Pith. sign in

REVIEW 4 major objections 6 minor 36 references

Strong Convergence of Forward-Backward-Forward Methods for Pseudo-monotone Variational Inequalities with Applications to Dynamic User Equilibrium in Traffic Networks

T0 review · 4 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read For pseudo-monotone variational inequalities in Hilbert spaces, a cheaply modified forward-backward-forward algorithm converges strongly to the minimal-norm solution, and an adaptive step-size variant does so without knowledge of the…

desk verdict Anchored Tseng forward-backward-forward for pseudo-monotone VIs is a clean and genuinely new combination, but Lemma 3.6 has a load-bearing gap—pseudo-monotonicity is applied at points outside X—so the main theorem needs a stronger assumption or a repaired proof. read the letter →

arxiv 1908.07211 v2 pith:T2TBD3OH submitted 2019-08-20 math.OC

Benoit Duvocelle , Dennis Meier , Mathias Staudigl , Phan Tu Vuong This is my paper · ORCID
classification math.OC MSC 47J2047J2549J4065K1590C33
keywords variationalinequalitiespseudo-monotonemapsforward-backward-forwardmethodstrongconvergenceminimal-normsolutionadaptivestep-sizedynamicuserequilibriumtrafficnetworks
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 claims that adding one convex-combination step to the forward-backward-forward algorithm turns weak convergence into strong convergence for variational inequalities defined by a pseudo-monotone map on a Hilbert space. The modification is computationally cheap: the iteration still uses only one projection per step, and the same guarantee holds for an adaptive step-size version that does not need to know the Lipschitz constant. A sympathetic reader would care because strong convergence is stable under discretization in infinite-dimensional settings, while pseudo-monotonicity is the realistic assumption for dynamic user equilibrium in traffic networks, where strong monotonicity is known to fail. The paper shows the method working on two standard traffic networks, reporting that it matches or beats the projected-gradient baseline while providing stronger guarantees.

What carries the argument

The central object is the augmented forward-backward-forward iteration. In the base step, z_k is the projection of x_k - gamma F(x_k) onto X, and r_k = z_k + gamma(F(x_k)-F(z_k)) is the forward-backward-forward correction; the new point x_{k+1} is then the convex combination (1-alpha_k-beta_k)x_k + beta_k r_k. Relative to the target p, this update carries a small pull term -alpha_k p, and the conditions on alpha_k and beta_k make that pull vanish slowly enough to force strong convergence while preserving the weak-convergence structure inherited from the base method. The proof chains three ingredients: a basic recursion (Lemma 3.3) that contracts ||r_k - x*|| by the gap (1-(gamma L)^2)||x_k - z_k||^2, a quasi-Fejér inequality (Lemma 3.5) that converts that gap into a perturbed recursion for ||x_{k+1} - p||^2, and a cluster-point lemma (Lemma 3.6) that uses pseudo-monotonicity plus weak-to-weak continuity to identify weak limits with vanishing gap as solutions. In the adaptive variant, the same contraction holds with (1 - $gamma_k^{2}$ $rho^{2}$ / gamma_{k+1}^2) in place of (1-(gamma L)^2).

What would settle it

Search for a bounded, Lipschitz, weak-to-weak continuous map F that is pseudo-monotone on X but fails pseudo-monotonicity at every point outside X, with X* nonempty; if running Algorithm 1 with parameters satisfying (3.1) yields a trajectory whose weak cluster point is not in X*, or whose ||x_k - z_k|| does not vanish, then the omitted neighborhood condition is essential and Theorem 3.1 as stated is false. A quicker check: compute the points y + epsilon_j F(z_{N_j})/||F(z_{N_j})||^2 from the proof and test whether the implication defining pseudo-monotonicity holds there; if it fails, the proof step (3.18) has no justification.

Watch

Extended reading notes

Core claim

Under Lipschitz continuity, sequential weak-to-weak continuity, and pseudo-monotonicity of F, with a nonempty closed convex solution set X*, Algorithm 1 produces a sequence (x_k) that converges strongly to p = argmin{||z|| : z in X*}, the minimal-norm solution of VI(X,F). The update is x_{k+1} = (1-alpha_k-beta_k)x_k + beta_k r_k, with z_k = P_X(x_k - gamma F(x_k)) and r_k = z_k + gamma(F(x_k)-F(z_k)), and the parameter sequences satisfy alpha_k -> 0, sum alpha_k = infinity, and beta_k bounded away from 0 and from 1-alpha_k. Theorem 3.2 shows the same strong convergence when the constant step-size is replaced by the recursive rule (3.2), which shrinks gamma_k according to the ratio ||z_k - x_k||/||F(z_k)-F(x_k)||, so no global Lipschitz constant is needed. The proof establishes a perturbed Fejér-type recursion for ||x_k - p||^2 whose perturbation vanishes suitably, then shows that every weak cluster point of the sequence must be a solution, and concludes strong convergence via a classical lemma on perturbed contractions.

Load-bearing premise

The proof's load-bearing premise is that F is pseudo-monotone not only on the feasible set X but on a neighborhood of X or all of H, because Lemma 3.6 applies the pseudo-monotonicity definition to the auxiliary point y + epsilon_j F(z_{N_j})/||F(z_{N_j})||^2, which need not lie in X.

Editorial extensions

If this is right

  • For any pseudo-monotone variational inequality satisfying the paper's assumptions, the minimal-norm solution can now be computed with one projection per iteration and guaranteed strong convergence in the Hilbert-space norm.
  • The adaptive step-size rule removes the need to estimate a global Lipschitz constant, so the method can be applied to black-box operators such as the effective-delay operator in dynamic traffic assignment.
  • Because strong convergence is stable under discretization, the method's iterates from any finite-dimensional approximation will not drift away from the true solution set.
  • The experiments indicate the method is at least competitive with the projected-gradient baseline on the Nguyen and Sioux Falls networks, while carrying stronger theoretical guarantees.
  • Pseudo-monotonicity is strictly weaker than monotonicity, so the method covers equilibrium problems such as dynamic user equilibrium where strong monotonicity provably fails.

Reading between the lines

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

  • The same 'pull toward the origin' augmentation could be grafted onto other single-projection algorithms (for example, extragradient variants or proximal-point methods) to force strong convergence for pseudo-monotone maps, as long as a quasi-Fejér relation with a vanishing gap can be established.
  • A testable refinement would be to replace the unverified neighborhood pseudo-monotonicity with the weaker and checkable condition that F is pseudo-monotone on the convex hull of X and the iterates; if that suffices, the theorem could be proved without extending the assumption off X.
  • In the traffic application, the delay operator is only evaluated at finitely many discretization points through the dynamic network loading subroutine; the strong-convergence guarantee would carry over to the discretized algorithm only if the approximate operators are uniformly pseudo-monotone on the relevant region, which is an empirical question the paper does not settle.
  • Converging to the minimal-norm solution gives a principled selection rule among multiple dynamic user equilibria—the one with least total departure-flow energy—which could matter for policy evaluation beyond the paper's examples.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 6 minor

Summary. This paper proposes a forward-backward-forward algorithm for solving variational inequalities VI(X,F) in real Hilbert spaces under Lipschitz continuity, sequential weak-to-weak continuity, and pseudo-monotonicity of F. The update adds an anchoring extrapolation x_{k+1}=(1-alpha_k-beta_k)x_k+beta_k r_k to the classical Tseng iteration, and the paper claims strong convergence to the minimal-norm solution p=P_{X*}(0). An adaptive step-size variant is also claimed. The final section applies the algorithm to dynamic user equilibrium in traffic networks and reports numerical comparisons on Nguyen and Sioux Falls instances.

Significance. The intended contribution is timely: strong convergence with a single projection per iteration under pseudo-monotonicity would improve over weakly convergent Tseng methods and over strongly convergent methods requiring extra projections. The non-adaptive part of the proof is largely self-contained and uses standard tools (Xu's lemma, quasi-Fejer inequalities, two-case Mange argument), and the numerical study is useful empirical evidence. However, the central lemma that identifies weak cluster points as solutions is not proved under the stated assumptions: it applies pseudo-monotonicity at points outside X and passes to the limit in an inner product of two weakly convergent sequences. The adaptive theorem is only sketched and its displayed estimate is not sufficient. These gaps are load-bearing; the main theorems do not yet follow. The DUE application also does not verify the assumptions.

major comments (4)
  1. [§3.2, Lemma 3.6, Eq. (3.18)] The proof of Lemma 3.6 applies the pseudo-monotonicity implication to the pair (y+epsilon_j u_Nj, z_Nj) immediately before Eq. (3.18). Assumption 2 defines pseudo-monotonicity only for pairs in X, and the point y+epsilon_j u_Nj is never shown to belong to X. This is not a technicality: for X={x in R^2 : x_1 >= 0}, y=0, z_Nj=0, and F(x)=(x_1-1, x_2), the map F is Lipschitz, sequentially weak-to-weak continuous, and monotone (hence pseudo-monotone on X), yet y+epsilon_j u_Nj = (-epsilon_j, 0) is outside X. Therefore the inequality (3.18) is not justified. Since Lemma 3.6 is invoked in both cases of the proof of Theorem 3.1 and the proof of Theorem 3.2 is said to follow analogously, Theorems 3.1 and 3.2 are not established under the stated assumptions. The proof can be repaired by assuming pseudo-monotonicity on a neighborhood of X or on all of H, or by providing a valid argument that keeps the perturbed point inside X.
  2. [§3.2, Lemma 3.6, limit after Eq. (3.18)] Even if (3.18) were valid, the passage to the limit to conclude 0 <= <F(y), y-x_hat> is not justified: the first factor F(y+epsilon_j u_Nj) converges weakly to F(y), and the second factor y+epsilon_j u_Nj - z_Nj converges weakly to y-x_hat, but weak convergence of both factors does not imply convergence of their inner product to the inner product of the limits. A strong convergence of at least one factor (or a different proof structure) is required. Consequently the conclusion x_hat in X* is not obtained from the displayed argument.
  3. [§3.2, Proof of Theorem 3.2] The proof of the adaptive variant is a one-line sketch. The displayed bound ||r_k-x*||^2 <= ||x_k-x*||^2 - (1 - gamma_k^2 rho^2 / gamma_{k+1}^2) ||x_k-z_k||^2 does not establish the analogue of Lemma 3.3, because gamma_{k+1} <= gamma_k implies gamma_k^2 rho^2 / gamma_{k+1}^2 can exceed 1, making the alleged contraction coefficient negative. The adaptive rule (3.2) does not prevent this. Thus the assertion that 'the rest of the proofs follows analogously' is not sufficient; Theorem 3.2 requires a complete proof.
  4. [§4.2-4.3] The DUE application does not verify that the effective delay operator Psi satisfies the hypotheses of Theorems 3.1-3.2, namely Lipschitz continuity, sequential weak-to-weak continuity, pseudo-monotonicity on the feasible set (or on a neighborhood), and nonempty solution set. Section 4.2 states only that Algorithm 3 is equivalent to Algorithm 1 'if' Psi is Lipschitz continuous and pseudomonotone, and the numerics are run without establishing these conditions. As a result, the convergence guarantee does not formally apply to the DUE computations reported.
minor comments (6)
  1. [Abstract] The abstract contains two typos: 'device' should be 'devise' and 'pseudomonote' should be 'pseudomonotone'.
  2. [Theorem 3.2 statement] The statement of Theorem 3.2 says the sequence is generated by Algorithm 1; it should refer to Algorithm 2.
  3. [Proof of Theorem 3.1, Case 2] The quantity y_{tau(k)} is used in the Case 2 argument but never defined; the surrounding formulas indicate z_{tau(k)} is intended.
  4. [§3.1, after Eq. (3.2)] The sentence claiming the limit point is 'not smaller than {gamma_0, rho/L}' should read min{gamma_0, rho/L}.
  5. [Lemma 3.6] The assertion that 'for each j >= 1, F(z_Nj) != 0' is not justified; if F(z_Nj)=0 then z_Nj is already in X* and this case should be handled separately before defining u_Nj.
  6. [§4.3, figures] The figure numbering is inconsistent: Figure 2 appears with different captions, and the caption referring to 'four test networks' includes networks not discussed in the numerical comparisons.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the strong-convergence theorem is derived from stated assumptions with standard Fejér-type arguments; the minimal-norm solution is a consequence, not an input.

full rationale

The convergence analysis in Section 3 is a self-contained derivation. Lemma 3.3 applies pseudo-monotonicity only to pairs in X (x* and z_k), exactly as Assumption 2 defines it. Lemma 3.4 bounds the iterates using the anchor x* and the Lipschitz constant. Lemma 3.5 is an algebraic recursion, and Lemma 3.6 obtains X* membership of a weak cluster point from the gap condition via weak-to-weak continuity; p is then defined as P_X*(0), i.e., the minimum-norm element of X*, not a fitted or pre-imposed target. No parameter in Theorem 3.1 or Theorem 3.2 is fitted to the solution, and no equation is equivalent to its own input by construction. Self-citations such as [5], [6], and [30] are contextual and not load-bearing; the proof does not invoke them. The DUE experiments compare the algorithm's relative gap with a benchmark solver [17] on test networks; the reported O-D gaps are computed after termination, not used to define the method, so this is benchmarking, not prediction from fitted constants. The manuscript does contain two non-circular weaknesses: the step before (3.18) invokes pseudo-monotonicity at a point y + eps_j u_Nj that is not shown to lie in X, which is a possible gap in Lemma 3.6 under Assumption 2 as stated, and the proof of Theorem 3.2 is explicitly left to the reader. Both are correctness and completeness concerns; neither makes the derivation circular.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The theoretical claim rests on standard Hilbert-space facts plus explicit regularity and monotonicity assumptions on F. The proof of Lemma 3.6 implicitly requires pseudo-monotonicity at perturbed points outside the feasible set, which is stronger than the stated Assumption 2. The numerical claims rest on user-chosen parameters that are not reported. No new physical or mathematical entities are introduced.

free parameters (3)
  • Experimental step-size and momentum parameters (alpha_k, beta_k, gamma) = Not reported (chosen per instance)
    Section 4.3 states these were chosen for each instance separately to guarantee the best convergence; no values or selection rule are given, so the numerical comparison is not reproducible and may favor the proposed method.
  • Stopping tolerance epsilon = 1e-4 for Nguyen and Sioux Falls; 1e-3 mentioned for other networks in pasted text
    Section 4.3 and the inserted Section 5.1 use a relative-gap stopping criterion; the pasted text also applies a different threshold to networks not tested in this paper.
  • Adaptive parameters gamma_0 and rho = Not specified in experiments
    Algorithm 2 requires an initial step-size and a shrinkage factor; their values are not reported, and although asymptotic convergence does not depend on them, convergence speed and the reported iteration counts do.
assumptions (5)
  • domain assumption F is L-Lipschitz and sequentially weak-to-weak continuous on bounded subsets (Assumption 1)
    Used for the contraction estimate in Lemma 3.3 and for passing to weak limits in Lemma 3.6; not verified for the DUE delay operator.
  • domain assumption F is pseudo-monotone on X, and implicitly at perturbed points outside X (Assumption 2 plus an unstated strengthening)
    Pseudo-monotonicity is used in Lemmas 3.3 and 3.6; the proof of Lemma 3.6 requires it at points y + epsilon F(z)/||F(z)||^2, which are not shown to be in X.
  • domain assumption The solution set X* is nonempty, closed, and convex (Assumption 3)
    Ensures the minimal-norm solution p exists and is unique; convexity of solution sets for pseudo-monotone VIs is not generally guaranteed.
  • standard math Projection characterization lemma (Lemma 2.1)
    Standard Hilbert-space projection property used in Lemma 3.3.
  • standard math Xu's lemma for quasi-Fejer recursions (Lemma 2.3)
    External standard lemma used to convert the limsup inequality into strong convergence.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Strong Convergence of Forward-Backward-Forward Methods for Pseudo-monotone Variational Inequalities with Applications to Dynamic User Equilibrium in Traffic Networks." pith.science (2026). https://pith.science/paper/T2TBD3OH

@misc{pith2026190807211,
  author       = {Pith},
  title        = {Pith review of: Strong Convergence of Forward-Backward-Forward Methods for Pseudo-monotone Variational Inequalities with Applications to Dynamic User Equilibrium in Traffic Networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T2TBD3OH}},
  note         = {Machine review of arXiv:1908.07211}
}
read the original abstract

In infinite-dimensional Hilbert spaces we device a class of strongly convergent primal-dual schemes for solving variational inequalities defined by a Lipschitz continuous and pseudomonote map. Our novel numerical scheme is based on Tseng's forward-backward-forward scheme, which is known to display weak convergence, unless very strong global monotonicity assumptions are made on the involved operators. We provide a simple augmentation of this algorithm which is computationally cheap and still guarantees strong convergence to a minimal norm solution of the underlying problem. We provide an adaptive extension of the algorithm, freeing us from requiring knowledge of the global Lipschitz constant. We test the performance of the algorithm in the computationally challenging task to find dynamic user equilibria in traffic networks and verify that our scheme is at least competitive to state-of-the-art solvers, and in some case even improve upon them.

Figures

Figures reproduced from arXiv: 1908.07211 by the authors.

Figure 1
Figure 1. The Nguyen and Sioux Falls network. LWR model is used, generating the delay operator Ψ(h) at flow profile h ∈ H. By adapting this toolbox to Algorithm 3, we compute dynamic user equilibria for the Nguyen and the Sioux fall network (see [PITH_FULL_IMAGE:figures/full_fig_p020_1.png] view at source ↗
Figure 2
Figure 2. The four test networks for DUE algorithms. until the stopping criterion is reached. It can be seen from this Figure that both [PITH_FULL_IMAGE:figures/full_fig_p020_2.png] view at source ↗
Figure 2
Figure 2. Relative gap (4.2) (called epsilon in the figure) computed under the forward-backward iteration of [17] and Algorithm 3, using the same parameter values 0 10 20 30 40 50 60 70 Iteration number 10-4 10-3 10-2 10-1 Relative gap epsilon FB epsilon FBF (a) Ngyuen Network 0 10 20 30 40 50 60 Iteration number 10-4 10-3 10-2 10-1 100 Relative gap epsilon FB epsilon FBF (b) Sioux Fall Network 0 2 4 6 Time (hr) 0 100 200 300… view at source ↗
Figures from the paper (2 more)
Figure 3
Figure 3. Figure 3: Path departure rates and corresponding effective path delays of selected paths in the DUE solutions. 21 [PITH_FULL_IMAGE:figures/full_fig_p021_3.png]
Figure 4
Figure 4. Figure 4: Distributions of O-D gaps corresponding to the DUE solutions. The O-D gap is calculated according to (4.3). In an exact DUE, we should have Γw = 0 for all w ∈ W [PITH_FULL_IMAGE:figures/full_fig_p022_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

36 extracted references · 31 canonical work pages

  1. [1]

    Method of convex programming using a symmetric modification of lagrange function

    AS Antipin. Method of convex programming using a symmetric modification of lagrange function. Matekon, 14(2):23–38, 1978

    work page 1978

  2. [2]

    Asymptotic cones and functions in optimization and variational inequalities

    Alfred Auslender and Marc Teboulle. Asymptotic cones and functions in optimization and variational inequalities. Springer Science & Business Media, 2006

    work page 2006

  3. [3]

    Bauschke and Patrick L

    Heinz H. Bauschke and Patrick L. Combettes. A weak-to-strong convergence principle for fejér-monotone methods in hilbert spaces. Mathematics of Operations Research, 26(2): 248–264, 2019/07/28 2001. doi: 10.1287 /moor.26.2.248.10558. URL https://doi.org/ 10.1287/moor.26.2.248.10558

    work page doi:10.1287/moor.26.2.248.10558 2019

  4. [4]

    Bauschke and Patrick L

    Heinz H. Bauschke and Patrick L. Combettes. Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer - CMS Books in Mathematics, 2016

    work page 2016

  5. [5]

    The Forward-Backward-Forward Method from continuous and discrete perspective for pseudo-monotone variational inequalities in Hilbert spaces

    Radu Ioan Bot, Ernö Robert Csetnek, and Phan Tu Vuong. The forward-backward- forward method from discrete and continuous perspective for pseudo-monotone vari- ational inequalities in hilbert spaces. arXiv preprint arXiv:1808.08084, 2018

    work page Pith review arXiv 2018

  6. [6]

    Forward-backward-forward methods with variance reduction for stochastic varia- tional inequalities

    Radu Ioan Bot, Panayotis Mertikopoulos, Mathias Staudigl, and Phan Tu Vuong. Forward-backward-forward methods with variance reduction for stochastic varia- tional inequalities. arXiv preprint arXiv:1902.03355, 2019

    arXiv 1902

  7. [7]

    Flows on networks: recent results and perspectives

    Alberto Bressan, Sunˇ cica ˇCani´ c, Mauro Garavello, Michael Herty, and Benedetto Pic- coli. Flows on networks: recent results and perspectives. EMS Surveys in Mathematical Sciences, 1(1):47–111

    work page

  8. [8]

    Finite-Dimensional Variational Inequalities and Complementarity Problems - Volume I and Volume II

    Francisco Facchinei and Jong-shi Pang. Finite-Dimensional Variational Inequalities and Complementarity Problems - Volume I and Volume II . Springer Series in Operations Re- search, 2003

    work page 2003

Show all 36 references
  1. [9]

    Friesz and Ke Han

    Terry L. Friesz and Ke Han. The mathematical foundations of dynamic user equilib- rium. Transportation Research Part B: Methodological , 126:309–328, 2019. doi: https: //doi.org/10.1016/j.trb.2018.08.015. URL http://www.sciencedirect.com/science/ article/pii/S0191261517301960

    2019 doi

  2. [10]

    Friesz, David Bernstein, Tony E

    Terry L. Friesz, David Bernstein, Tony E. Smith, Roger L. Tobin, and B. W. Wie. Vari- ational inequality formulation of the dynamic network user equilibrium. Operations Research, 41(1):179–191, 1993

    1993

  3. [11]

    Traffic flow on networks , volume 1

    Mauro Garavello and Benedetto Piccoli. Traffic flow on networks , volume 1. American institute of mathematical sciences Springfield, 2006

    2006

  4. [12]

    Tseng type methods for solving inclusion problems and its applications

    Aviv Gibali and Duong Viet Thong. Tseng type methods for solving inclusion problems and its applications. Calcolo, 55(4):49, 2018. doi: 10.1007 /s10092-018-0292-1. URL https://doi.org/10.1007/s10092-018-0292-1 . 23

    2018 doi

  5. [13]

    Goebel and S

    K. Goebel and S. Reich. Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Marcel Dekker, New York, 1984

    1984

  6. [14]

    O. Güler. On the convergence of the proximal point algorithm for convex minimization. SIAM Journal on Control and Optimization, 29(2):403–419, 2019/07/28 1991. doi: 10.1137 / 0329022. URL https://doi.org/10.1137/0329022

    2019 doi

  7. [15]

    Two new self-adaptive projection methods for variational inequality problems

    Deren Han and Hong K Lo. Two new self-adaptive projection methods for variational inequality problems. Computers & Mathematics with Applications , 43(12):1529–1537, 2002

    2002

  8. [16]

    Friesz, W

    Ke Han, Terry L. Friesz, W. Y. Szeto, and Hongcheng Liu. Elastic demand dy- namic network user equilibrium: Formulation, existence and computation. Trans- portation Research Part B: Methodological , 81:183–209, 2015. doi: https: //doi.org/10. 1016/j.trb.2015.07.008. URL http:/...

    2015

  9. [17]

    Computing dynamic user equilibria on large- scale networks with software implementation

    Ke Han, Gabriel Eve, and Terry L Friesz. Computing dynamic user equilibria on large- scale networks with software implementation. Networks and Spatial Economics , pages 1–34, 2019

    2019

  10. [18]

    Sur les inéquations variationnelles et la minimisation de fonction- nelles convexes

    Yves Haugazeau. Sur les inéquations variationnelles et la minimisation de fonction- nelles convexes. These, Universite de Paris, 1968

    1968

  11. [19]

    B. S. He and L. Z. Liao. Improvements of some projection methods for monotone nonlinear variational inequalities. Journal of Optimization Theory and Applications , 112 (1):111–128, 2002. doi: 10.1023 /A:1013096613105. URL https://doi.org/10.1023/A: 1013096613105

    2002 doi

  12. [20]

    Optimization with PDE constraints, volume 23

    Michael Hinze, René Pinnau, Michael Ulbrich, and Stefan Ulbrich. Optimization with PDE constraints, volume 23. Springer Science & Business Media, 2008

    2008

  13. [21]

    An Introduction to Variational Inequalities and their Applications

    David Kinderlehrer and Guido Stampacchia. An Introduction to Variational Inequalities and their Applications. Academic Press, New York, 1980

    1980

  14. [22]

    The extragradient method for finding saddle points and other prob- lems

    GM Korpelevich. The extragradient method for finding saddle points and other prob- lems. Matecon, 12:747–756, 1976

    1976

  15. [23]

    Lo and W

    Hong K. Lo and W. Y. Szeto. A cell-based variational inequality formulation of the dynamic user optimal assignment problem. Transportation Research Part B: Method- ological, 36(5):421–443, 2002. doi: https: //doi.org/10.1016/S0191-2615(01)00011-X. URL http://www.sciencedirect....

    2002 doi

  16. [24]

    Jiancheng Long, Hai-Jun Huang, Ziyou Gao, and W. Y. Szeto. An intersection- movement-based dynamic user optimal route choice problem. Operations Research, 61(5):1134–1147, 2019 /08/09 2013. doi: 10.1287 /opre.2013.1202. URL https://doi. org/10.1287/opre.2013.1202

    2019

  17. [25]

    P . Maingé. A hybrid extragradient-viscosity method for monotone operators and fixed point problems. SIAM Journal on Control and Optimization, 47(3):1499–1515, 2019/07/28

    2019

  18. [26]

    Régularisation d’inéquations variationnelles par approximations successives

    Bernard Martinet. Régularisation d’inéquations variationnelles par approximations successives. Revue française d’informatique et de recherche opérationnelle. Série rouge, 4(R3): 154–158, 1970

    1970

  19. [27]

    Uniqueness of equilibrium in steady state and dynamic traffic networks

    Richard Mounce and Mike Smith. Uniqueness of equilibrium in steady state and dynamic traffic networks. In Transportation and Traffic Theory 2007. Papers Selected for Presentation at ISTTT17 Engineering and Physical Sciences Research Council (Great Britain), 2007. 24

    2007

  20. [28]

    Monotone operators and the proximal point algorithm

    R Tyrrell Rockafellar. Monotone operators and the proximal point algorithm. SIAM journal on control and optimization, 14(5):877–898, 1976

    1976

  21. [29]

    M. V . Solodov and B. F. Svaiter. Forcing strong convergence of proximal point it- erations in a hilbert space. Mathematical Programming , 87(1):189–202, 2000. doi: 10.1007/s101079900113. URL https://doi.org/10.1007/s101079900113

    doi

  22. [30]

    Convergent noisy forward-backward- forward algorithms in non-monotone variational inequalities

    Mathias Staudigl and Panayotis Mertikopoulos. Convergent noisy forward-backward- forward algorithms in non-monotone variational inequalities. In 15th IFAC Sym- posium on Large Scale Complex Systems LSS 2019 , volume 52, pages 120–125, 2019. ISBN 2405-8963. doi: https: //doi.or...

    2019 doi

  23. [31]

    A strong convergence theorem for tseng’s extragradient method for solving variational inequality prob- lems

    Duong Viet Thong, Nguyen The Vinh, and Yeol Je Cho. A strong convergence theorem for tseng’s extragradient method for solving variational inequality prob- lems. Optimization Letters , 2019. doi: 10.1007 /s11590-019-01391-3. URL https: //doi.org/10.1007/s11590-019-01391-3

    2019 doi

  24. [32]

    P . Tseng. A modified forward-backward splitting method for maximal monotone map- pings. SIAM Journal on Control and Optimization , 38(2):431–446, 2018/09/13 2000. doi: 10.1137/S0363012998338806. URL https://doi.org/10.1137/S0363012998338806

    2018 doi

  25. [33]

    On the weak convergence of the extragradient method for solv- ing pseudo-monotone variational inequalities

    Phan Tu Vuong. On the weak convergence of the extragradient method for solv- ing pseudo-monotone variational inequalities. Journal of Optimization Theory and Applications, 176(2):399–409, 2018. doi: 10.1007 /s10957-017-1214-0. URL https: //doi.org/10.1007/s10957-017-1214-0

    2018 doi

  26. [34]

    John G. Wardrop. Some theoretical aspects of road tra ffic research. Proceedings of the Institution of Civil Engineers II, 1:325–378, 1952

    1952

  27. [35]

    Iterative algorithms for nonlinear operators

    Hong-Kun Xu. Iterative algorithms for nonlinear operators. Journal of the London Mathematical Society, 66(1):240–256 25

  28. [2008]

    URL https://doi.org/10.1137/060675319

    doi: 10.1137 /060675319. URL https://doi.org/10.1137/060675319

    doi

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.