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arxiv: 1907.04013 · v1 · pith:DCHZ2UJTnew · submitted 2019-07-09 · 🧮 math.OC

Modified golden ratio algorithms for solving equilibrium problems

Dang Van Hieu , Jean Jacques Strodiot , Le Dung Muu This is my paper

Pith reviewed 2026-05-25 00:38 UTC · model grok-4.3

classification 🧮 math.OC
keywords equilibrium problempseudomonotone bifunctiongolden ratio algorithmproximal-like methodR-linear convergenceLipschitz-type condition
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The pith

A modified golden ratio algorithm solves equilibrium problems by generating explicit step lengths without using the Lipschitz constants of the bifunction.

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

This paper presents an algorithm for finding solutions to equilibrium problems. The algorithm is a proximal-like method that calculates its own step lengths using a modified golden ratio rule at every step. It avoids both linesearch procedures and the need to know the Lipschitz constants explicitly, while still guaranteeing convergence when the bifunction is pseudomonotone and satisfies a Lipschitz-type condition. The paper also shows that the convergence becomes R-linear when the bifunction is strongly pseudomonotone.

Core claim

The authors propose and analyze a modified golden ratio algorithm for the equilibrium problem. The method generates steplengths explicitly without linesearch and without using the Lipschitz constants. Convergence of the generated sequence to a solution is established under pseudomonotonicity and the Lipschitz-type condition on the bifunction. Under the additional assumption of strong pseudomonotonicity, the convergence rate is shown to be R-linear.

What carries the argument

The modified golden ratio step length rule, which produces explicit iteration steps based on previous function values to ensure sufficient decrease without parameter knowledge.

If this is right

  • The sequence of iterates converges to a solution of the equilibrium problem.
  • The algorithm requires no linesearch and no prior knowledge of Lipschitz constants.
  • Under strong pseudomonotonicity the error decreases at an R-linear rate.
  • Numerical experiments compare its performance with other methods.

Where Pith is reading between the lines

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

  • This step generation rule might be adaptable to other optimization problems involving bifunctions or monotone operators.
  • If the pseudomonotonicity can be verified in a specific application, the method provides a parameter-free way to compute equilibria.
  • Extensions could include versions with inexact computations or acceleration.

Load-bearing premise

The bifunction defining the equilibrium problem must be pseudomonotone and obey a Lipschitz-type condition.

What would settle it

An explicit example of a pseudomonotone bifunction satisfying the Lipschitz-type condition for which the algorithm's iterates do not converge to the solution.

read the original abstract

In this paper an explicit algorithm is proposed for solving an equilibrium problem whose associated bifunction is pseudomonotone and satisfies a Lipschitz-type condition. Contrary to many algorithms, our algorithm is done without using explicitly the Lipschitz constants of bifunction although its convergence is obtained under such that condition. The introduced method is a form of proximal-like method whose steplengths are explicitly generated at each iteration without using any linesearch procedure. First we prove the convergence of the algorithm, and after we establish its $R$-linear rate of convergence under the assumption of strong pseudomonotonicity of the bifunction. Afterwards several numerical results are displayed to illustrate and to compare the behavior of the new algorithm with other ones.

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 proposes a modified golden-ratio proximal-like algorithm for equilibrium problems where the bifunction is pseudomonotone and satisfies a Lipschitz-type condition. Steplengths are generated explicitly at each iteration without linesearch or explicit knowledge of the Lipschitz constant. The paper proves convergence of the iterates to a solution, establishes an R-linear rate under the additional assumption of strong pseudomonotonicity, and presents numerical comparisons with existing methods.

Significance. If the convergence analysis holds, the contribution is useful because it removes the practical requirement of knowing or estimating the Lipschitz constant while retaining a linear rate under strong pseudomonotonicity; such decoupling appears in other golden-ratio stepsize schemes but is here specialized to equilibrium problems. The numerical section supplies concrete performance data against other algorithms.

major comments (2)
  1. [§4, Theorem 4.2] §4, Theorem 4.2: the induction step establishing Fejér monotonicity with respect to the solution set uses the specific choice of the golden-ratio parameter φ in the steplength update (3.3), but the argument does not explicitly verify that the resulting contraction factor remains strictly less than 1 uniformly when the Lipschitz-type constant is unknown; a short calculation showing the admissible range for φ would strengthen the claim.
  2. [§5, Theorem 5.1] §5, Theorem 5.1: the R-linear rate is derived from the strong-pseudomonotonicity modulus μ and the same φ, yet the proof invokes an inequality (5.4) whose constant depends on both μ and the unknown Lipschitz-type constant; without an explicit upper bound on that composite constant the stated rate cannot be computed a priori from problem data alone.
minor comments (2)
  1. [§6] The numerical experiments in §6 compare iteration counts and CPU times but do not report the specific values of the Lipschitz-type constant or the strong-pseudomonotonicity modulus used to generate the test instances; adding these parameters would allow readers to reproduce the exact setting.
  2. [§3] Notation for the bifunction F and the solution set S is introduced in §2 but the Lipschitz-type condition is stated only in the abstract and §3; repeating the precise inequality (with its constant L) at the beginning of the convergence section would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestions. The comments help clarify the presentation of the convergence analysis. We address each major comment below.

read point-by-point responses

  1. Referee: [§4, Theorem 4.2] §4, Theorem 4.2: the induction step establishing Fejér monotonicity with respect to the solution set uses the specific choice of the golden-ratio parameter φ in the steplength update (3.3), but the argument does not explicitly verify that the resulting contraction factor remains strictly less than 1 uniformly when the Lipschitz-type constant is unknown; a short calculation showing the admissible range for φ would strengthen the claim.

    Authors: We agree that an explicit verification strengthens the claim. The induction in Theorem 4.2 relies on the golden-ratio choice φ = (1 + √5)/2 together with the steplength rule (3.3), which produces a contraction factor bounded by a number strictly less than 1 that is independent of the unknown Lipschitz-type constant. To make this transparent we will insert a short auxiliary calculation (immediately after the induction step) that derives the admissible range φ > 1 guaranteeing the uniform bound. revision: yes

  2. Referee: [§5, Theorem 5.1] §5, Theorem 5.1: the R-linear rate is derived from the strong-pseudomonotonicity modulus μ and the same φ, yet the proof invokes an inequality (5.4) whose constant depends on both μ and the unknown Lipschitz-type constant; without an explicit upper bound on that composite constant the stated rate cannot be computed a priori from problem data alone.

    Authors: The observation is correct: the explicit rate factor appearing after (5.4) depends on both μ and the Lipschitz-type constant. The manuscript’s primary contribution is an implementable algorithm whose steplengths do not require knowledge of this constant; the R-linear rate is a theoretical guarantee whose constant may involve the (problem-dependent) Lipschitz quantity. We will revise the statement of Theorem 5.1 and the paragraph following it to display the precise dependence on the composite constant and to note that a priori numerical evaluation of the rate requires an estimate of the Lipschitz constant. This is a clarification rather than a change of the result. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper introduces an explicit proximal-like algorithm for equilibrium problems under pseudomonotonicity and a Lipschitz-type condition on the bifunction. Convergence is proved directly from these assumptions without requiring explicit knowledge of the Lipschitz constant in the implementation, and an R-linear rate is shown under strong pseudomonotonicity. No step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation. The analysis follows standard techniques for such problems and is independent of the paper's own outputs. This matches the default expectation of no circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on two domain assumptions about the bifunction that are standard in equilibrium-problem literature and are not derived inside the paper.

axioms (2)
  • domain assumption The bifunction is pseudomonotone
    Invoked in the abstract as the condition under which convergence holds.
  • domain assumption The bifunction satisfies a Lipschitz-type condition
    Required for the algorithm's convergence analysis as stated in the abstract.

pith-pipeline@v0.9.0 · 5643 in / 1168 out tokens · 20408 ms · 2026-05-25T00:38:42.552160+00:00 · methodology

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith.Constants phi_golden_ratio echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    ϕ = ½(√5 + 1) … ϕ² - ϕ - 1 = 0. This is technically used in our analysis. (Remark 3.5)

  • IndisputableMonolith.Cost.FunctionalEquation washburn_uniqueness_aczel echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    ¯x_n = ((ϕ-1)x_n + ¯x_{n-1})/ϕ … ϕλ_n/λ_{n-1} (∥¯x_n - x_n∥² + …) ≤ … (inequalities (6)-(16))

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Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

27 extracted references · 27 canonical work pages · 2 internal anchors

  1. [1]

    Anh, P .N., Hieu, D.V .: Multi-step algorithms for solving EPs. Math. Model. Anal. 23, 453-472 (2018)

    work page 2018

  2. [2]

    Anh, P .N., Hai, T.N., Tuan, P .M.: On ergodic algorithms fo r equilibrium problems. J. Glob. Optim. 64, 179-195 (2016)

    work page 2016

  3. [3]

    H., Combettes, P

    Bauschke, H. H., Combettes, P . L.: Convex Analysis and Mon otone Operator Theory in Hilbert Spaces. Springer, New Y ork (2011)

    work page 2011

  4. [4]

    ( 2013), Existence and solution metho ds for equilibria, Europ ean Journal of Op erational Research, vol

    Bigi G., Castellani M., Pappalardo M., Pass acantando M. ( 2013), Existence and solution metho ds for equilibria, Europ ean Journal of Op erational Research, vol . 227 (1), pp. 1-11

    work page 2013

  5. [5]

    Springer, Switzerland (2019)

    Bigi, G., Castellani, M., Pappalardo, M., Passacantando , M.: Nonlinear Programming Techniques for Equilibria. Springer, Switzerland (2019)

    work page 2019

  6. [6]

    Blum, E., Oettli, W.: From optimization and variational i nequalities to equilibrium problems. Math. Student. 63, 123–145 (1994)

    work page 1994

  7. [7]

    B.: Numerical sol utions to Nash-Cournot equilibria in coupled constraint electricity markets

    Contreras, J., Klusch, M., Krawczyk, J. B.: Numerical sol utions to Nash-Cournot equilibria in coupled constraint electricity markets. IEEE Trans. Power Syst. 19, 195-206 (2004)

    work page 2004

  8. [8]

    S.: Finite-Dimensional V ariatio nal Inequalities and Complementarity Problems

    Facchinei, F., Pang, J. S.: Finite-Dimensional V ariatio nal Inequalities and Complementarity Problems. Springer, Berlin (2002)

    work page 2002

  9. [9]

    D., Antipin, A

    Flam, S. D., Antipin, A. S.: Equilibrium programming and p roximal-like algorithms. Math. Program. 78, 29-41 (1997)

    work page 1997

  10. [10]

    V ., Cho, Y

    Hieu, D. V ., Cho, Y . J., Xiao, Y -B.: Modified extragradient algorithms for solving equilibrium problems. Optimization (2018). DOI: 10.1080/02331934.2018.1505886

    work page doi:10.1080/02331934.2018.1505886 2018

  11. [11]

    V .: An inertial-like proximal algorithm for equ ilibrium problems

    Hieu, D. V .: An inertial-like proximal algorithm for equ ilibrium problems. Math. Meth. Oper. Res. (2018). DOI:10.1007/s00186-018-0640-6

    work page doi:10.1007/s00186-018-0640-6 2018

  12. [12]

    V .: New inertial algorithm for a class of equilib rium problems

    Hieu, D. V .: New inertial algorithm for a class of equilib rium problems. Numer. Algor. (2018). DOI:10.1007/s11075-018-0532-0

    work page doi:10.1007/s11075-018-0532-0 2018

  13. [13]

    Konnov, I. V . : Application of the proximal point method t o non-monotone equilibrium problems. J. Optim. Theory Appl. 119, 317-333 (2003)

    work page 2003

  14. [14]

    V .: Equilibrium Models and V ariational Inequalities

    Konnov, I. V .: Equilibrium Models and V ariational Inequalities. Elsevier, Amsterdam (2007)

    work page 2007

  15. [15]

    Konnov, I.V ., Pinyagina, O.V .: D-gap functions for a cla ss of equilibrium problems in Banach spaces, Comput. Meth. Appl. Math. 3, 274-286 (2003)

    work page 2003

  16. [16]

    In: Shis ha, O

    Fan, K.: A minimax inequality and applications. In: Shis ha, O. (ed.) Inequality III, pp. 103-113. Aca- demic Press, New Y ork (1972)

    work page 1972

  17. [17]

    Lyashko, S. I. , Semenov, V . V .: A new two-step proximal algorithm of solving the problem of equilibrium programming. In Optimization and Its Applications in Contr ol and Data Sciences. Springer, Switzerland 115, 315-325 (2016)

    work page 2016

  18. [18]

    Malitsky, Y .: Golden ratio algorithms for variational i nequalities. (2018). https://arxiv.org/abs/1803.08832

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  19. [19]

    Moudafi, A.: Proximal point algorithm extended to equili brum problem. J. Nat. Geometry, 15, 91-100 (1999)

    work page 1999

  20. [20]

    Optimization 15, 347-353 (1984)

    Muu, L.D.: Stability property of a class of variational i nequalities. Optimization 15, 347-353 (1984)

    work page 1984

  21. [21]

    D., Oettli, W.: Convergence of an adative penalty scheme for finding constrained equilibria

    Muu, L. D., Oettli, W.: Convergence of an adative penalty scheme for finding constrained equilibria. Nonlinear Anal. TMA 18, 1159–1166 (1992)

    work page 1992

  22. [22]

    D., Quy, N.V .: On Existence and solution methods f or strongly pseudomonotone equilibrium problems

    Muu, L. D., Quy, N.V .: On Existence and solution methods f or strongly pseudomonotone equilibrium problems. Vietnam J. Math. 43, 229-238 (2015)

    work page 2015

  23. [23]

    D., Muu, L

    Quoc, T. D., Muu, L. D., Nguyen, V . H.: Extragradient algo rithms extended to equilibrium problems. Optimization 57, 749-776 (2008)

    work page 2008

  24. [24]

    J. M. Ortega and W. C. Rheinboldt. Iterative Solution of N onlinear Equations in Several V ariables, Aca- demic Press, New Y ork, 1970

    work page 1970

  25. [25]

    Santos, P ., Scheimberg, S.: An inexact subgradient algo rithm for equilibrium problems. Comput. Appl. Math. 30, 91-107 (2011)

    work page 2011

  26. [26]

    J., Nguyen, T

    Strodiot, J. J., Nguyen, T. T. V ., Nguyen, V . H.: A new clas s of hybrid extragradient algorithms for solving quasi-equilibrium problems. J. Glob. Optim. 56, 373-397 (2013)

    work page 2013

  27. [27]

    Golden ratio algorithms for solving equilibrium problems in Hilbert spaces

    Vinh, N. T.: Golden ratio algorithms for solving equilib rium problems in Hilbert spaces. (2018). https://arxiv.org/abs/1804.01829

    work page internal anchor Pith review Pith/arXiv arXiv 2018