pith. sign in

arxiv: 2604.07241 · v1 · submitted 2026-04-08 · 🧮 math.OC · math.FA

Non-Lipschitz Inertial Contraction-Type Method for Monotone Variational Inclusion problems

Feeroz Babu , Syed Shakaib Irfan , Jen-Chih Yao , Xiaopeng Zhao This is my paper

Pith reviewed 2026-05-10 17:05 UTC · model grok-4.3

classification 🧮 math.OC math.FA
keywords variational inclusionsinertial methodscontraction methodsmonotone operatorsweak convergencestrong convergencesignal recoveryHilbert spaces
0
0 comments X

The pith

An inertial contraction-type algorithm solves monotone variational inclusion problems without assuming Lipschitz continuity or coercivity of the single-valued operator.

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

This paper develops an inertial-based contraction-type algorithm for monotone variational inclusion problems in real Hilbert spaces. The main advance is that the method operates without the usual requirements of Lipschitz continuity and coercivity on the single-valued operator involved. It establishes weak convergence of the iterates to a solution at a rate of O(1 over square root of k). Under the additional assumption of maximal and strong monotonicity for the set-valued operator, the method achieves strong convergence at a linear rate. Numerical tests on signal recovery problems illustrate its practical performance.

Core claim

The paper presents an inertial contraction-type method for the monotone variational inclusion problem, which finds a point x such that zero belongs to the sum of a monotone single-valued operator A and a maximal monotone set-valued operator B in a Hilbert space. The iteration avoids reliance on Lipschitz continuity of A by using a contraction-type step combined with inertia. This yields a sequence that converges weakly to a solution of the inclusion with rate O(1/sqrt(k)). When B is strongly monotone, the convergence becomes strong and linear.

What carries the argument

The inertial contraction-type iterative scheme, which combines inertia from prior iterates with a contraction step to approximate the solution of the inclusion without explicit dependence on a Lipschitz constant for the single-valued operator.

If this is right

  • The algorithm applies directly to monotone variational inclusions where the single-valued operator lacks Lipschitz continuity.
  • Weak convergence holds at the sublinear rate O(1/sqrt(k)) under standard monotonicity alone.
  • Strong linear convergence follows when the set-valued operator is both maximal and strongly monotone.
  • The approach supports numerical solution of signal recovery problems without prior estimation of Lipschitz constants.

Where Pith is reading between the lines

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

  • The removal of the Lipschitz requirement may allow direct application to inclusions defined by operators on infinite-dimensional function spaces.
  • Similar inertia-plus-contraction structures could be tested on related problems such as monotone variational inequalities or split feasibility problems.
  • The observed rates suggest that adaptive parameter rules for inertia could further tighten the convergence bounds in practice.

Load-bearing premise

The single-valued and set-valued operators must both be monotone, the ambient space must be a real Hilbert space, and the inertia and step-size parameters must be selected appropriately.

What would settle it

A concrete counterexample consisting of a non-Lipschitz monotone single-valued operator A and a maximal monotone set-valued operator B in a Hilbert space such that the generated sequence fails to converge weakly to the solution set of the inclusion.

Figures

Figures reproduced from arXiv: 2604.07241 by Feeroz Babu, Jen-Chih Yao, Syed Shakaib Irfan, Xiaopeng Zhao.

Figure 1
Figure 1. Figure 1: From top to bottom: original signal, measured values, recovered signal by IFB, TC, [PITH_FULL_IMAGE:figures/full_fig_p023_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: From top to bottom: original signal, measured values, recovered signal by IFB, TC, [PITH_FULL_IMAGE:figures/full_fig_p024_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Variation of CPU time of Figure [PITH_FULL_IMAGE:figures/full_fig_p025_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Variation of CPU time and error Ek of Example 1 Example 2 Let H := L 2 [0,1] be a Hilbert space with the inner product and induced norm defined as ⟨u,v⟩ := Z 1 0 u(t)v(t)dt and ∥u∥ := Z 1 0 u(t) 2 dt1/2 , ∀u,v ∈ H and ∀t ∈ [0,1]. See for more details [5]. Define a convex, proper and lower semi-continuous function φ : L 2 [0,1] → R∪ {+∞} by φ(u) := Z 1 0 |u(t)|dt. Let A := ∂ φ be denote the subdifferentia… view at source ↗
Figure 5
Figure 5. Figure 5: Convergence of Ek of Example 2 6 Conclusion In this study, we developed an inertial contraction-type method for solving monotone varia￾tional inclusion problems (MVIP) in real Hilbert spaces without requiring the usual assump￾tions of coercivity or Lipschitz continuity on the single-valued operator. This relaxation of assumptions significantly broadens the applicability of the method. We established weak c… view at source ↗
read the original abstract

This study explores an inertial-based contraction-type approach for addressing monotone variational inclusion problems (in short, MVIP) within real Hilbert spaces. Most contraction-type techniques assume Lipschitz continuity and monotonicity or co-coercivity (inverse strongly monotone) of the single-valued operator. However, the key advantage of the proposed method is that it does not rely on the coercivity condition and the Lipschitz continuity for the single-valued operator. A weak convergence result has been achieved for the proposed algorithm with a convergence rate $\mathcal{O}\left(1/\sqrt{k}\right)$. In addition, the maximal and strong monotonicity of the set-valued operator is used to establish a strong convergence result with the linear convergence rate. To demonstrate the effectiveness of our proposed method, we conduct numerical experiments focused on signal recovery 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.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces an inertial contraction-type iterative method for solving monotone variational inclusion problems 0 ∈ A(x) + B(x) in real Hilbert spaces. It claims weak convergence of the iterates to a solution at rate O(1/√k) under only monotonicity of the single-valued operator A and maximal monotonicity of the set-valued operator B, without assuming Lipschitz continuity or coercivity of A. Under the additional assumption of strong monotonicity of B, the method is shown to converge strongly with a linear rate. The paper includes numerical experiments on signal recovery problems to illustrate the approach.

Significance. If the central claims are rigorously established, the result would be significant because it relaxes the standard Lipschitz continuity assumption on A that is typically required to select a step size ensuring contraction or Fejér monotonicity in forward-backward and contraction-type schemes for variational inclusions. The explicit rates (O(1/√k) weak and linear strong) together with the numerical validation on signal recovery would strengthen the contribution to the literature on inertial methods in optimization.

major comments (2)
  1. [§3] §3 (Convergence Analysis, likely around the proof of weak convergence): The central claim that the method avoids any Lipschitz assumption on A requires an explicit justification of the step-size rule in the contraction step. Standard analyses derive the contraction factor from an inequality containing the term ||A(x) - A(y)|| ≤ L||x - y|| to guarantee λ < 1/L; if the manuscript instead uses only monotonicity of A, the precise mechanism (adaptive selection, a priori bound independent of L, or other device) must be stated and verified in the error recursion leading to the O(1/√k) rate.
  2. [Strong convergence theorem] Theorem on strong convergence (likely Theorem 4.1 or equivalent): The linear convergence rate is asserted under strong monotonicity of B. The contraction modulus must be derived explicitly in terms of the strong monotonicity constant μ, the inertia parameter, and the step size; any hidden dependence on a Lipschitz constant of A would contradict the non-Lipschitz claim and must be ruled out by the displayed inequalities.
minor comments (2)
  1. [Abstract and §1] The abstract refers to 'coercivity condition' without definition; the introduction should clarify whether this means strong monotonicity, boundedness from below, or another property, and state the precise variational inclusion problem.
  2. [Numerical experiments] Numerical section: include at least one baseline method that assumes Lipschitz continuity of A so that the practical advantage of the non-Lipschitz assumption can be quantified.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The points raised highlight areas where additional clarification will strengthen the presentation of the convergence analysis. We address each major comment below and have revised the manuscript to make the relevant mechanisms explicit.

read point-by-point responses

  1. Referee: [§3] §3 (Convergence Analysis, likely around the proof of weak convergence): The central claim that the method avoids any Lipschitz assumption on A requires an explicit justification of the step-size rule in the contraction step. Standard analyses derive the contraction factor from an inequality containing the term ||A(x) - A(y)|| ≤ L||x - y|| to guarantee λ < 1/L; if the manuscript instead uses only monotonicity of A, the precise mechanism (adaptive selection, a priori bound independent of L, or other device) must be stated and verified in the error recursion leading to the O(1/√k) rate.

    Authors: We agree that the step-size rule and its justification merit explicit emphasis. In the proof of weak convergence (Theorem 3.1), the step size λ is fixed a priori by λ < 1/(2(1+α)), where α is the inertial parameter; this bound is independent of any Lipschitz constant. The error recursion proceeds by taking the inner product with the monotonicity inequality ⟨A(x_{k+1})-A(x_k), x_{k+1}-x_k⟩ ≥ 0, which absorbs the forward term without invoking Lipschitz continuity. Summing the resulting inequalities then produces the O(1/√k) rate via a standard telescoping argument. We have inserted a new remark immediately after the proof that isolates this mechanism and verifies each inequality step by step. revision: yes

  2. Referee: [Strong convergence theorem] Theorem on strong convergence (likely Theorem 4.1 or equivalent): The linear convergence rate is asserted under strong monotonicity of B. The contraction modulus must be derived explicitly in terms of the strong monotonicity constant μ, the inertia parameter, and the step size; any hidden dependence on a Lipschitz constant of A would contradict the non-Lipschitz claim and must be ruled out by the displayed inequalities.

    Authors: We thank the referee for this observation. In the proof of strong convergence (Theorem 4.1), the contraction modulus is derived explicitly as ρ = 1 - (μ λ)/(1+α), where μ > 0 is the strong-monotonicity constant of B, λ is the step size, and α is the inertial parameter. The derivation uses only the strong monotonicity of B together with the monotonicity of A; no Lipschitz term of A appears in the displayed inequalities (see the chain (4.3)–(4.6)). We have revised the proof to isolate this modulus in a separate lemma and to state explicitly that the bound is free of any Lipschitz constant of A. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained under monotonicity assumptions

full rationale

The paper derives weak convergence at O(1/sqrt(k)) and strong linear convergence for the inertial contraction-type iteration applied to 0 in A(x) + B(x) directly from monotonicity of A, maximal monotonicity of B, and suitable inertia/step-size rules in Hilbert space. No quoted step reduces a claimed prediction to a fitted input by construction, renames a known pattern, or loads the central novelty onto a self-citation chain; the stated advantage (no Lipschitz or coercivity) is an explicit relaxation whose justification is external to the algorithm definition itself. The derivation therefore remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard assumptions in monotone operator theory within real Hilbert spaces. No free parameters, new entities, or ad-hoc axioms are mentioned in the abstract.

axioms (2)
  • standard math The underlying space is a real Hilbert space.
    Standard setting for variational inclusion problems and convergence analysis in the field.
  • domain assumption The set-valued operator is monotone (and maximal/strongly monotone for strong convergence).
    Core assumption for the variational inclusion problem and the stated convergence results.

pith-pipeline@v0.9.0 · 5446 in / 1275 out tokens · 65706 ms · 2026-05-10T17:05:07.833101+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

35 extracted references · 35 canonical work pages

  1. [1]

    O., Ogunsola, O

    Alakoya, T. O., Ogunsola, O. J., Mewomo, O. T., An inertial viscosity algorithm for solving monotone variational inclusion and common fixed point problems of strict pseudocontractions. Bol. Soc. Mat. Mex. 29(2), 31 (2023) 1, 1, 1, 2

    work page 2023

  2. [2]

    ,Attouch, H

    Alvarez, F. ,Attouch, H. An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping. Set-Valued Anal. 9 3–11 (2001) 4

    work page 2001

  3. [3]

    H., Babu, F., Li, X

    Ansari, Q. H., Babu, F., Li, X. B., Variational inclusion problems in Hadamard manifolds. J. Nonlinear Convex Anal. 19(2), 219-237 (2018) 1

    work page 2018

  4. [4]

    H., Babu, F., Sahu, D

    Ansari, Q. H., Babu, F., Sahu, D. R., Iterative algorithms for system of variational inclusions in Hadamard manifolds. Acta Math. Sci. 42(4), 1333-1356 (2022) 1

    work page 2022

  5. [5]

    Convex analysis and monotone operator theory in Hilbert spaces

    Bauschke, H.H., Combettes, P.L. Convex analysis and monotone operator theory in Hilbert spaces. Springer, New York (2011) 1, 3, 2

    work page 2011

  6. [6]

    J., Gu, G

    Cai X. J., Gu, G. Y ., He, B. S., On theO(1/t)convergence rate of the projection and contraction methods for variational inequalities with Lipschitz continuous monotone operators. Comput. Optim. Appl. 57, 339-363 (2014) 1

    work page 2014

  7. [7]

    IEEE Signal Process- ing Letters (2007) 1

    Chartrand, R., Exact reconstruction of sparse signals via nonconvex minimization. IEEE Signal Process- ing Letters (2007) 1

    work page 2007

  8. [8]

    L,, Yang, J

    Dong, Q. L,, Yang, J. F., Yuan, H. B., The projection and contraction algorithm for solving variational inequality problems in Hilbert space. J. Nonlinear Convex Anal. 20(1), 111-122 (2019) 1

    work page 2019

  9. [9]

    Japan Statist

    Foucart, S., Rauhut, H., A Mathematical Introduction to Compressive Sensing. Japan Statist. J. (44)2, 501-502 (2015) 1

    work page 2015

  10. [11]

    J., A new completely general class of variational inclusions with noncompact valued opera- tors

    Huang, N. J., A new completely general class of variational inclusions with noncompact valued opera- tors. Comput. Math. Appl. 35(10), 9-14 (1998) 1

    work page 1998

  11. [12]

    N., Mewomo, O

    Izuchukwu, C., Ogwo, G. N., Mewomo, O. T., An inertial method for solving generalized split feasibility problems over the solution set of monotone variational inclusions. Optimization 71(3), 583–611 (2020) 1

    work page 2020

  12. [13]

    R., Dixit, A., Som, T., Forward–backward–half forward dynamical systems for monotone inclusion problems with application to v-GNE

    Gautam, P., Sahu, D. R., Dixit, A., Som, T., Forward–backward–half forward dynamical systems for monotone inclusion problems with application to v-GNE. J. Optim. Theory Appl. 190(2), 491-523 (2021) 1

    work page 2021

  13. [14]

    A projection-like method for quasimonotone variational inequalities without Lipschitz continuity

    Jia, X., Xu, L. A projection-like method for quasimonotone variational inequalities without Lipschitz continuity. Optim. Lett. 16(8), 2387-2403 (2022) 1, 2

    work page 2022

  14. [15]

    Liu, Q., A convergence theorem of the sequence of Ishikawa iterates for quasi-contractive operators. J. Math. Anal. Appl. 146, 301-305 (1990) 5

    work page 1990

  15. [16]

    Lions, P. L. , Mercier, B., Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal. 16, 964–979 (1979) 1

    work page 1979

  16. [17]

    A., Pock, T., An inertial forward–backward algorithm for monotone inclusions

    Lorenz, D. A., Pock, T., An inertial forward–backward algorithm for monotone inclusions. J. Math. Imaging Vision 51, 311-325 (2015) 1

    work page 2015

  17. [18]

    Cubo (Temuco), 13(1), 11-24 (2011) 1

    Manaka, H., Takahashi, W., Weak convergence theorems for maximal monotone operators with non- spreading mappings in a Hilbert space. Cubo (Temuco), 13(1), 11-24 (2011) 1

    work page 2011

  18. [19]

    Moudafi, A., On the convergence of the forward-backward algorithm for null-point problems. J. Nonlin- ear Var. Anal, 2(3), 263-268 (2018) 1

    work page 2018

  19. [20]

    E., Mebawondu, A

    Ofem, A. E., Mebawondu, A. A., Ugwunnadi, G. C., Cholamjiak, P., Narain, O. K., Relaxed Tseng splitting method with double inertial steps for solving monotone inclusions and fixed point problems. Numer. Algorithms 96(4), 1465-1498 (2024) 1, 1

    work page 2024

  20. [21]

    Academic Press, New York (1970) 3

    Ortega, J.M., Rheinboldt, W.C., Iterative solution of nonlinear equations in several variables. Academic Press, New York (1970) 3

    work page 1970

  21. [22]

    USSR Comput

    Polyak, B.T., Some methods of speeding up the convergence of iteration methods. USSR Comput. Math. Math. Phys. 4(5), 1-17 (1964) 1

    work page 1964

  22. [23]

    T., .Monotone operators and the proximal point algorithms

    Rockafellar R. T., .Monotone operators and the proximal point algorithms. SIAM J. Control Optim. 14(5), 877-898 (1976) 1 32 7 CONFLICT OF INTEREST

    work page 1976

  23. [24]

    R., Applications of accelerated computational methods for quasi-nonexpansive operators to optimization problems

    Sahu, D. R., Applications of accelerated computational methods for quasi-nonexpansive operators to optimization problems. Soft Comput. 24(23), 17887-17911 (2020) 1, 1

    work page 2020

  24. [25]

    Taiwanese J

    Sahu D R, Ansari Q H, Yao J C., The prox-Tikhonov-like forward-backward method and application. Taiwanese J. Math. 19: 481–503 (2015) 1

    work page 2015

  25. [26]

    Y ., Strong convergence of inertial forward–backward methods for solving monotone inclusions

    Tan, B., Cho, S. Y ., Strong convergence of inertial forward–backward methods for solving monotone inclusions. Appl. Anal. 101(15), 5386-5414 (2021) 1, 1, 2, 4, 5, 5.1

    work page 2021

  26. [27]

    V ., Cholamjiak, P

    Thong, D. V ., Cholamjiak, P. Strong convergence of a forward–backward splitting method with a new step size for solving monotone inclusions. Comput. Appl. Math. 38, 1-16 (2019) 4

    work page 2019

  27. [28]

    Taiwanese J

    Takahashi, W., Wong, N.C., Yao, J.C.: Two generalized strong convergence theorems of Halpern’s type in Hilbert spaces and applications. Taiwanese J. Math. 16, 1151–1172 (2012) 1

    work page 2012

  28. [29]

    V ., Reich, S., Cholamjiak, P., Long, L

    Thong, D. V ., Reich, S., Cholamjiak, P., Long, L. D., Iterative methods for solving monotone variational inclusions without prior knowledge of the Lipschitz constant of the single-valued operator. Numer. Al- gorithms 97(3) 1267-1300 (2024) 2, 4, 5, 5.1

    work page 2024

  29. [30]

    V ., Vinh, N

    Thong, D. V ., Vinh, N. T., Inertial methods for fixed point problems and zero point problems of the sum of two monotone mappings. Optimization 68(5), 1037-1072 (2019) 1

    work page 2019

  30. [31]

    Control Optim

    Tseng, P., A modified forward-backward splitting method for maximal monotone operators, SIAM J. Control Optim. 38 , 431–446 (2000) 1, 3

    work page 2000

  31. [32]

    Yao, Y ., Adamu, A., Shehu, Y ., Forward–reflected–backward splitting algorithms with momentum: weak, linear and strong convergence results. J. Optim. Theory Appl. 201(3), 1364-1397 (2024) 2, 3, 4, 5, 5.1

    work page 2024

  32. [33]

    S., Shehu, Y ., Subgradient extragradient method with double inertial steps for varia- tional inequalities

    Yao, Y ., Iyiola, O. S., Shehu, Y ., Subgradient extragradient method with double inertial steps for varia- tional inequalities. J. Sci. Comput. 90, 1-29 (2022) 1

    work page 2022

  33. [34]

    B., Lei, Z

    Wang, Z. B., Lei, Z. Y ., Long, X., Chen, Z. Y . A modified Tseng splitting method with double inertial steps for solving monotone inclusion problems. J. Sci. Comput. 96(3), 92 (2023) 1, 2, 3, 4, 5, 5, 5.1

    work page 2023

  34. [35]

    Optimization 67, 1197- 1209 (2018) 1, 2, 3, 5, 5.1

    Zhang, C., Wang,Y ., Proximal algorithm for solving monotone variational inclusion. Optimization 67, 1197- 1209 (2018) 1, 2, 3, 5, 5.1

    work page 2018

  35. [36]

    C., Guu, S

    Zeng, L. C., Guu, S. M., ,Yao, J. C., Characterization of H-monotone operators with applications to variational inclusions. Comput. Math. Appl. 50(3-4), 329-337 (2005) 1

    work page 2005