Splitting Algorithms of Common Solutions Between Equilibrium and Inclusion Problems on Hadamard Manifolds

Konrawut Khammahawong; Parin Chaipunya; Poom Kumam

arxiv: 1907.00364 · v1 · pith:HGMAPGVKnew · submitted 2019-06-30 · 🧮 math.FA

Splitting Algorithms of Common Solutions Between Equilibrium and Inclusion Problems on Hadamard Manifolds

Konrawut Khammahawong , Poom Kumam , Parin Chaipunya This is my paper

Pith reviewed 2026-05-25 12:38 UTC · model grok-4.3

classification 🧮 math.FA

keywords equilibrium probleminclusion problemHadamard manifoldsplitting algorithmiterative methodconvergence analysisminimization problemminimax problem

0 comments

The pith

An iterative splitting algorithm finds common solutions to equilibrium and inclusion problems on Hadamard manifolds with proven convergence.

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

The paper develops an iterative algorithm to identify points that simultaneously solve an equilibrium problem defined by a bifunction and an inclusion problem on a Hadamard manifold. The generated sequence is shown to converge to such a common solution under mild assumptions. Special cases of the algorithm are examined along with direct applications to minimization problems and minimax problems. A reader would care because the method unifies two distinct problem classes into one convergent procedure in non-Euclidean geometry.

Core claim

The authors introduce an iterative algorithm for finding a common solution from the set of an equilibrium point for a bifunction and the set of a singularity of an inclusion problem on an Hadamard manifold. The convergence of a sequence generated by the proposed algorithm is proved under mild assumptions. Moreover, the results are applied to solving minimization problems and minimax problems, and some particular cases of the problem are discussed by the proposed algorithm.

What carries the argument

A splitting iterative algorithm that alternates between equilibrium and inclusion steps on the manifold to produce the common solution sequence.

If this is right

The algorithm produces a sequence that converges to the desired common solution set.
Particular cases of the combined equilibrium-inclusion problem are directly solvable by the same procedure.
Minimization problems reduce to the framework and inherit the convergence guarantee.
Minimax problems likewise reduce and are solved by the algorithm.

Where Pith is reading between the lines

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

Implementation on concrete Hadamard manifolds such as hyperbolic space could reveal the practical speed of convergence.
The splitting structure may adapt to related variational inequalities or fixed-point problems in the same geometric setting.
Relaxing the mild assumptions or adding error tolerance could widen applicability to approximate data.

Load-bearing premise

The convergence proof depends on unspecified mild assumptions about the bifunction and the inclusion operator whose exact form, necessity, and verifiability are not provided.

What would settle it

A numerical run on the hyperbolic plane in which the generated sequence fails to approach any common solution while the bifunction and operator satisfy the mild assumptions used in the proof.

read the original abstract

The aim of this article is to introduce an iterative algorithm for finding a common solution from the set of an equilibrium point for a bifunction and the set of a singularity of an inclusion problem on an Hadamard manifold. We also discuss some particular cases of the problem by the proposed algorithm. The convergence of a sequence generated by the proposed algorithm is proved under mild assumptions. Moreover, we apply our results to solving minimization problems and minimax problems.

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 gives a splitting algorithm for equilibrium and inclusion problems on Hadamard manifolds but leaves the key convergence assumptions too vague to check.

read the letter

The paper introduces an iterative splitting method to find a common solution to an equilibrium problem for a bifunction and an inclusion problem on a Hadamard manifold. It proves convergence under mild assumptions and recovers minimization and minimax problems as special cases. The extension of splitting methods to this combined setting on manifolds is the main step forward from prior Euclidean work. They adapt the iteration using the manifold geometry and use Fejér monotonicity plus the Opial property in that space, which is a competent application of existing tools. The discussion of particular cases helps show the framework is not empty. The soft spot is the assumptions. The proof is said to hold under mild conditions on the bifunction and the set-valued operator, yet the paper never isolates exactly what monotonicity, continuity, or maximality properties are required. Without those statements it is impossible to see whether the claimed applications satisfy them or whether the result is useful. The stress-test note is correct on this point; the central claim cannot be verified from what is given. This is for specialists already working on variational problems in non-Euclidean spaces. A reader in that narrow area might extract an algorithm worth testing, but would have to supply the missing conditions themselves. I would send it for peer review so the authors can make the hypotheses explicit and perhaps add a concrete example or comparison. The thinking is straightforward and the geometry is handled honestly.

Referee Report

1 major / 0 minor

Summary. The manuscript introduces a splitting iterative algorithm to compute a common solution of an equilibrium problem (defined via a bifunction) and an inclusion problem (defined via a set-valued operator) on a Hadamard manifold. Convergence of the generated sequence is asserted under unspecified 'mild assumptions,' particular cases are discussed, and the results are applied to minimization and minimax problems.

Significance. If the assumptions turn out to be standard (e.g., pseudomonotonicity of the bifunction and maximal monotonicity of the operator) and the proof is correct, the work would extend Euclidean splitting methods to Hadamard manifolds, offering a tool for variational problems in non-Euclidean geometry.

major comments (1)

[Abstract] Abstract: the central convergence claim is stated to hold 'under mild assumptions,' yet neither the abstract nor the main theorem isolates the precise hypotheses on the bifunction (monotonicity type, continuity, etc.) and on the set-valued operator (maximal monotonicity, resolvent properties, etc.). This is load-bearing because every subsequent step (Fejér monotonicity, Opial property on the manifold) is invoked only after these conditions; without an explicit list it is impossible to verify applicability to the claimed minimization and minimax examples.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need for greater precision in stating the hypotheses. We address the single major comment below.

read point-by-point responses

Referee: [Abstract] Abstract: the central convergence claim is stated to hold 'under mild assumptions,' yet neither the abstract nor the main theorem isolates the precise hypotheses on the bifunction (monotonicity type, continuity, etc.) and on the set-valued operator (maximal monotonicity, resolvent properties, etc.). This is load-bearing because every subsequent step (Fejér monotonicity, Opial property on the manifold) is invoked only after these conditions; without an explicit list it is impossible to verify applicability to the claimed minimization and minimax examples.

Authors: We agree that the abstract and the statement of the principal convergence theorem refer only to 'mild assumptions' without enumerating the required properties of the bifunction and the set-valued operator. In the revised manuscript we will explicitly list these hypotheses (pseudomonotonicity and continuity of the bifunction, maximal monotonicity of the operator together with the standard resolvent properties) both in the abstract and in the theorem statement itself, so that the subsequent arguments and the applicability to the minimization and minimax examples become immediately verifiable. revision: yes

Circularity Check

0 steps flagged

No circularity: standard convergence proof with no self-referential reductions

full rationale

The manuscript introduces a splitting iteration for the intersection of an equilibrium set (defined by a bifunction) and a singularity set (defined by an inclusion) on a Hadamard manifold, then states that convergence holds under unspecified mild assumptions and lists particular cases (minimization, minimax). No equation is shown to equal its own input by construction, no parameter is fitted on a subset and then relabeled a prediction, and no load-bearing premise is justified solely by a self-citation whose content reduces to the present claim. The derivation chain therefore remains self-contained as an ordinary existence-plus-convergence argument; the vagueness of the assumptions affects verifiability but does not create a definitional loop.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no free parameters, axioms, or invented entities can be extracted or audited.

pith-pipeline@v0.9.0 · 5603 in / 1114 out tokens · 47489 ms · 2026-05-25T12:38:16.704839+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

28 extracted references · 28 canonical work pages

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work page doi:10.1016/j.jmaa.2011.11.001 2012
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An inexact proximal point algorithm for maxim al monotone vec- tor ﬁelds on Hadamard manifolds

Tang Gj, Huang Nj. An inexact proximal point algorithm for maxim al monotone vec- tor ﬁelds on Hadamard manifolds. Oper Res Lett. 2013;41(6):586–5 91. Available from: https://doi.org/10.1016/j.orl.2013.08.003

work page doi:10.1016/j.orl.2013.08.003 2013
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Equilibrium problems on Riemannian mani- folds with applications

Wang X, L´ opez G, Li C, Yao JC. Equilibrium problems on Riemannian mani- folds with applications. J Math Anal Appl. 2019;473(2):866–891. Ava ilable from: https://doi.org/10.1016/j.jmaa.2018.12.073

work page doi:10.1016/j.jmaa.2018.12.073 2019
[22]

Monotone vector ﬁelds a nd the proximal point algo- rithm on Hadamard manifolds

Li C, L´ opez G, Mart ´ ın-M´ arquez V. Monotone vector ﬁelds a nd the proximal point algo- rithm on Hadamard manifolds. J Lond Math Soc (2). 2009;79(3):663– 683. Available from: https://doi.org/10.1112/jlms/jdn087

work page doi:10.1112/jlms/jdn087 2009
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Riemannian geometry

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Metric spaces of non-positive curvatu re

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Monotone p oint-to-set vector ﬁelds

da Cruz Neto JX, Ferreira OP, Lucambio P´ erez LR. Monotone p oint-to-set vector ﬁelds. Balkan J Geom Appl. 2000;5(1):69–79. Dedicated to Professor Constantin Udri¸ ste

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Halpern- and Mann-Type Algor ithms for Fixed Points and Inclusion Problems on Hadamard Manifolds

Al-Homidan S, Ansari QH, Babu F. Halpern- and Mann-Type Algor ithms for Fixed Points and Inclusion Problems on Hadamard Manifolds. Numerical Functional An alysis and Optimization. 2019 feb;p. 1–33. Available from: https://doi.org/10.1080%2F01630563.2018.1553887

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[1] [1]

A minimax inequality and applications

Fan K. A minimax inequality and applications. 1972;p. 103–113

work page 1972

[2] [2]

From optimization and variational inequalities to eq uilibrium problems

Blum E, Oettli W. From optimization and variational inequalities to eq uilibrium problems. Math Student. 1994;63(1-4):123–145

work page 1994

[3] [3]

Equilibrium models and variational inequalities

Konnov IV. Equilibrium models and variational inequalities. vol. 210 o f Mathematics in Science and Engineering. Elsevier B. V., Amsterdam; 2007

work page 2007

[4] [4]

Convergence of an adaptive penalty scheme fo r ﬁnd- ing constrained equilibria

Muu LD, Oettli W. Convergence of an adaptive penalty scheme fo r ﬁnd- ing constrained equilibria. Nonlinear Anal. 1992;18(12):1159–1166. A vailable from: https://doi.org/10.1016/0362-546X(92)90159-C

work page doi:10.1016/0362-546x(92)90159-c 1992

[5] [5]

A class of diﬀerential inverse qua si-variational inequali- ties in ﬁnite dimensional spaces

Li W, Xiao YB, Huang NJ, Cho YJ. A class of diﬀerential inverse qua si-variational inequali- ties in ﬁnite dimensional spaces. J Nonlinear Sci Appl. 2017;10(8):453 2–4543. Available from: https://doi.org/10.22436/jnsa.010.08.45

work page doi:10.22436/jnsa.010.08.45 2017

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Equivalence of well-posedness b etween systems of hemivari- ational inequalities and inclusion problems

Wang YM, Xiao YB, Wang X, Cho YJ. Equivalence of well-posedness b etween systems of hemivari- ational inequalities and inclusion problems. J Nonlinear Sci Appl. 2016;9 (3):1178–1192. Available from: https://doi.org/10.22436/jnsa.009.03.44

work page doi:10.22436/jnsa.009.03.44 2016

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Monotone operators and the proximal point algo rithm

Rockafellar RT. Monotone operators and the proximal point algo rithm. SIAM J Control Opti- mization. 1976;14(5):877–898. Available from: https://doi.org/10.1137/0314056

work page doi:10.1137/0314056 1976

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R´ egularisation d’in´ equations variationnelles par app roximations successives

Martinet B. R´ egularisation d’in´ equations variationnelles par app roximations successives. Rev Fran¸ caise Informat Recherche Op´ erationnelle. 1970;4(S´ er. R-3):154–158. SPLITTING ALGORITHMS OF COMMON SOLUTIONS TO EQUILIBRIUM AN D INCLUSION 15

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Weak convergence theorems for maxim al monotone operators with nonspreading mappings in a Hilbert space

Manaka H, Takahashi W. Weak convergence theorems for maxim al monotone operators with nonspreading mappings in a Hilbert space. Cubo. 2011;13(1):11– 24. Available from: https://doi.org/10.4067/s0719-06462011000100002

work page doi:10.4067/s0719-06462011000100002 2011

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Set-valued variational inclusions in Banach spaces

Chang SS. Set-valued variational inclusions in Banach spaces. J Math Anal Appl. 2000;248(2):438–

work page 2000

[11] [11]

Available from: https://doi.org/10.1006/jmaa.2000.6919

work page doi:10.1006/jmaa.2000.6919 2000

[12] [12]

Existence and approximation of solutions for set-va lued variational inclusions in Banach space

Chang Ss. Existence and approximation of solutions for set-va lued variational inclusions in Banach space. In: Proceedings of the Third World Congress of Nonlinear An alysts, Part 1 (Catania, 2000). vol. 47; 2001. p. 583–594. Available from: https://doi.org/10.1016/S0362-546X(01)00203-6

work page doi:10.1016/s0362-546x(01)00203-6 2000

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Generalized set-valued var iational inclu- sions in Banach spaces

Chang SS, Cho YJ, Lee BS, Jung IH. Generalized set-valued var iational inclu- sions in Banach spaces. J Math Anal Appl. 2000;246(2):409–422. Av ailable from: https://doi.org/10.1006/jmaa.2000.6795

work page doi:10.1006/jmaa.2000.6795 2000

[14] [14]

Approximating solutions of maximal mo notone op- erators in Hilbert spaces

Kamimura S, Takahashi W. Approximating solutions of maximal mo notone op- erators in Hilbert spaces. J Approx Theory. 2000;106(2):226–240 . Available from: https://doi.org/10.1006/jath.2000.3493

work page doi:10.1006/jath.2000.3493 2000

[15] [15]

Optimization techniques on Riemannian manifolds

Smith ST. Optimization techniques on Riemannian manifolds. In: Ha miltonian and gradient ﬂows, algorithms and control. vol. 3 of Fields Inst. Commun. Amer. Math. S oc., Providence, RI; 1994. p. 113–136

work page 1994

[16] [16]

Newton’s m ethod on Riemannian manifolds and a geometric model for the human spine

Adler RL, Dedieu JP, Margulies JY, Martens M, Shub M. Newton’s m ethod on Riemannian manifolds and a geometric model for the human spine. IMA J Numer An al. 2002;22(3):359–390. Available from: https://doi.org/10.1093/imanum/22.3.359

work page doi:10.1093/imanum/22.3.359 2002

[17] [17]

Resolvents of s et-valued monotone vec- tor ﬁelds in Hadamard manifolds

Li C, L´ opez G, Mart ´ ın-M´ arquez V, Wang JH. Resolvents of s et-valued monotone vec- tor ﬁelds in Hadamard manifolds. Set-Valued Var Anal. 2011;19(3):36 1–383. Available from: https://doi.org/10.1007/s11228-010-0169-1

work page doi:10.1007/s11228-010-0169-1 2011

[18] [18]

Proximal point algorithm on Riemannian ma nifolds

Ferreira OP, Oliveira PR. Proximal point algorithm on Riemannian ma nifolds. Optimization. 2002;51(2):257–270. Available from: https://doi.org/10.1080/02331930290019413

work page doi:10.1080/02331930290019413 2002

[19] [19]

Equilibrium pro blems in Hadamard manifolds

Colao V, L´ opez G, Marino G, Mart ´ ın-M´ arquez V. Equilibrium pro blems in Hadamard manifolds. J Math Anal Appl. 2012;388(1):61–77. Available from: https://doi.org/10.1016/j.jmaa.2011.11.001

work page doi:10.1016/j.jmaa.2011.11.001 2012

[20] [20]

An inexact proximal point algorithm for maxim al monotone vec- tor ﬁelds on Hadamard manifolds

Tang Gj, Huang Nj. An inexact proximal point algorithm for maxim al monotone vec- tor ﬁelds on Hadamard manifolds. Oper Res Lett. 2013;41(6):586–5 91. Available from: https://doi.org/10.1016/j.orl.2013.08.003

work page doi:10.1016/j.orl.2013.08.003 2013

[21] [21]

Equilibrium problems on Riemannian mani- folds with applications

Wang X, L´ opez G, Li C, Yao JC. Equilibrium problems on Riemannian mani- folds with applications. J Math Anal Appl. 2019;473(2):866–891. Ava ilable from: https://doi.org/10.1016/j.jmaa.2018.12.073

work page doi:10.1016/j.jmaa.2018.12.073 2019

[22] [22]

Monotone vector ﬁelds a nd the proximal point algo- rithm on Hadamard manifolds

Li C, L´ opez G, Mart ´ ın-M´ arquez V. Monotone vector ﬁelds a nd the proximal point algo- rithm on Hadamard manifolds. J Lond Math Soc (2). 2009;79(3):663– 683. Available from: https://doi.org/10.1112/jlms/jdn087

work page doi:10.1112/jlms/jdn087 2009

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Riemannian geometry

Sakai T. Riemannian geometry. vol. 149 of Translations of Math ematical Monographs. American Mathematical Society, Providence, RI; 1996. Translated from th e 1992 Japanese original by the author

work page 1996

[24] [24]

Riemannian geometry

do Carmo MPa. Riemannian geometry. Mathematics: Theory & Ap plications. Birkh˝ auser Boston, Inc., Boston, MA; 1992. Translated from the second Portuguese edition by Francis Flaherty. Avail- able from: https://doi.org/10.1007/978-1-4757-2201-7

work page doi:10.1007/978-1-4757-2201-7 1992

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Convex functions and optimization methods on Riem annians manifolds

Udri¸ ste C. Convex functions and optimization methods on Riem annians manifolds. vol. 297 of Mathematics and its Applications. Kluwer Academic Publishers Group, Dordrecht; 1994. Available from: https://doi.org/10.1007/978-94-015-8390-9

work page doi:10.1007/978-94-015-8390-9 1994

[26] [26]

Metric spaces of non-positive curvatu re

Bridson MR, Haeﬂiger A. Metric spaces of non-positive curvatu re. vol. 319 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathe matical Sciences]. Springer- Verlag, Berlin; 1999. Available from: https://doi.org/10.1007/978-3-662-12494-9

work page doi:10.1007/978-3-662-12494-9 1999

[27] [27]

Monotone p oint-to-set vector ﬁelds

da Cruz Neto JX, Ferreira OP, Lucambio P´ erez LR. Monotone p oint-to-set vector ﬁelds. Balkan J Geom Appl. 2000;5(1):69–79. Dedicated to Professor Constantin Udri¸ ste

work page 2000

[28] [28]

Halpern- and Mann-Type Algor ithms for Fixed Points and Inclusion Problems on Hadamard Manifolds

Al-Homidan S, Ansari QH, Babu F. Halpern- and Mann-Type Algor ithms for Fixed Points and Inclusion Problems on Hadamard Manifolds. Numerical Functional An alysis and Optimization. 2019 feb;p. 1–33. Available from: https://doi.org/10.1080%2F01630563.2018.1553887

work page arXiv 2019