Convergence Rates of Tseng's Splitting Method and Its Acceleration Schemes for Monotone Inclusion Problem with a Sum of H\"older Continuous Operators

Yi Zhang

arxiv: 2606.22392 · v1 · pith:IQ4UKJ2Mnew · submitted 2026-06-21 · 🧮 math.OC

Convergence Rates of Tseng's Splitting Method and Its Acceleration Schemes for Monotone Inclusion Problem with a Sum of H\"older Continuous Operators

Yi Zhang This is my paper

Pith reviewed 2026-06-26 10:22 UTC · model grok-4.3

classification 🧮 math.OC

keywords Tseng's splitting methodHölder continuitymonotone inclusionconvergence ratesaccelerated methodsoperator splitting

0 comments

The pith

Tseng's splitting method and its accelerations converge at rates set by the Hölder exponent for monotone inclusions with summed Hölder continuous operators.

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

The paper derives convergence rates for Tseng's splitting method and two accelerated variants applied to monotone inclusion problems where the operators are Hölder continuous rather than Lipschitz. This removes a common restriction and shows that the rates depend explicitly on the Hölder exponent. A reader would care because many practical problems satisfy Hölder continuity but not the stronger Lipschitz condition. The work also includes numerical checks that the observed performance matches the derived rates.

Core claim

For the monotone inclusion problem formed as the sum of two monotone operators that are each Hölder continuous with exponent in (0,1], Tseng's splitting method converges at a rate governed by that exponent. The composite extra anchored gradient method and the symplectic composite extra gradient method improve the rate in a manner consistent with the same exponent.

What carries the argument

Tseng's splitting method extended to Hölder continuous monotone operators, which produces convergence rates controlled by the Hölder exponent.

Load-bearing premise

The two operators in the inclusion problem are monotone and Hölder continuous with some fixed exponent in (0,1].

What would settle it

A concrete monotone inclusion problem whose operators are Hölder continuous but for which Tseng's method or its accelerations fail to meet the predicted rate would disprove the claim.

read the original abstract

The monotone inclusion problem is fundamental in applied mathematics and is closely related to a wide range of practical applications. However, existing solution methods typically require the underlying operator to be Lipschitz continuous. Recently, H\"older continuity, a weaker condition than Lipschitz continuity, has proven useful in characterizing certain real-world problems. To bridge this gap, we investigate the convergence rates of the Tseng's splitting method (a fundamental algorithm for monotone inclusion problem) and its two accelerated variants, the composite extra anchored gradient method and the symplectic composite extra gradient method, under the H\"older continuity assumption. Our numerical experiments demonstrate that the numerical performance of these algorithms aligns with their respective theoretical convergence rates.

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.

Desk Editor's Note private letter to a colleague

The paper derives Hölder-dependent rates for Tseng's splitting and two accelerations, but the abstract leaves the maximal monotone operator unspecified so the method is not clearly defined.

read the letter

The main thing here is that the work extends rate analysis for Tseng's splitting method and its two accelerated schemes (composite extra anchored gradient and symplectic composite extra gradient) to the case of Hölder continuous operators instead of the standard Lipschitz assumption. The rates are stated to depend on the Hölder exponent, and the numerics are said to match those rates.

What the paper does reasonably is to target a weaker continuity condition that can cover more applied problems and to run experiments that check consistency with the claimed rates. That part is straightforward incremental work in the monotone inclusion setting.

The soft spot is the one flagged in the stress-test. Tseng's method requires one operator to be maximal monotone so the resolvent can be evaluated in the backward step, while the other operator is monotone and Hölder continuous for the forward step. The title and abstract describe only a sum of Hölder continuous operators without naming which one supports the resolvent. If the full paper does not make this split explicit and prove the rates under that structure, the claims do not hold because the algorithm itself is not well-defined. This is not a small wording issue; it affects whether the results apply at all.

The proofs themselves are not visible from the abstract, so the derivations cannot be checked for gaps or errors. No circularity is obvious from the framing, but that does not substitute for seeing the actual arguments.

This is for specialists working on convergence rates for operator splitting methods. A reader already following the Lipschitz-to-Hölder extensions in monotone operator theory would get some value from the specific rates and the numerics.

It deserves peer review because the topic is relevant and the Hölder extension is a real (if narrow) step forward, but only if the operator roles are clarified and the proofs hold up. I would send it out with a request to fix the setup description first.

Referee Report

1 major / 0 minor

Summary. The paper claims to derive convergence rates (determined by the Hölder exponent) for Tseng's splitting method and its two accelerated variants—the composite extra anchored gradient method and the symplectic composite extra gradient method—applied to the monotone inclusion problem 0 ∈ A(x) + B(x) where both operators are monotone and Hölder continuous (with exponent in (0,1]), and reports that numerical experiments align with the theoretical rates.

Significance. If the rates are correctly derived under the stated assumptions, the work would extend splitting-method theory from the standard Lipschitz setting to the weaker Hölder-continuous case, which is relevant for applications where Lipschitz continuity fails. The presence of numerical validation is a strength.

major comments (1)

[Problem statement / abstract and §1] Problem statement / abstract and §1: the setting is described only as a 'sum of Hölder Continuous Operators' without identifying which operator is maximal monotone. Tseng's method (and its accelerations) requires the resolvent (I + λA)^{-1} of one maximal monotone operator; without this explicit structural split the algorithm is not well-defined and the claimed rates cannot hold.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the thorough review and constructive comments. We address the major comment point by point below.

read point-by-point responses

Referee: [Problem statement / abstract and §1] Problem statement / abstract and §1: the setting is described only as a 'sum of Hölder Continuous Operators' without identifying which operator is maximal monotone. Tseng's method (and its accelerations) requires the resolvent (I + λA)^{-1} of one maximal monotone operator; without this explicit structural split the algorithm is not well-defined and the claimed rates cannot hold.

Authors: We agree with the referee that the problem statement requires clarification on the structural assumptions. In the manuscript, we consider the inclusion 0 ∈ A(x) + B(x), where A is maximal monotone and B is monotone and Hölder continuous. However, we acknowledge that this distinction was not explicitly highlighted in the abstract and introduction. We will revise the abstract, §1, and the problem formulation section to clearly specify that A is maximal monotone (to ensure the resolvent is defined) and B is monotone and Hölder continuous with exponent in (0,1]. This revision will make the setting unambiguous without changing the results or proofs. revision: yes

Circularity Check

0 steps flagged

No circularity: rates derived from stated monotonicity + Hölder assumptions

full rationale

The paper states the monotone inclusion problem with Hölder continuous operators as the setting and derives convergence rates for Tseng's method and accelerations from those assumptions. No self-definitional reduction, fitted-input-as-prediction, or load-bearing self-citation is present in the abstract or described claims. The derivation chain remains independent of the target rates; the result is not forced by definition or prior author work.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The analysis rests on standard assumptions from monotone operator theory together with the Hölder continuity condition; no free parameters or invented entities are indicated in the abstract.

axioms (1)

domain assumption The operators are monotone and Hölder continuous with some exponent alpha in (0,1].
This is the central modeling assumption stated in the abstract under which the convergence rates are investigated.

pith-pipeline@v0.9.1-grok · 5641 in / 1107 out tokens · 55449 ms · 2026-06-26T10:22:13.514196+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

48 extracted references · 18 canonical work pages

[1]

D. G. Anderson. Iterative procedures for nonlinear integ ral equations. Journal of the ACM (JACM) , 12(4): 547–560, 1965. doi: 10.1145/321296.321305

work page doi:10.1145/321296.321305 1965
[2]

A. S. Antipin. On a method for convex programs using a symme trical modiﬁcation of the Lagrange function. Economika i Matem. Metody , (6):1164–1173, 1976

1976
[3]

Ben-Tal, L

A. Ben-Tal, L. E. Ghaoui, and A. Nemirovski. Robust optimization, volume 28. Princeton university press, Princeton, 2009

2009
[4]

R. I. Bot ¸, E. R. Csetnek, and D.-K. Nguyen. Fast optimisti c gradient descent ascent (OGDA) method in continuous and discrete time. F oundations of Computational Mathematics , pages 1–60, 2023. doi: https: //doi.org/10.1007/s10208-023-09636-5

work page doi:10.1007/s10208-023-09636-5 2023
[5]

Cai, Q.-L

G. Cai, Q.-L. Dong, and Y . Peng. Strong convergence theore ms for solving variational inequality problems with pseudo-monotone and non-lipschitz operators. Journal of Optimization Theory and Applications, 188(2): 447–472, 2021. doi: 10.1007/s10957-020-01792-w

work page doi:10.1007/s10957-020-01792-w 2021
[6]

Y . Cai, A. Oikonomou, and W. Zheng. Accelerated algorithm s for constrained nonconvex-nonconcave min-max optimization and comonotone inclusion. In R. Salak hutdinov, Z. Kolter, K. Heller, A. Weller, N. Oliver, J. Scarlett, and F. Berkenkamp, editors, Proceedings of the 41st International Conference on Machine Learning , volume 235 of Proceedings of Machi...

2024
[7]

Chakraborty and A

A. Chakraborty and A. Nedi´ c. Popov mirror-prox method fo r variational inequalities, 2025. URL https://arxiv.org/abs/2507.23395

arXiv 2025
[8]

K. Chen, D. Sun, Y . Y uan, G. Zhang, and X. Zhao. HPR-LP: An im plementation of an HPR method for solving linear programming. Mathematical Programming Computation , pages 1–28, 2025. doi: 10.1007/ s12532-025-00292-0

2025
[9]

C. D. Dang and G. Lan. On the convergence properties of non- euclidean extragradient methods for variational inequalities with generalized monotone operators. Computational Optimization and applications , 60(2):277– 310, 2015. doi: 10.1007/s10589-014-9673-9

work page doi:10.1007/s10589-014-9673-9 2015
[10]

Daskalakis and I

C. Daskalakis and I. Panageas. The limit points of (optim istic) gradient descent in min-max optimization. In Proceedings of the 32nd International Conference on Neural Information Processing Systems, NIPS’18, page 9256–9266, 2018. URL https://dl.acm.org/doi/pdf/10.5555/3327546.3327597

work page doi:10.5555/3327546.3327597 2018
[11]

Daskalakis, A

C. Daskalakis, A. Ilyas, V . Syrgkanis, and H. Zeng. Train ing GANs with optimism. In International Conference on Learning Representations, 2018. URL https://openreview.net/forum?id=SJJySbbAZ

2018
[12]

Diakonikolas, C

J. Diakonikolas, C. Daskalakis, and M. I. Jordan. Efﬁcie nt methods for structured nonconvex-nonconcave min-max optimization. In International Conference on Artiﬁcial Intelligence and St atistics, pages 2746–2754. PMLR, 2021

2021
[13]

Y . Fan, Y . Li, and B. Chen. Weaker MVI condition: Extragra dient methods with multi-step exploration. In The Twelfth International Conference on Learning Represen tations, 2023

2023
[14]

Gorbunov, H

E. Gorbunov, H. Berard, G. Gidel, and N. Loizou. Stochast ic extragradient: General analysis and improved rates. In International Conference on Artiﬁcial Intelligence and St atistics, pages 7865–7901. PMLR, 2022

2022
[15]

B. Halpern. Fixed points of nonexpanding maps. Bulletin of the American Mathematical Society, 73:957–961,
[16]

26 YI ZHANG

URL https://api.semanticscholar.org/CorpusID:120539954. 26 YI ZHANG
[17]

M. Ito, Z. Lu, and C. He. A parameter-free conditional gra dient method for composite minimization under H¨ older condition. Journal of Machine Learning Research , 24(166):1–34, 2023. URL http://jmlr.org/papers/v24/22-0983.html

2023
[18]

A. N. Iusem, A. Jofr´ e, R. I. Oliveira, and P . Thompson. Ex tragradient method with variance reduction for stochastic variational inequalities. SIAM Journal on Optimization , 27(2):686–724, 2017. doi: 10.1137/ 15M1031953

2017
[19]

Kannan and U

A. Kannan and U. V . Shanbhag. Optimal stochastic extragr adient schemes for pseudomonotone stochastic variational inequality problems and their variants. Computational Optimization and Applications , 74(3):779– 820, 2019. doi: 10.1007/s10589-019-00120-x

work page doi:10.1007/s10589-019-00120-x 2019
[20]

Kinderlehrer and G

D. Kinderlehrer and G. Stampacchia. An introduction to variational inequalities and their applications. SIAM, 2000

2000
[21]

G. M. Korpelevich. The extragradient method for ﬁnding s addle points and other problems. Matecon, 12: 747–756, 1976

1976
[22]

Lee and D

S. Lee and D. Kim. Fast extra gradient methods for smooth s tructured nonconvex-nonconcave minimax problems. In A. Beygelzimer, Y . Dauphin, P . Liang, and J. W. V aughan, editors, Advances in Neural Information Processing Systems, 2021

2021
[23]

F. Lieder. On the convergence rate of the Halpern-iterat ion. Optimization letters, 15(2):405–418, 2021. doi: 10.1007/s11590-020-01617-9

work page doi:10.1007/s11590-020-01617-9 2021
[24]

Lu and J

H. Lu and J. Y ang. Restarted Halpern PDHG for linear progr amming, 2024. URL https://arxiv.org/abs/2407.16144

arXiv 2024
[25]

G. J. Minty. Monotone (nonlinear) operators in Hilbert s pace. Duke Mathematical Journal, 29(3):341 – 346,
[26]

URL https://doi.org/10.1215/S0012-7094-62-02933-2

doi: 10.1215/S0012-7094-62-02933-2. URL https://doi.org/10.1215/S0012-7094-62-02933-2

work page doi:10.1215/s0012-7094-62-02933-2
[27]

Mishchenko, D

K. Mishchenko, D. Kovalev, E. Shulgin, P . Richt´ arik, and Y . Malitsky. Revisiting stochastic extragradient. In International Conference on Artiﬁcial Intelligence and St atistics, pages 4573–4582. PMLR, 2020

2020
[28]

Nemirovski

A. Nemirovski. Prox-method with rate of convergence o(1 /t) for variational inequalities with lipschitz continuous monotone operators and smooth convex-concave s addle point problems. SIAM Journal on Optimization, 15(1):229–251, 2004. doi: 10.1137/S1052623403425629

work page doi:10.1137/s1052623403425629 2004
[29]

Nesterov

Y . Nesterov. A method of solving a convex programming pro blem with convergence rate o(1/k2). Doklady Akademii Nauk SSSR, 269(3):543, 1983

1983
[30]

Ouyang and Y

Y . Ouyang and Y . Xu. Lower complexity bounds of ﬁrst-orde r methods for convex-concave bilinear saddle- point problems. Mathematical Programming, 185(1):1–35, 2021. doi: 10.1007/s10107-019-01420-0

work page doi:10.1007/s10107-019-01420-0 2021
[31]

Pethick, P

T. Pethick, P . Latafat, P . Patrinos, O. Fercoq, and V . Cev her. Escaping limit cycles: Global convergence for constrained nonconvex-nonconcave minimax problems. I n International Conference on Learning Representations, 2022. URL https://openreview.net/forum?id=2 vhkAMARk

2022
[32]

L. D. Povov. A modiﬁcation of the Arrow-Hurwitz method of search for saddle points. Mat. Zametki, 28(5): 777–784, 1980

1980
[33]

Qi and H.-K

H. Qi and H.-K. Xu. Convergence of Halpern’s iteration me thod with applications in optimization. Numerical Functional Analysis and Optimization, 42(15):1839–1854, 2021. doi: https://doi.org/10.1080/ 01630563.2021. 2001826

arXiv 2021
[34]

X. Qu, W. Bian, and X. Chen. An extra gradient Anderson-ac celerated algorithm for pseudomonotone variational inequalities. Mathematics of Computation, 95(359):1361–1387, 2026. doi: 10.1090/mcom/4095

work page doi:10.1090/mcom/4095 2026
[35]

Sedlmayer, D.-K

M. Sedlmayer, D.-K. Nguyen, and R. I. Bot ¸. A fast optimis tic method for monotone variational inequalities. In Proceedings of the 40th International Conference on Machin e Learning, ICML’23, 2023

2023
[36]

Stonyakin, A

F. Stonyakin, A. Gasnikov, P . Dvurechensky, A. Titov, an d M. Alkousa. Generalized mirror prox algorithm for monotone variational inequalities: Universality and i nexact oracle. Journal of Optimization Theory and Applications, 194(3):988–1013, 2022. doi: 10.1007/s10957-022-02062- 7

work page doi:10.1007/s10957-022-02062- 2022
[37]

D. Sun, Y . Y uan, G. Zhang, and X. Zhao. Accelerating preco nditioned ADMM via degenerate proximal point mappings. SIAM Journal on Optimization, 35(2):1165–1193, 2025. doi: 10.1137/24M1650053

work page doi:10.1137/24m1650053 2025
[38]

J. E. Taylor and M. P . Bendsøe. An interpretation for min- max structural design problems including a method for relaxing constraints. International Journal of Solids and Structures , 20(4):301–314, 1984. doi: 10.1016/ ALGORITHMS FOR MONOTONE INCLUSION PROBLEM WITH SUMS OF H ¨OLDER OPERA TORS 27 0020-7683(84)90041-6

1984
[39]

Toyasaki, P

F. Toyasaki, P . Daniele, and T. Wakolbinger. A variation al inequality formulation of equilibrium models for end-of-life products with nonlinear constraints. European Journal of Operational Research, 236(1):340–350,
[40]

doi: 10.1016/j.ejor.2013.12.006

work page doi:10.1016/j.ejor.2013.12.006 2013
[41]

Tran-Dinh

Q. Tran-Dinh. Sublinear convergence rates of extragrad ient-type methods: A survey on classical and recent developments, 2023. URL https://arxiv.org/abs/2303.17192

arXiv 2023
[42]

Tran-Dinh

Q. Tran-Dinh. Extragradient-type methods with O(1/k) last-iterate convergence rates for co-hypomonotone inclusions. Journal of Global Optimization , 89(1):197–221, 2024. doi: 10.1007/s10898-023-01347-z

work page doi:10.1007/s10898-023-01347-z 2024
[43]

Tran-Dinh and N

Q. Tran-Dinh and N. Nguyen-Trung. Accelerated extragra dient-type methods – part 2: Generalization and sublinear convergence rates under co-hypomonotonicity, 2 025. URL https://arxiv.org/abs/2501.04585

arXiv
[44]

P . Tseng. A modiﬁed forward-backward splitting method f or maximal monotone mappings. SIAM Journal on Control and Optimization, 38(2):431–446, 2000. doi: 10.1137/S0363012998338806

work page doi:10.1137/s0363012998338806 2000
[45]

T. Ui. Bayesian Nash equilibrium and variational inequa lities. Journal of Mathematical Economics , 63: 139–146, 2016. ISSN 0304-4068. doi: 10.1016/j.jmateco.20 16.02.004

work page doi:10.1016/j.jmateco.20 2016
[46]

xiang Y uan and Y

Y . xiang Y uan and Y . Zhang. Symplectic extra-gradient ty pe method for solving general non-monotone inclusion problem, 2025. URL https://arxiv.org/abs/2406.10793

arXiv 2025
[47]

Y oon and E

T. Y oon and E. K. Ryu. Accelerated algorithms for smooth c onvex-concave minimax problems with O(1/k2) rate on squared gradient norm. In International Conference on Machine Learning , pages 12098–12109. PMLR, 2021

2021
[48]

Y uan and Y

Y .-x. Y uan and Y . Zhang. Symplectic discretization appr oach for developing new proximal point algorithm. Computational Optimization and Applications , 93(2):651–687, 2026. doi: 10.1007/s10589-025-00728-2

work page doi:10.1007/s10589-025-00728-2 2026

[1] [1]

D. G. Anderson. Iterative procedures for nonlinear integ ral equations. Journal of the ACM (JACM) , 12(4): 547–560, 1965. doi: 10.1145/321296.321305

work page doi:10.1145/321296.321305 1965

[2] [2]

A. S. Antipin. On a method for convex programs using a symme trical modiﬁcation of the Lagrange function. Economika i Matem. Metody , (6):1164–1173, 1976

1976

[3] [3]

Ben-Tal, L

A. Ben-Tal, L. E. Ghaoui, and A. Nemirovski. Robust optimization, volume 28. Princeton university press, Princeton, 2009

2009

[4] [4]

R. I. Bot ¸, E. R. Csetnek, and D.-K. Nguyen. Fast optimisti c gradient descent ascent (OGDA) method in continuous and discrete time. F oundations of Computational Mathematics , pages 1–60, 2023. doi: https: //doi.org/10.1007/s10208-023-09636-5

work page doi:10.1007/s10208-023-09636-5 2023

[5] [5]

Cai, Q.-L

G. Cai, Q.-L. Dong, and Y . Peng. Strong convergence theore ms for solving variational inequality problems with pseudo-monotone and non-lipschitz operators. Journal of Optimization Theory and Applications, 188(2): 447–472, 2021. doi: 10.1007/s10957-020-01792-w

work page doi:10.1007/s10957-020-01792-w 2021

[6] [6]

Y . Cai, A. Oikonomou, and W. Zheng. Accelerated algorithm s for constrained nonconvex-nonconcave min-max optimization and comonotone inclusion. In R. Salak hutdinov, Z. Kolter, K. Heller, A. Weller, N. Oliver, J. Scarlett, and F. Berkenkamp, editors, Proceedings of the 41st International Conference on Machine Learning , volume 235 of Proceedings of Machi...

2024

[7] [7]

Chakraborty and A

A. Chakraborty and A. Nedi´ c. Popov mirror-prox method fo r variational inequalities, 2025. URL https://arxiv.org/abs/2507.23395

arXiv 2025

[8] [8]

K. Chen, D. Sun, Y . Y uan, G. Zhang, and X. Zhao. HPR-LP: An im plementation of an HPR method for solving linear programming. Mathematical Programming Computation , pages 1–28, 2025. doi: 10.1007/ s12532-025-00292-0

2025

[9] [9]

C. D. Dang and G. Lan. On the convergence properties of non- euclidean extragradient methods for variational inequalities with generalized monotone operators. Computational Optimization and applications , 60(2):277– 310, 2015. doi: 10.1007/s10589-014-9673-9

work page doi:10.1007/s10589-014-9673-9 2015

[10] [10]

Daskalakis and I

C. Daskalakis and I. Panageas. The limit points of (optim istic) gradient descent in min-max optimization. In Proceedings of the 32nd International Conference on Neural Information Processing Systems, NIPS’18, page 9256–9266, 2018. URL https://dl.acm.org/doi/pdf/10.5555/3327546.3327597

work page doi:10.5555/3327546.3327597 2018

[11] [11]

Daskalakis, A

C. Daskalakis, A. Ilyas, V . Syrgkanis, and H. Zeng. Train ing GANs with optimism. In International Conference on Learning Representations, 2018. URL https://openreview.net/forum?id=SJJySbbAZ

2018

[12] [12]

Diakonikolas, C

J. Diakonikolas, C. Daskalakis, and M. I. Jordan. Efﬁcie nt methods for structured nonconvex-nonconcave min-max optimization. In International Conference on Artiﬁcial Intelligence and St atistics, pages 2746–2754. PMLR, 2021

2021

[13] [13]

Y . Fan, Y . Li, and B. Chen. Weaker MVI condition: Extragra dient methods with multi-step exploration. In The Twelfth International Conference on Learning Represen tations, 2023

2023

[14] [14]

Gorbunov, H

E. Gorbunov, H. Berard, G. Gidel, and N. Loizou. Stochast ic extragradient: General analysis and improved rates. In International Conference on Artiﬁcial Intelligence and St atistics, pages 7865–7901. PMLR, 2022

2022

[15] [15]

B. Halpern. Fixed points of nonexpanding maps. Bulletin of the American Mathematical Society, 73:957–961,

[16] [16]

26 YI ZHANG

URL https://api.semanticscholar.org/CorpusID:120539954. 26 YI ZHANG

[17] [17]

M. Ito, Z. Lu, and C. He. A parameter-free conditional gra dient method for composite minimization under H¨ older condition. Journal of Machine Learning Research , 24(166):1–34, 2023. URL http://jmlr.org/papers/v24/22-0983.html

2023

[18] [18]

A. N. Iusem, A. Jofr´ e, R. I. Oliveira, and P . Thompson. Ex tragradient method with variance reduction for stochastic variational inequalities. SIAM Journal on Optimization , 27(2):686–724, 2017. doi: 10.1137/ 15M1031953

2017

[19] [19]

Kannan and U

A. Kannan and U. V . Shanbhag. Optimal stochastic extragr adient schemes for pseudomonotone stochastic variational inequality problems and their variants. Computational Optimization and Applications , 74(3):779– 820, 2019. doi: 10.1007/s10589-019-00120-x

work page doi:10.1007/s10589-019-00120-x 2019

[20] [20]

Kinderlehrer and G

D. Kinderlehrer and G. Stampacchia. An introduction to variational inequalities and their applications. SIAM, 2000

2000

[21] [21]

G. M. Korpelevich. The extragradient method for ﬁnding s addle points and other problems. Matecon, 12: 747–756, 1976

1976

[22] [22]

Lee and D

S. Lee and D. Kim. Fast extra gradient methods for smooth s tructured nonconvex-nonconcave minimax problems. In A. Beygelzimer, Y . Dauphin, P . Liang, and J. W. V aughan, editors, Advances in Neural Information Processing Systems, 2021

2021

[23] [23]

F. Lieder. On the convergence rate of the Halpern-iterat ion. Optimization letters, 15(2):405–418, 2021. doi: 10.1007/s11590-020-01617-9

work page doi:10.1007/s11590-020-01617-9 2021

[24] [24]

Lu and J

H. Lu and J. Y ang. Restarted Halpern PDHG for linear progr amming, 2024. URL https://arxiv.org/abs/2407.16144

arXiv 2024

[25] [25]

G. J. Minty. Monotone (nonlinear) operators in Hilbert s pace. Duke Mathematical Journal, 29(3):341 – 346,

[26] [26]

URL https://doi.org/10.1215/S0012-7094-62-02933-2

doi: 10.1215/S0012-7094-62-02933-2. URL https://doi.org/10.1215/S0012-7094-62-02933-2

work page doi:10.1215/s0012-7094-62-02933-2

[27] [27]

Mishchenko, D

K. Mishchenko, D. Kovalev, E. Shulgin, P . Richt´ arik, and Y . Malitsky. Revisiting stochastic extragradient. In International Conference on Artiﬁcial Intelligence and St atistics, pages 4573–4582. PMLR, 2020

2020

[28] [28]

Nemirovski

A. Nemirovski. Prox-method with rate of convergence o(1 /t) for variational inequalities with lipschitz continuous monotone operators and smooth convex-concave s addle point problems. SIAM Journal on Optimization, 15(1):229–251, 2004. doi: 10.1137/S1052623403425629

work page doi:10.1137/s1052623403425629 2004

[29] [29]

Nesterov

Y . Nesterov. A method of solving a convex programming pro blem with convergence rate o(1/k2). Doklady Akademii Nauk SSSR, 269(3):543, 1983

1983

[30] [30]

Ouyang and Y

Y . Ouyang and Y . Xu. Lower complexity bounds of ﬁrst-orde r methods for convex-concave bilinear saddle- point problems. Mathematical Programming, 185(1):1–35, 2021. doi: 10.1007/s10107-019-01420-0

work page doi:10.1007/s10107-019-01420-0 2021

[31] [31]

Pethick, P

T. Pethick, P . Latafat, P . Patrinos, O. Fercoq, and V . Cev her. Escaping limit cycles: Global convergence for constrained nonconvex-nonconcave minimax problems. I n International Conference on Learning Representations, 2022. URL https://openreview.net/forum?id=2 vhkAMARk

2022

[32] [32]

L. D. Povov. A modiﬁcation of the Arrow-Hurwitz method of search for saddle points. Mat. Zametki, 28(5): 777–784, 1980

1980

[33] [33]

Qi and H.-K

H. Qi and H.-K. Xu. Convergence of Halpern’s iteration me thod with applications in optimization. Numerical Functional Analysis and Optimization, 42(15):1839–1854, 2021. doi: https://doi.org/10.1080/ 01630563.2021. 2001826

arXiv 2021

[34] [34]

X. Qu, W. Bian, and X. Chen. An extra gradient Anderson-ac celerated algorithm for pseudomonotone variational inequalities. Mathematics of Computation, 95(359):1361–1387, 2026. doi: 10.1090/mcom/4095

work page doi:10.1090/mcom/4095 2026

[35] [35]

Sedlmayer, D.-K

M. Sedlmayer, D.-K. Nguyen, and R. I. Bot ¸. A fast optimis tic method for monotone variational inequalities. In Proceedings of the 40th International Conference on Machin e Learning, ICML’23, 2023

2023

[36] [36]

Stonyakin, A

F. Stonyakin, A. Gasnikov, P . Dvurechensky, A. Titov, an d M. Alkousa. Generalized mirror prox algorithm for monotone variational inequalities: Universality and i nexact oracle. Journal of Optimization Theory and Applications, 194(3):988–1013, 2022. doi: 10.1007/s10957-022-02062- 7

work page doi:10.1007/s10957-022-02062- 2022

[37] [37]

D. Sun, Y . Y uan, G. Zhang, and X. Zhao. Accelerating preco nditioned ADMM via degenerate proximal point mappings. SIAM Journal on Optimization, 35(2):1165–1193, 2025. doi: 10.1137/24M1650053

work page doi:10.1137/24m1650053 2025

[38] [38]

J. E. Taylor and M. P . Bendsøe. An interpretation for min- max structural design problems including a method for relaxing constraints. International Journal of Solids and Structures , 20(4):301–314, 1984. doi: 10.1016/ ALGORITHMS FOR MONOTONE INCLUSION PROBLEM WITH SUMS OF H ¨OLDER OPERA TORS 27 0020-7683(84)90041-6

1984

[39] [39]

Toyasaki, P

F. Toyasaki, P . Daniele, and T. Wakolbinger. A variation al inequality formulation of equilibrium models for end-of-life products with nonlinear constraints. European Journal of Operational Research, 236(1):340–350,

[40] [40]

doi: 10.1016/j.ejor.2013.12.006

work page doi:10.1016/j.ejor.2013.12.006 2013

[41] [41]

Tran-Dinh

Q. Tran-Dinh. Sublinear convergence rates of extragrad ient-type methods: A survey on classical and recent developments, 2023. URL https://arxiv.org/abs/2303.17192

arXiv 2023

[42] [42]

Tran-Dinh

Q. Tran-Dinh. Extragradient-type methods with O(1/k) last-iterate convergence rates for co-hypomonotone inclusions. Journal of Global Optimization , 89(1):197–221, 2024. doi: 10.1007/s10898-023-01347-z

work page doi:10.1007/s10898-023-01347-z 2024

[43] [43]

Tran-Dinh and N

Q. Tran-Dinh and N. Nguyen-Trung. Accelerated extragra dient-type methods – part 2: Generalization and sublinear convergence rates under co-hypomonotonicity, 2 025. URL https://arxiv.org/abs/2501.04585

arXiv

[44] [44]

P . Tseng. A modiﬁed forward-backward splitting method f or maximal monotone mappings. SIAM Journal on Control and Optimization, 38(2):431–446, 2000. doi: 10.1137/S0363012998338806

work page doi:10.1137/s0363012998338806 2000

[45] [45]

T. Ui. Bayesian Nash equilibrium and variational inequa lities. Journal of Mathematical Economics , 63: 139–146, 2016. ISSN 0304-4068. doi: 10.1016/j.jmateco.20 16.02.004

work page doi:10.1016/j.jmateco.20 2016

[46] [46]

xiang Y uan and Y

Y . xiang Y uan and Y . Zhang. Symplectic extra-gradient ty pe method for solving general non-monotone inclusion problem, 2025. URL https://arxiv.org/abs/2406.10793

arXiv 2025

[47] [47]

Y oon and E

T. Y oon and E. K. Ryu. Accelerated algorithms for smooth c onvex-concave minimax problems with O(1/k2) rate on squared gradient norm. In International Conference on Machine Learning , pages 12098–12109. PMLR, 2021

2021

[48] [48]

Y uan and Y

Y .-x. Y uan and Y . Zhang. Symplectic discretization appr oach for developing new proximal point algorithm. Computational Optimization and Applications , 93(2):651–687, 2026. doi: 10.1007/s10589-025-00728-2

work page doi:10.1007/s10589-025-00728-2 2026