Second-order dynamical systems with a smoothing effect for solving paramonotone variational inequalities

Pham Viet Hai; Trinh Ngoc Hai

REVIEW 3 major objections 4 minor 25 references

Second-order dynamical systems with a smoothing effect for solving paramonotone variational inequalities

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper proves that a second-order dynamical system with a smoothed projection term converges strongly to a solution of a paramonotone variational inequality, and that its time discretization yields a convergent inertial…

desk verdict A genuinely new second-order smoothing projection system for paramonotone VIs, with a coherent Lyapunov proof, but the continuous-time convergence theorem is conditional on an unproved global-existence assumption for the non-Lipschitz case. read the letter →

arxiv 2411.14651 v1 pith:7XCY6TVC submitted 2024-11-22 math.OC

Pham Viet Hai , Trinh Ngoc Hai This is my paper · ORCID

classification math.OC MSC 47H0565K1590C25

keywords paramonotonevariationalinequalitiessecond-orderdynamicalsystemsinertialprojectionalgorithmsmoothingeffectnon-Lipschitzoperatorstrajectoryconvergencetimediscretizationacceleratedalgorithms

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper aims to extend second-order inertial dynamics from strongly monotone, Lipschitz problems to variational inequalities governed by paramonotone, merely continuous operators in finite-dimensional spaces, a class where few methods exist. The proposed system adds a damping term $\lambda(t)x'(t)$ inside the projection step to smooth the trajectory, and the paper proves that, under coefficient conditions, the trajectory $x(t)$ stays in the feasible set and converges to a solution of the variational inequality. A time discretization of the same system yields an inertial projection-type iteration whose convergence the paper also establishes. If the results hold, second-order acceleration methods become available for non-Lipschitz monotone problems, and the discretized algorithm is reported to outperform the direct projection method in numerical tests.

What carries the argument

The load-bearing object is the smoothed projected point $y(t) = P_\Omega\bigl(x(t) + \lambda(t)x'(t) - \tfrac{\alpha_0(t)}{\max\{1,\|U(x(t)+\lambda(t)x'(t))\|\}}U(x(t)+\lambda(t)x'(t))\bigr)$, which acts as the target of the second-order forcing term. The smoothing term $\lambda(t)x'(t)$ inside the projection annihilates oscillations and permits a factorization of the second-order equation into two coupled first-order systems, which is why the strict coefficient condition (2.4) appears and why the trajectory can be shown to remain in $\Omega$. The convergence proof then runs on the energy identities for $v(t)=\tfrac{1}{2}\|x(t)-x^\star\|^2$ and $b(t)=\tfrac{1}{2}\|x'(t)\|^2$, with the integrability conditions on $\delta(t)\alpha_0(t)$ and $\delta(t)\alpha_0(t)^2$ forcing the key monotonicity gap $\langle U(x+\lambda x'), x+\lambda x' - x^\star\rangle$ to vanish along a subsequence; paramonotonicity converts that subsequential limit into a solution.

What would settle it

A concrete check is to run the discretized algorithm (2.10) on the paper's own test problem, with $\Omega$ the unit ball in $\mathbb{R}^3$ and $U(x)=Ax$ for the $3\times 3$ matrix given in Section 5, using parameter families $\beta_0(n)=(n+\omega)^{-q}$, $\beta_1(n)=1+\delta(n+\omega)^{-p}$, $\xi(n)=(n+\omega)^{-p}$, $\eta(n)=-\theta(n+\omega)^{-\lambda}$ that satisfy conditions (4.21)-(4.26); if the residual $\|w(n)-z(n)\|$ fails to tend to zero for any admissible parameter choice, Theorem 4.3 would be contradicted.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem 3.4: for a closed convex feasible set $\Omega$, a continuous paramonotone operator $U$ with nonempty solution set, and damping and step parameters obeying Assumption 3.1, every global strong solution of the second-order system (2.1) satisfies $\|x''(t)\| \in L^2$, $x'(t) \to 0$, and $x(t) \to x^\star \in \mathrm{Sol}(U,\Omega)$. The corresponding discrete inertial algorithm (2.10) likewise converges to a solution under Assumption 4.1. The smoothing term $\lambda(t)x'(t)$ inside the projection is what lets the analysis proceed without Lipschitz continuity or strong monotonicity: it controls oscillations in the error and, together with the coefficient condition $\delta(t) < \tfrac{1}{4}(\alpha_1(t)^2 + 2\alpha_1'(t))$, keeps the trajectory inside the feasible set. The proof works through Lyapunov-type estimates on $v = \tfrac{1}{2}\|x-x^\star\|^2$ and $b = \tfrac{1}{2}\|x'\|^2$, and uses paramonotonicity to identify any cluster point as a solution.

Load-bearing premise

The load-bearing premise is that the second-order system admits a global strong solution for all $t \ge t_0$; the paper proves this only for Lipschitz operators, so for the target class of merely continuous paramonotone operators the convergence statement is conditional on a solution that is assumed rather than shown to exist.

Editorial extensions

If this is right

Second-order inertial dynamics are shown to be compatible with variational inequalities whose operators are merely continuous and paramonotone, removing the Lipschitz and strong-monotonicity assumptions used by earlier second-order schemes.
A practical inertial projection algorithm is obtained by time discretization, with a wider admissible exponent range for the step size than the direct method ($q \in (0,1)$ versus $\tau \in (0.5,1]$).
Under coefficient condition (2.4), trajectories remain in the feasible set $\Omega$ for all time, an invariance property that the proof needs and that is of independent interest.
Explicit parameter choices in Propositions 3.5 and 3.6 give ready-to-use coefficient schedules, such as $\alpha_0(t)=(t+1)^{-q}$, $\alpha_1(t)=h+(t+1)^{-s}$, and $\delta(t)=(t+1)^{-p}$, for which convergence is guaranteed.
In the reported numerical experiments, the new inertial algorithm converges faster than the direct paramonotone projection method, reaching the same residual in fewer iterations.

Reading between the lines

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

The paper leaves open whether the second-order ODE actually has a global strong solution when $U$ is merely continuous; closing this existence gap would turn the conditional convergence theorem into an unconditional one, and a blow-up example would delineate exactly where the theory stops.
Because the proof only needs the monotonicity gap to vanish along a subsequence, the same smoothing construction may extend to pseudomonotone or quasimonotone operators, provided a substitute for the paramonotone characterization lemma is available.
A natural testable extension is to quantify convergence rates under stronger modulus conditions such as weak sharpness or strong monotonicity, since the paper establishes asymptotic convergence but no rate.
Applying the same smoothing-and-projection idea in stochastic or online settings, where exact projections are replaced by noisy ones, would be a direct algorithmic outgrowth of this paper's construction.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

The genuinely new thing here is the system (2.1): a second-order damped projection method with a smoothing term λ(t)x'(t) inside the projection, analyzed for paramonotone, continuous, not necessarily Lipschitz operators. That combination is not in Hai's first-order system, Vuong's second-order system, or Alecsa et al.'s smoothing analysis. The continuous-time convergence proof (Theorem 3.4) is a standard but careful Lyapunov argument, and Lemma 2.9 on invariance of Ω is a nice piece of work. The discrete counterpart (Theorem 4.3) is also coherent, and the numerical experiments, while simple, support the claims.

The soft spot is the one the stress-test flags. Theorem 3.4 assumes global existence of the trajectory, while Proposition 2.7 proves existence and uniqueness only for Lipschitz U, and even that proof is omitted ("similar to [25]"). Since the paper's advertised target is non-Lipschitz U, the main convergence result is conditional on an unproved premise. I think the gap is closable: the vector field is continuous, and the normalized operator U/max{1,‖U‖} is bounded by 1, so a Peano existence step plus a linear-growth/Gronwall bound should give global existence under mild integrability on the coefficients. But as written, it's missing. Also, the vector field is only defined for arguments in Ω, so local existence needs a short argument that the solution stays in the domain; the stress-test note worries about this, and that worry is legitimate.

Two smaller issues. Proposition 3.6's statement says conditions (3.25)–(3.28) but the list is actually (3.29)–(3.31); that's a labeling typo, easy to fix. And the proof of Proposition 2.7 should be included or at least carefully cited, since the existence claim carries real weight here.

Overall, the mathematics is plausible and the paper does real work. It is not a desk reject. It needs a revision that either proves global existence for continuous U or states the continuous-time result in a form that makes the existence assumption explicit and discusses when it holds. Then it would be a solid contribution.

This is for researchers working on continuous-time methods for monotone variational inequalities and on inertial projection algorithms. A serious referee should engage with it; I'd send it to review.

Referee Report

3 major / 4 minor

Summary. The paper proposes a second-order dynamical system with a smoothing term for solving paramonotone variational inequalities without Lipschitz continuity, proves convergence of trajectories under a set of integrability and parameter conditions, derives a discrete inertial projection algorithm by time discretization, proves its convergence, and presents numerical experiments. The main continuous-time result, Theorem 3.4, shows that if the system admits a global strong solution, then the trajectory converges to a solution of the variational inequality. The discrete analogue, Theorem 4.3, establishes convergence of the inertial projection iteration under Assumption 4.1.

Significance. If fully established, the paper would extend second-order dynamical-system methods to paramonotone non-Lipschitz variational inequalities, a class for which few algorithms exist, and it would provide a corresponding inertial projection algorithm with convergence guarantees. The Lyapunov analysis in the continuous-time proof is detailed, the discrete convergence proof is coherent, and the parameter examples in Propositions 3.5, 3.6, and 4.4 give concrete admissible choices. However, the central continuous-time theorem is conditional on a global-existence assumption that is not established for the advertised non-Lipschitz class, and the well-posedness of the vector field is not fully justified. The discrete part is more self-contained and appears sound, but the overall contribution is currently limited by the continuous-time existence gap.

major comments (3)

[Section 3, Theorem 3.4; Section 2, Proposition 2.7] Theorem 3.4 assumes that the dynamical system (2.1) admits a global solution, but Proposition 2.7 establishes existence and uniqueness only when U is Lipschitz continuous, and its proof is omitted with the remark 'similar to [25]'. Since Assumption 2.1 assumes only continuity and paramonotonicity, the theorem as stated has no guaranteed applicability to the paper's advertised non-Lipschitz case. The a priori estimates in the proof suggest that a global-existence result may be obtainable, but it must be proved explicitly or the theorem must be restricted to a class for which global existence is known.
[Section 2, equations (2.1) and (2.2); Remark 3.3(2)] The vector field in (2.2) is defined only when x(t)+λ(t)x'(t) belongs to Ω, because U is not extended outside Ω. The invariance of x(t)+λ(t)x'(t) in Ω is established in Remark 3.3(2) only after a solution is assumed and under condition (3.9). Consequently, even local existence cannot be obtained directly from a standard Cauchy problem without knowing that the argument of U remains in Ω. The manuscript should either specify an extension of U to the whole space or prove a local existence result together with the invariance property on the existence interval.
[Theorem 3.4, Step 2] In Step 2 of the proof of Theorem 3.4, the inequality M1 ≥ v'(s) + (C1/2)v(s) does not by itself imply that v is bounded, because v' may be negative with large magnitude while the sum remains bounded. To conclude boundedness, the proof must additionally use the bound |v'(s)| ≤ 2‖x'(s)‖‖x(s)-x*‖ together with the already established b∈L∞, or provide an analogous Gronwall-type argument. Please add the missing step.

minor comments (4)

[Definition 2.5(1)] In item (1), 'The functions x, x', x'' : [t0,∞) → H is locally absolutely continuous' should read 'are locally absolutely continuous'.
[Equation (2.1)] The initial condition x'(t0) = (1/4)α1(t0)(x1 - x0) is stated as belonging to Ω, but x1 - x0 need not be an element of Ω; this condition should be stated as an element of H (or R^d).
[Proposition 3.6] Proposition 3.6 states that convergence holds 'if conditions (3.25)-(3.28) hold', but the listed conditions are numbered (3.29)-(3.31); the cross-reference should be corrected.
[Abstract and text] Several grammatical and typographical errors remain, such as 'the trajectories of this dynamical system converges' in the abstract; a careful proofreading pass is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation found; the convergence proof derives the limit from the stated assumptions, and the unproved global-existence premise is an incompleteness risk, not a circular step.

full rationale

The paper's central claim, Theorem 3.4(iii), is a conditional convergence result: under Assumptions 2.1 and 3.1, and assuming the dynamical system (2.1) admits a global solution, the trajectory converges to a solution. The convergence proof is a forward derivation from Lyapunov inequalities (Lemma 3.1, Step 1, Step 2, Step 3) and the parameter conditions in Assumption 3.1; it does not use the conclusion as an input. The damping term is introduced as a design choice motivated by reference [2], not as a derived consequence, so no ansatz is smuggled in via citation. The only self-citation of possible note is Lemma 2.2, cited to [16] by one of the authors, but it is a standard, independently verifiable property of paramonotone operators and is used only as an auxiliary identification step, not as the source of the convergence mechanism. The main caveat is that Proposition 2.7 proves global existence only for Lipschitz U and with a proof omitted as 'similar to [25]', while Theorem 3.4 explicitly assumes global existence for the non-Lipschitz case; this is a genuine gap in justification, but it is not circular reasoning because the theorem is stated as a conditional statement. No fitted parameter is relabeled as a prediction, no known result is merely renamed, and no uniqueness or existence theorem is imported from the authors' prior work to force the conclusion.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The convergence theorems rest on the standard assumptions of the VI problem (closed convex Ω, nonempty solution set, paramonotone continuous U), on the existence of a global solution to the ODE (assumed, not proved for non-Lipschitz U), and on standard Lyapunov lemmas from Abbas, Attouch, and Svaiter [1]. No invented entities are introduced. The coefficient functions are user-chosen inputs constrained by the inequalities in Assumptions 3.1 and 4.1; they are not fitted to data.

free parameters (1)

coefficient functions α0, α1, δ, λ
User-chosen time-dependent step and damping functions. Convergence is conditional on them satisfying Assumption 3.1 (e.g., (2.4), (3.5), (3.11), (3.12)); the paper gives admissible parameter families in Propositions 3.5, 3.6, and 4.4 but does not fit them to data.

assumptions (6)

domain assumption Sol(U,Ω) is nonempty
Assumption 2.1(ii); without a target solution the convergence statement is vacuous.
domain assumption U is paramonotone and continuous on Ω
Assumption 2.1(iii); paramonotonicity is used via Lemma 2.2 to identify limit points as solutions, and continuity is used to pass limits through U.
domain assumption Ω is a nonempty, closed, convex subset of R^d
Assumption 2.1(i); needed for projection and convex combination arguments in Lemmas 2.8 and 2.9.
domain assumption A global strong solution x(t) to (2.1) exists
Theorem 3.4 assumes existence; Proposition 2.7 proves it only for Lipschitz U and its proof is omitted ('similar to [25]'). For continuous non-Lipschitz U, global existence is not established in the paper.
standard math Lyapunov lemmas (Lemmas 2.3 and 2.4) from Abbas, Attouch, and Svaiter
Used in Theorem 3.4 steps (ii) and (iii) to infer limits from integrable derivative bounds; accepted background results cited from [1].
standard math Existence of γ and μ in Lemma 2.9 via Picard-Lindelöf and comparison
The proof of Lemma 2.9 relies on Picard-Lindelöf for local existence of the Riccati equation (2.6) and an argument excluding finite blow-up; this is standard but the write-up contains a garbled line ('γ(t) and µ(t) ≥ t0').

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Cite this review

Pith. "Pith review of Second-order dynamical systems with a smoothing effect for solving paramonotone variational inequalities." pith.science (2026). https://pith.science/paper/7XCY6TVC

@misc{pith2026241114651,
  author       = {Pith},
  title        = {Pith review of: Second-order dynamical systems with a smoothing effect for solving paramonotone variational inequalities},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7XCY6TVC}},
  note         = {Machine review of arXiv:2411.14651}
}

read the original abstract

In this paper, we propose a second-order dynamical system with a smoothing effect for solving paramonotone variational inequalities. Under standard assumptions, we prove that the trajectories of this dynamical system converges to a solution of the variational inequality problem. A time discretization of this dynamical system provides an iterative inertial projection-type method. Our result generalizes and improves the existing results. Some numerical examples are given to confirm the theoretical results and illustrate the effectiveness of the proposed algorithms.

Figures

Figures reproduced from arXiv: 2411.14651 by the authors.

**Figure 1.** Performance of Algorithm (2.1) with different parameter • In Algorithm (2.8), we choose the parameters as in Proposition 4.4 with different p, q, δ, θ, λ. Set ω = max e ln(δ+1) p , e ln θ λ + 1 and z(0) = (1; 0; 0)T , z(1) = (0; 1; 0)T . The results are presented in [PITH_FULL_IMAGE:figures/full_fig_p020_1.png] view at source ↗

**Figure 2.** Performance of Algorithm (2.8) with different parameters 0 10 20 30 40 50 60 70 80 90 100 0 0.5 1 1.5 Iterations ||z(k)−P C[z(k)−Uz(k)]|| Our algorithm with δ=0.9; θ=0.9; λ=0.9 Our algorithm with δ=1; θ=1; λ=1 Our algorithm with δ=1.5; θ=1.5; λ=1.5 Direct method with τ=1 Direct method with τ=0.7 Direct method with τ=0.9 [PITH_FULL_IMAGE:figures/full_fig_p021_2.png] view at source ↗

**Figure 3.** Comparison results of Algorithm (2.8) and Dir.Method 6. Conclusion We have proposed a second order dynamical system for solving paramonotone VIs in a finite-dimensional space and studied its global convergence. A discretization of this dynamical system gives rise to an inertial iterative projection-type algorithm for which we establish the convergence of the iterations. Some numerical examples are given to confirm … view at source ↗

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [25]

Vuong: A second order dynamical system and its disc retization for strongly pseudo-monotone variational inequalities

P.T. Vuong: A second order dynamical system and its disc retization for strongly pseudo-monotone variational inequalities. SIAM J. Control Optim., 59(4), 2 875-2897 (2021) 23 F aculty of Ma thema tics and Informa tics, Hanoi University of Science and Technology, 1 Dai Co Viet, Hanoi, Vietnam. Email address : hai.phamviet@hust.edu.vn; hai.trinhngoc@hust.edu.vn

work page 2021

[2] [1]

Abbas, H

B. Abbas, H. Attouch, Benar F. Svaiter: Newton-like dyna mics and forward-backward methods for structured monotone inclusions in Hilbert spaces. J. Optim . Theory Appl., 161, 331-360 (2014)

work page 2014

[3] [2]

Alecsa, S.C

C.D. Alecsa, S.C. Laszlo, T. Pinta: An Extension of the Se cond Order Dynamical System that Models Nesterov’s Convex Gradient Method. Appl. Math Optim., 84, 1 687-1716 (2021)

work page 2021

[4] [3]

Anh, N.T

P.K. Anh, N.T. Vinh: Self-adaptive gradient projection algorithms for variational inequalities involving non-Lipschitz continuous operators. Numer. Algorithms, 8 1, 983-1001 (2019)

work page 2019

[5] [4]

Attouch, F

H. Attouch, F. Alvarez: The heavy ball with friction dyna mical system for convex constrained mini- mization problems, in: Optimization (Namur, 1998), Lectur e Notes in Economics and Mathematical Systems 481, Springer, Berlin, 25-35, (2000)

work page 2000

[6] [5]

Attouch, Z

H. Attouch, Z. Chbani, H. Riahi: Combining fast inertial dynamics for convex optimization with Tikhonov regularization. J. Math. Anal. Appl., 457, 1065-1 094 (2018)

work page 2018

[7] [6]

Attouch, Z

H. Attouch, Z. Chbani, H. Riahi: Rate of convergence of th e Nesterov accelerated gradient method in the subcritical case α ≤ 3. ESAIM, Control Optim. Calc. Var., 25, 1292-8119 (2019)

work page 2019

[8] [7]

Bao, P.Q

T.Q. Bao, P.Q. Khanh: A projection-type algorithm for ps eudomonotone nonlipschitzian multi-valued variational inequalities. Nonconvex Optim. Appl., 77, 113 -129 (2005)

work page 2005