Convergence Analysis of the Self-Adaptive Projection Method for Variational Inequalities with Non-Lipschitz Continuous Operators

Hongwei Liu; Jun Yang; Meiying Wang

REVIEW 3 major objections 6 minor 1 cited by

Convergence Analysis of the Self-Adaptive Projection Method for Variational Inequalities with Non-Lipschitz Continuous Operators

T0 review · 3 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper proves that a Tseng-type extragradient method with a self-adaptive stepsize converges for variational inequalities whose operator is quasimonotone and uniformly continuous, dropping the usual Lipschitz assumption.

desk verdict Worthwhile algorithm paper whose main theorem has a gap in the positive-λ case, but the gap is a short fix and the λ_n→0 analysis is genuinely new. read the letter →

arxiv 2506.07349 v1 pith:SCIMAP57 submitted 2025-06-09 math.OC

Meiying Wang , Hongwei Liu , Jun Yang This is my paper

classification math.OC MSC 47H0547J2047J2590C25

keywords variationalinequalityquasimonotoneoperatornon-Lipschitzcontinuousself-adaptivestepsizeTsengextragradientmethodstrongconvergenceHilbertspaceprojection

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 establishes convergence for a projection-based iterative method that solves variational inequalities of the form: find $x^* \in C$ such that $\langle F(x^*), w - x^*\rangle \ge 0$ for all $w \in C$, where $F$ is quasimonotone and uniformly continuous but need not be Lipschitz. The method combines the extragradient update with a self-adaptive stepsize, so it avoids both the Lipschitz constant and the linesearch procedures used in earlier non-Lipschitz methods. In finite-dimensional spaces, the generated sequence converges to a solution under four mild assumptions. In infinite-dimensional Hilbert spaces, strong convergence is also obtained provided an additional ratio condition holds. The significance is practical as well as theoretical: without Lipschitz control or linesearch, each iteration needs only one projection and two operator evaluations.

What carries the argument

The central object is the self-adaptive stepsize rule $\lambda_{n+1} = \min\{\mu\|z_n - w_n\|/\|F(z_n) - F(w_n)\|, \lambda_n + \xi_n\}$ when $F$ changes, together with the extragradient correction $z_{n+1} = w_n - \lambda_n(F(w_n) - F(z_n))$. This rule keeps the stepsize bounded and convergent even without a Lipschitz constant, and the correction step contracts distances toward the dual solution set $S_D$, the set of points $z \in C$ with $\langle F(v), v - z\rangle \ge 0$ for all $v \in C$. The proof also leans on the fact that uniform continuity is almost Lipschitz, which controls the difference $F(z_n) - F(w_n)$ when $\lambda_n$ shrinks to zero.

What would settle it

A concrete test of the finite-dimensional theorem is to run Algorithm 3.1 on the quasimonotone Lipschitz example $C = [0,1]$, $F(w) = w(w-1)$, from initial points near $0$: the point $0$ is a solution with $F(0) = 0$ and $0 \notin S_D$, so the paper's nondegeneracy assumption is violated, and whichever limit the orbit picks reveals whether that assumption is essential. For the infinite-dimensional theorem, compute the liminf in (A5) along the generated sequence $\{w_n\}$; if it is nonpositive while all other assumptions hold and the orbit still converges strongly, then (A5) is stronger than necessary.

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Extended reading notes

Core claim

The central claim is that the self-adaptive extragradient algorithm converges for variational inequalities with quasimonotone, uniformly continuous operators. Under assumptions (A1)-(A4) and the nondegeneracy condition $F(z) \neq 0$ for any $z \in S \setminus S_D$, the sequence generated by Algorithm 3.1 converges to a solution of $VI(C,F)$ in finite dimensions; under the additional condition (A5), the convergence is strong in infinite-dimensional Hilbert spaces. The proof handles the delicate case where the stepsize $\lambda_n$ tends to zero, using uniform continuity to emulate a Lipschitz bound up to an arbitrarily small error. This removes the Lipschitz requirement that underpins earlier self-adaptive projection methods.

Load-bearing premise

The load-bearing premise is the extra ratio condition (A5): once the iterates converge weakly to a dual solution, the operator must push back strongly enough, with the inner product against the displacement exceeding a positive multiple of the distance to the power $2+\epsilon$; this condition is not implied by quasimonotonicity and must be verified separately for each application.

Editorial extensions

If this is right

Finite-dimensional convergence now applies to a strictly larger class of variational inequalities: quasimonotone and uniformly continuous operators, not just Lipschitz or pseudomonotone ones.
The algorithm avoids linesearch, so each iteration costs one projection and two function evaluations, making it cheaper per iteration than linesearch-based predecessors.
In infinite-dimensional Hilbert spaces, strong convergence holds whenever the operator satisfies the ratio condition (A5), which includes $\eta$-strongly pseudomonotone operators with $\epsilon = 0$.
The stepsize sequence $\lambda_n$ remains bounded and convergent even when the operator is non-Lipschitz, and the stopping criterion $\|z_n - w_n\|/\lambda_n < \epsilon$ is implemented without an extra projection.
The proof template suggests that other projection methods can drop Lipschitz assumptions by pairing adaptive steps with uniform-continuity estimates.

Reading between the lines

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

A practical consequence not discussed in the paper is that the hardest regime is a stepsize $\lambda_n$ decaying to zero; a restart or averaging strategy designed to keep the stepsize bounded away from zero could improve observed performance.
Condition (A5) behaves like a quantitative angle or error-bound condition, and connecting it to Hölderian error bounds might yield simpler verifiable sufficient conditions for strong convergence.
The numerical example in the paper appears to satisfy (A5) because the operator is strongly pseudomonotone near the solution; checking the displayed liminf explicitly for that example would show when the strong-convergence theorem is active.
The dual solution set $S_D$ plays a role analogous to a gauge of the operator's coercivity direction; one could test whether replacing $S_D$ by a larger set weakens the nondegeneracy assumption in the finite-dimensional theorem.

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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

Here's the short version: this is a worthwhile algorithm paper with a real hole in the main proof as written. The λ_n→0 case is handled carefully and is new; the λ_n→λ>0 case is skipped by referring to the authors' earlier Lipschitz-only paper, and that reference doesn't carry over. The fix is a few lines, but without it the convergence theorems are not proven.

What's actually new: a self-adaptive stepsize for Tseng-type extragradient under quasimonotonicity plus uniform continuity, with no Lipschitz assumption and no line search. The authors properly treat the difficult case λ_n→0, showing that even though the stepsize vanishes, the stopping ratio ||z_n-w_n||/λ_n goes to zero along the decreasing subsequence and cluster points land in SD. That is a genuine extension of Liu-Yang [13].

Soft spots, in order:

1. Positive-λ branch. Lemma 4.2(i),(ii) and Lemma 4.4 say 'same as [13]' for λ_n→λ>0, but [13] assumes Lipschitz. The paper's own stepsize rule gives λ_{n+1} ≤ μ||z_n-w_n||/||F(z_n)-F(w_n)||, so ||F(z_n)-F(w_n)|| ≤ (μ/λ_{n+1})||z_n-w_n|| ~ (μ/λ)||z_n-w_n|| for large n. That dominates the troublesome λ_n^2 term in (4.4). The gap is real but easily closed; still, Theorems 4.1 and 4.2 as written are conditional on an unproved case.

2. Numerical example. The operator F(z)=((||z||+1)/(||z||+θ))z is Lipschitz for every θ>0; the scalar factor has bounded derivative. The claim that it is not Lipschitz, citing [21], is wrong. The experiments therefore do not demonstrate the non-Lipschitz advantage. They do show the self-adaptive rule beats line search on that test problem, which is evidence of efficiency but not of applicability to the advertised class.

3. Minor: Lemma 4.4 asserts 'limsup < +∞' without proof; it's true because a uniformly continuous map is bounded on a bounded set, but a line of justification would help.

Overall: the central idea is sound, the λ_n→0 analysis is a real contribution, and the missing positive-λ proof is a short fix. This deserves peer review; a serious referee should ask for the fix and a corrected example. I'd engage with it once it's properly patched.

Referee Report

3 major / 6 minor

Summary. The paper proposes a Tseng-type extragradient method with a self-adaptive stepsize for solving variational inequalities governed by quasimonotone and uniformly continuous (not necessarily Lipschitz) operators in real Hilbert spaces. The main theoretical results are Theorem 4.1, which asserts convergence to a solution in finite-dimensional spaces under assumptions (A1)-(A4) plus the condition that F does not vanish on S\SD, and Theorem 4.2, which asserts strong convergence in general Hilbert spaces under an additional assumption (A5). The convergence proof is split into the cases λ_n→λ>0 and λ_n→0; the λ_n→0 branch is analyzed in detail, while the positive-λ branch is delegated to results from the authors' earlier Lipschitz-based paper [13]. Numerical experiments compare the method with two linesearch-based methods from [21].

Significance. If the proof gap described below is repaired, the paper would make a useful contribution: it extends a self-adaptive Tseng extragradient scheme to non-Lipschitz uniformly continuous quasimonotone operators, avoiding the per-iteration linesearch used in comparable methods, and it provides both finite-dimensional convergence and a strong-convergence result under an additional growth condition. The λ_n→0 branch of the analysis is largely self-contained and appears sound, and the algorithmic idea is clearly motivated. However, the positive-λ branch of the main convergence theorem is currently not proved in the manuscript, and the numerical example misstates the Lipschitz property of the chosen operator, so the claims as written are not fully supported.

major comments (3)

[§4, Lemma 4.2, Case A] The proof of Lemma 4.2(i)-(ii) in the positive-λ case is dismissed with 'the subsequent proof is the same as that of Lemma 3.2 in [13]', but [13] analyzes the same stepsize rule only for Lipschitz-continuous F. Under assumption (A3), inequality (4.4) contains the term λ_n^2‖F(z_n)-F(w_n)‖^2, which is not controlled by ‖z_n-w_n‖^2 without a Lipschitz bound. A short argument is available: since λ_n→λ>0 and λ_{n+1}≤μ‖z_n-w_n‖/‖F(z_n)-F(w_n)‖ whenever F(z_n)≠F(w_n), for all large n one has ‖F(z_n)-F(w_n)‖≤(μ/λ_{n+1})‖z_n-w_n‖ (and trivially if F(z_n)=F(w_n)), so the coefficient of ‖z_n-w_n‖^2 in (4.4) is eventually 1-μ^2λ_n^2/λ_{n+1}^2, whose limit is 1-μ^2>0. This argument is not supplied, so as written Lemma 4.2(i)-(ii) is unproved in the positive-λ branch, and Theorem 4.1 and Theorem 4.2 inherit the gap.
[§4, Lemma 4.4, positive-λ case] The same delegation problem occurs in Lemma 4.4: the claim that all weak cluster points u* satisfy u*∈SD or F(u*)=0 is justified by saying 'As shown in the proof of Lemma 3.3 in [13]', but that proof is in the Lipschitz setting. The transfer to uniform continuity needs an explicit argument using the eventual lower bound on λ_n and the stepsize rule to obtain a local Lipschitz-type estimate along the tail of the sequence. Without this, the statement of Lemma 4.4 is not proved in the λ_n→λ>0 case, and the subsequent use of Lemma 4.4 in Theorem 4.1 and Theorem 4.2 is unsupported.
[§5, Example 4.1] The claim that F(z)=((‖z‖+1)/(‖z‖+θ))z with θ>0 'does not satisfy Lipschitz continuity on H' is incorrect for θ>0. With φ(r)=(r+1)/(r+θ) on [0,∞), both φ and φ' are bounded, so the Fréchet derivative of F has bounded norm on H; hence F is Lipschitz continuous on H (and certainly on the bounded set C). The numerical experiments therefore do not test the non-Lipschitz regime that the paper advertises. The example should be replaced by a genuinely non-Lipschitz uniformly continuous operator, or the claim should be corrected.

minor comments (6)

[§4, Lemma 4.1] The proof of Lemma 4.1 is a direct citation to Lemma 3.1 of [13]; since the stepsize rule is identical this is acceptable, but a short self-contained proof would make the paper more readable.
[§4, Theorem 4.2, Case 1] The text says 'By Lemma 4.2 (ii) and (iii)', but Lemma 4.2 has only items (i) and (ii); the intended reference is probably Lemma 4.1(iii) for the summability of the negative parts of the stepsize increments.
[§3, assumption (A5)] Assumption (A5) should state explicitly that w_n≠u for all sufficiently large n, since the denominator ‖w_n-u‖^(2+ε) is undefined when w_n=u.
[§4, Lemma 4.2, Case B] In the sentence introducing N with λ_n<μ/(2M), the constant M from Lemma 2.1 should be chosen positive; Lemma 2.1 as stated allows M=0, which would make the expression μ/(2M) undefined.
[§4, Lemma 4.3 and Remark 4.2] Several limits are written as 'lim_{n→∞}' when the index is actually k→∞ along the subsequence n_k; for example, Lemma 4.3(i) should read 'lim_{k→∞} ‖z_{n_k}-w_{n_k}‖/λ_{n_k}=0'.
[§5, Example 4.1] The example is first posed in l2 and then implemented in R^m with a box constraint; the relation between these two settings should be described as a finite-dimensional truncation of the infinite-dimensional problem rather than as a separate problem.

Circularity Check

2 steps flagged · score 4.0 of 10

Positive-lambda branch of Lemmas 4.2 and 4.4 is delegated to the authors' prior Lipschitz-only paper [13] without a transfer proof to uniform continuity.

self citation load bearing [Section 4, Lemma 4.2, Case A]
"For clarity, the proof is organized into two parts based on different cases. Case A Suppose λn → λ > 0 as n → ∞. Then the subsequent proof is the same as that of Lemma 3.2 in [13]. Consequently, (i) and (ii) hold."

Lemma 4.2 supplies the existence of lim ||z_n - x|| and ||z_n - w_n|| -> 0 used by Theorems 4.1 and 4.2. In the positive-lambda case the proof is not written; it is referred to Lemma 3.2 of [13]. That reference (Liu, Yang, Comput. Optim. Appl. 2020) is by two of the present authors, and the paper itself (Remark 3.1 and the discussion around (1.3)) states that [13] treats Lipschitz continuous F. The present assumptions (A1)-(A4) give only quasimonotonicity and uniform continuity. The term lambda_n^2||F(z_n)-F(w_n)||^2 in (4.4) is precisely the term requiring control; no estimate is given showing how the positive-lambda branch closes under (A2)-(A4). Thus a main branch of the convergence proof rests on an unverified self-citation instead of a demonstrated derivation.
self citation load bearing [Section 4, Lemma 4.4]
"If λn → λ > 0 as n → ∞, then lim n→∞ λnk → λ > 0. As shown in the proof of Lemma 3.3 in [13], it follows that all weak cluster points u∗ of {zn} satisfy u∗ ∈ SD or F (u∗) = 0."

Lemma 4.4 is the key result converting weak cluster points into solutions of the dual VI (or zeros of F), and Theorem 4.1 and Theorem 4.2 both invoke it to conclude z_l ∈ SD. Its positive-lambda branch is asserted to follow from Lemma 3.3 of [13], the same authors' earlier Lipschitz-based analysis. Since the paper's advertised novelty is removing the Lipschitz assumption, citing [13] for this branch transfers an unproved case into the central strong-convergence theorem. The lambda -> 0 branch is proved in the text, so the circularity is partial, but the positive-lambda branch is load-bearing and not independently established.

full rationale

The manuscript is not definitionally circular: no quantity is defined in terms of the result it is supposed to establish, no fitted parameter is renamed as a prediction, and the lambda -> 0 branch of the proof is a genuine non-Lipschitz argument. The circularity concern is concentrated in the positive-lambda case. Lemma 4.2 Case A and Lemma 4.4 both delegate the proof to [13], a paper authored by two of the present authors (Liu and Yang) that, according to the manuscript's own Remark 3.1, analyzes the same self-adaptive Tseng method under Lipschitz continuity. The current paper's assumptions (A2)-(A3) replace Lipschitz continuity by quasimonotonicity plus uniform continuity, and no transfer argument is supplied for the branch lambda_n -> lambda > 0. Because Theorems 4.1 and 4.2 use Lemma 4.4 (and Lemma 4.2) in that branch, the central convergence claims inherit this self-citation. Weighing this: the lambda -> 0 case contains substantial independent content, so the paper is not wholly circular; but the positive-lambda branch is load-bearing and is neither proved nor shown to follow from the cited result under the weaker assumptions. This warrants a score of 4 rather than 0-2 or 6+. The numerical experiments are comparisons against independent algorithms and do not enter the circularity chain.

Assumptions & free parameters 0 free parameters · 9 assumptions · 0 invented entities

No parameters are fitted to data; the algorithm parameters (mu, lambda_1, xi_n) are free hyperparameters for which convergence is proven for any admissible value. The central claim rests on the domain assumptions (A1)-(A5) and on standard results cited from prior literature. The most distinctive assumption is (A5), which is introduced specifically to obtain strong convergence in infinite dimensions.

assumptions (9)

domain assumption Dual solution set SD is nonempty (A1).
Invoked in Lemma 4.2 to define a point x* in SD that makes the sequence ||z_n - x*|| monotone up to a summable error. SD may be empty for quasimonotone operators even when S is nonempty, so this is a real restriction.
domain assumption F is quasimonotone (A2).
Used in Lemma 4.4 to pass from a positive inner product at w_n to a nonnegative inner product at q, and in the definition of SD.
domain assumption F is uniformly continuous on H (A3).
Used throughout the proof, especially with Lemma 2.1 to get the approximate Lipschitz inequality (4.5) in the lambda_n -> 0 case.
domain assumption Weak sequential lower semicontinuity of ||F||: if z_n weakly converges to z then ||F(z)|| <= liminf ||F(z_n)|| (A4).
Used in Lemma 4.4 Case I to conclude F(u*)=0 when ||F(w_nk)|| -> 0.
ad hoc to paper There exists epsilon >= 0 such that liminf <F(w_n), w_n - u> / ||w_n - u||^(2+epsilon) > 0 for w_n weakly converging to u in SD (A5).
Introduced in Section 3 for the strong convergence theorem in infinite-dimensional Hilbert spaces. It is a nonstandard, strong coercivity-like condition that is not implied by the other assumptions.
domain assumption F(z) is nonzero for every z in S \ SD (assumption in Theorems 4.1 and 4.2).
Used to convert the disjunction in Lemma 4.4 (u* in SD or F(u*)=0) into u* in SD, which is needed to apply the Fejer-monotonicity Lemma 4.2.
standard math Lemma 2.1: uniform continuity gives an approximate Lipschitz bound with arbitrary additive epsilon (cited from [25]).
Used in the proof of Lemma 4.2 Case B to bound ||F(z_n) - F(w_n)|| when lambda_n -> 0.
standard math Lemma 2.3: the projection residual is nondecreasing in step size (cited from [27]).
Used in Remark 4.1 and Theorem 4.1 to relate ||e(z,1)|| to ||z_n - w_n||/lambda_n.
standard math Lemma 2.4: quasi-Fejer monotone sequences converge (cited from [28]).
Used to conclude that lim ||z_n - x*|| exists from the inequalities in Lemma 4.2.

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Pith. "Pith review of Convergence Analysis of the Self-Adaptive Projection Method for Variational Inequalities with Non-Lipschitz Continuous Operators." pith.science (2026). https://pith.science/paper/SCIMAP57

@misc{pith2026250607349,
  author       = {Pith},
  title        = {Pith review of: Convergence Analysis of the Self-Adaptive Projection Method for Variational Inequalities with Non-Lipschitz Continuous Operators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SCIMAP57}},
  note         = {Machine review of arXiv:2506.07349}
}

read the original abstract

In this paper, we employ Tseng's extragradient method with the self-adaptive stepsize to solve variational inequality problems involving non-Lipschitz continuous and quasimonotone operators in real Hilbert spaces. The convergence of the proposed method is analyzed under some mild assumptions. The key advantages of the method are that it does not require the operator associated with the variational inequality to be Lipschitz continuous and that it adopts the self-adaptive stepsize. Numerical experiments are also provided to illustrate the effectiveness and superiority of the method.

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

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The paper extends FBF dynamics and a Bregman golden-ratio algorithm to quasimonotone variational inequalities, claiming strong convergence under only uniform continuity, but the proofs require extra assumptions not in...

Reference graph

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [13]

Liu, H., Yang, J.: Weak convergence of iterative methods for so lving quasimono- tone variational inequalities. Comput. Optim. Appl. 77, 491–508 (2020)

work page 2020

[2] [21]

Optimization 43(4), 2207–2226 (2019)

Thong, D.V., Vuong, P.T.: Modiﬁed Tseng’s extragradient method s for solving pseudo-monotone variational inequalities. Optimization 43(4), 2207–2226 (2019)

work page 2019

[3] [1]

Cottle, R.W., Yao, J.C.: Pseudo-monotone complementarity proble ms in Hilbert space. J. Optimiz. Theory App. 75(2), 281–295 (1992)

work page 1992

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work page 2015

[5] [3]

Academic Press, New York (2000)

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work page 2000

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Springer, Berlin (2001)

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Facchinei, F., Pang, J.S.: Finite-dimensional Variational Inequalitie s and Com- plementarity Problems vol. 1-2. Springer, New York (2003)

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