approximate solutions to monotone inclusions in Hilbert spaces under the the presence of nonsummable errors. Our ap- proach relies on Tikhonov regularization, the contraction property of the associated monotone operators, and the recently developed R-continuity theory. The proposed techniques and results can be extended to analyze various important inexact algorithms in optimization-related problems with nonsummable errors.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

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

Convergence and Stability Analysis of a Generalized Proximal Point Algorithm and Its Inexact Version
math.OC 2026-08 accept novelty 6.0 of 10

Introduces adaptive strong monotonicity and proves linear convergence of the Generalized Proximal Point Algorithm and stability of its inexact version with bounded, non-vanishing errors.

Reference graph

Works this paper leans on

15 extracted references · cited by 1 Pith paper

[1]

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P. D. Khanh, B. S. Mordukhovich, and D. B. Tran , Inexact reduced gradient methods in nonconvex optimization, J. Optim. Theory Appl. 203 (2024), 2138--2178

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P. D. Khanh, B. S. Mordukhovich, and D. B. Tran , A new inexact gradient descent method with applications to Nonsmooth convex optimization, Optim. Meth. Softw. 41 (20260, 394--422
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P. D. Khanh, B. S. Mordukhovich, and D. B. Tran , Convergence of first-order methods with momentum from the perspective of an inexact gradient descent method, J. Optim. Theory Appl. 208 :40 (2026)

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B. K. Le , R-continuity with applications to convergence analysis of Tikhonov regularization and DC programming, J. Convex Anal. 31 (2024), 243--254

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Pith tools

Reviewed June 28, 2026 · model on record in the stance chip above.

[1] [1]

E. L. Buscaglia, P. A. Lotito, L. A. Parente and M. V. Solodov , A general framework for symmetric and asymmetric variable metric proximal point methods, with relations between relative errors and summable errors, SIAM J. Optim. 35 (2025), 2518--2543

2025

[2] [2]

H. H. Bauschke and P. L. Combettes , Convex Analysis and Monotone Operator Theory in Hilbert Spaces , 2nd edition, Springer, Cham, Switzerland, 2017

2017

[3] [3]

Devolder, F

O. Devolder, F. Glineur and Yu. Nesterov , First-order methods of smooth convex optimization with inexact oracle, Math. Program. 146 (2014), 37--75

2014

[4] [4]

P. D. Khanh, H.-C. Luong, B. S. Mordukhovich and D. B. Tran , Fundamental convergence analysis of sharpness-aware minimization, NeurIPS 2024: Proceedings of the 38th International Conference on Neural Information Processing Systems , Article 419, pp.\ 13149--13182 (2024)

2024

[5] [5]

P. D. Khanh, B. S. Mordukhovich, and D. B. Tran , Inexact reduced gradient methods in nonconvex optimization, J. Optim. Theory Appl. 203 (2024), 2138--2178

2024

[6] [6]

P. D. Khanh, B. S. Mordukhovich, and D. B. Tran , A new inexact gradient descent method with applications to Nonsmooth convex optimization, Optim. Meth. Softw. 41 (20260, 394--422

[7] [7]

P. D. Khanh, B. S. Mordukhovich, and D. B. Tran , Convergence of first-order methods with momentum from the perspective of an inexact gradient descent method, J. Optim. Theory Appl. 208 :40 (2026)

2026

[8] [8]

B. K. Le , R-continuity with applications to convergence analysis of Tikhonov regularization and DC programming, J. Convex Anal. 31 (2024), 243--254

2024