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REVIEW 3 major objections 4 minor 70 references

Inexact projected preconditioned gradient methods with variable metrics: a Lyapunov convergence theory (extended version)

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper proves exponential and linear convergence rates for inexact projected preconditioned gradient methods with variable metrics, via a new Lyapunov function, and derives a faster under-relaxed variant from the same ODE discretization.

desk verdict A genuinely new ODE model and inexact projection construction for variable-metric projected gradient methods, but the main Lyapunov function as stated is not the one being differentiated, so the central theorem is wrong as written. read the letter →

arxiv 2506.03671 v2 pith:ZMYT3WRN submitted 2025-06-04 math.OC cs.NAmath.NA

Ruchi Guo , Jun Zou This is my paper
classification math.OCcs.NAmath.NA MSC 65K0590C2565N55
keywords inexactprojectionprojectedgradientmethodsvariablepreconditionersLyapunovanalysisconstrainedoptimizationnonlinearellipticPDEsmultigridpreconditioning
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

The paper develops an ODE-based convergence theory for inexact projected preconditioned gradient methods (IPPGD) with metrics that change at every step. It proves that trajectories of the flow $u' + u - \tilde{P}_{M(t)}(u - \alpha(t)M^{-1}(t)\nabla f(u)) = 0$ converge exponentially to a unique fixed point $u^*_\varphi$ of the inexact operator, and that discretizing the flow gives linearly convergent iterations, under Assumption 4.1. The same discretization with relaxation parameter $\tau_k = 1$ recovers the original IPPGD method, while $\tau_k < 1$ gives a new method that is numerically faster at the same cost. The authors argue this matters because computing exact projections in PDE-constrained problems means solving costly ill-conditioned Schur complement systems, and the theory shows how much inexactness is tolerable and how step size and relaxation interact.

What carries the argument

The load-bearing object is the inexact projection operator $\tilde{P}_M = I - M^{-1}B^T\tilde{S}^{-1}B$, which replaces the Schur complement $S = BM^{-1}B^T$ by an easily inverted approximation $\tilde{S}$, chosen as a multigrid-based operator in the PDE experiments. The ODE (2.4) inserts $\tilde{P}_M$ into a Tanabe-type flow, and its forward-Euler discretization with relaxation $\tau_k$ is (2.5). The proof is carried by the Lyapunov function $E = \lambda\alpha D_f(u,u^*) + \tfrac{1}{2}\|(I-\tilde{P}_{M_*})(u-u^*_\varphi)\|^2_{M_*}$, where the first term is a Bregman divergence and the second measures deviation from the constraint set; Assumption (H4), which bounds the metric deviation $\Theta_m(t)$ by the current distance to $u^*_\varphi$, is what makes the $\Theta_m^2$ terms absorbable in the derivative estimates.

What would settle it

Run the discrete IPPGD (1.6) on a smooth strongly convex problem with a variable metric chosen to satisfy (H1)-(H3) but to violate (H4), for instance $\Theta_m(k) = c\|u_k - u^*_\varphi\|^{1/2}_{M_*}$, and monitor $E_k$: the linear bound of Theorem 5.4 should fail, while modifying the metric to make $\Theta_m(k) \le C\|u_k - u^*_\varphi\|$ should restore the predicted rate. A second, cheaper check is to compute $\Theta_m(k)/\|u_k - u^*_\varphi\|$ for the multigrid-constructed $\tilde{S}(\sigma_h)$ in Section 6; if the ratio is not bounded, the theorem's hypotheses are not met.

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Extended reading notes

Core claim

The central claim is that the interplay between inexact projections and variable preconditioners can be captured by the single ODE (2.4), whose equilibrium $u^*_\varphi$ is the actual limit of the method. For smooth strongly convex $f$ with the metric family satisfying (H1)-(H4), the designed Lyapunov function $E = \lambda\alpha D_f(u,u^*) + \tfrac{1}{2}\|(I-\tilde{P}_{M_*})(u-u^*_\varphi)\|^2_{M_*}$ satisfies $\frac{d}{dt}E \le -\omega E$, giving exponential convergence at the continuous level; the discrete analogue satisfies $E_{k+1} \le (1-\omega_k)E_k$, giving linear convergence. The analysis also bounds the distance between $u^*_\varphi$ and the true minimizer $u^*$ by a multiple of $\delta_*\sqrt{\alpha_*}$ times the residual gradient, so the accuracy of the limit is controlled by both inexactness and step size. Discretizing the flow with forward Euler and relaxation $\tau_k$ produces a family of methods (2.5) that is provably linearly convergent for $\tau_k=1$ and remains so for $\tau_k<1$, and the numerical experiments on quasilinear elliptic equations show fewer outer iterations, fewer inner multigrid cycles, and lower CPU time than exact projected gradient, inexact fixed-metric, and inexact variable-metric alternatives.

Load-bearing premise

The load-bearing premise is Assumption (H4): the deviation $\Theta_m(t)$ between the current metric family and its limiting metric must be bounded by a constant times the distance from the current iterate to the fixed point $u^*_\varphi$; the paper assumes this Lipschitz-type control on the metric family and does not verify it for the finite-element multigrid metrics used in the experiments.

Editorial extensions

If this is right

  • Setting $\tau_k = 1$ in (2.5) exactly recovers the original IPPGD method (1.6), so the new convergence proof covers the classical scheme; choosing $\tau_k < 1$ yields a new method with the same per-iteration cost.
  • The discrete rate depends only on the condition number $\kappa_{f,M_k}$ and on $\tau_k$; when $\kappa_{f,M_*}$ is independent of mesh size, so is the number of outer iterations, which the PDE experiments confirm.
  • The limit point $u^*_\varphi$ is not the true minimizer, but its distance to $u^*$ is bounded by $\delta_*\sqrt{\alpha_*}$ times the gradient at $u^*$, so one can deliberately run with larger inexactness and larger steps to match a prescribed mesh-scale accuracy.
  • The assumptions allow the metric sequence to oscillate without converging a priori, and no monotone Loewner decrease of the metrics is imposed.

Reading between the lines

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

  • The $\tau < 1$ relaxation acts as a stabilizer that trades step size for projection accuracy; this suggests that under-relaxing any iteration built on an inexact projection may restore linear rates even when inexactness cannot be controlled tightly, a principle that could be tested on other projection-based splitting methods.
  • The paper does not verify (H4) for the multigrid-based metric family used in its own experiments; a numerical check of $\Theta_m(k)$ against $\|u_k - u^*_\varphi\|$ would either close that gap or identify realistic regimes where the theorem's hypotheses fail.
  • Because the continuous flow carries the step size $\alpha(t)$ explicitly, the ODE model could be adapted to study adaptive step-size and adaptive inexactness schedules, or to stochastic projected gradient methods, by treating the Lyapunov inequality as a template.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The manuscript studies inexact projected preconditioned gradient descent (IPPGD) methods with variable metrics for minimizing a convex functional under a linear constraint. It proposes an inexact projection operator built from a Schur-complement approximation, models the method by the ODE (2.4), and proves exponential convergence of the continuous flow and linear convergence of the forward-Euler discretization (2.5) via a Lyapunov argument under Assumptions 4.1 and 5.1. The paper also presents finite-element experiments for a quasilinear elliptic problem, reporting mesh-independent outer iteration counts and computational gains from variable metrics and from the relaxation parameter τ<1 in the new method (2.5).

Significance. If the proof issues are repaired, the contribution is significant: the paper gives explicit convergence rates with all constants expressed in terms of problem parameters, an ODE model that recovers the original method at τ=1 and suggests a potentially faster variant, and numerical evidence of robustness with respect to mesh size. The proofs are detailed, the constants in the rates are explicit rather than fitted, and the numerical study is reproducible in structure. However, the main convergence theorems currently rest on an inconsistency in the definition of the first Lyapunov component and on a metric-regularity assumption that is not verified for the numerical examples; these points must be resolved before the central claims can be accepted.

major comments (3)
  1. [§4.1, Eq. (4.1); Lemma 4.3, Eq. (4.24)] The Lyapunov function E^(1) is defined as D_f(u,u⋆), with u⋆ the true minimizer, but the derivative computed in (4.24) is the derivative of D_f(u,u⋆_φ): the right-hand side contains ∇f(u)-∇f(u⋆_φ) and the term u⋆_φ - eP_M(u⋆_φ - αM^{-1}∇f(u⋆_φ)). For δ⋆>0, Lemma 3.6 gives u⋆_φ≠u⋆, so d/dt D_f(u,u⋆) contains the additional term ⟨∇f(u⋆_φ)-∇f(u⋆), u'⟩, which is not controlled by the estimates R1-R3 in (4.24). Consequently Lemma 4.3 and the exponential decay in Theorem 4.5 do not follow as stated. The same mismatch appears in the discrete setting in (5.1) and Lemma 5.1. The proof becomes internally consistent if E^(1) is redefined as D_f(u,u⋆_φ); then the distance from u⋆_φ to u⋆ must be recovered separately through Lemma 3.6. This correction is load-bearing because the claimed Lyapunov decay otherwise forces convergence to u⋆, while the ODE (2.4) has equilibrium u⋆_φ.
  2. [Assumption 4.1(H4), Eq. (4.5); Assumption 5.1(H4'), Eq. (5.5)] The bound Θ_m(t) ≤ μ^{1/2}_{f,M⋆} K_θ^{-1} ‖u(t)-u⋆_φ‖ is used in Theorems 4.5 and 5.3 to convert Θ_m^2 terms into multiples of E^(1); without it the exponential and linear rates do not follow. For the numerical application in Section 6, M(σ_h) is a weighted mass matrix and eS^{-1} is a multigrid approximation, but the paper does not verify (H4) for this metric family, nor does it report the quantities Θ_m/‖u_k-u⋆_φ‖ along the computed trajectories. A concrete verification, either analytical for the weighted-mass family or computational along the actual iterates, is needed before the numerical experiments can be read as evidence for the general theorem.
  3. [§4.2, Theorem 4.6 and §5, Theorem 5.4] The final accuracy statements (4.29b) and (5.14a) contain a garbled constant: the denominator is printed as "8(9κf,M⋆+4 + 4)^2" (and similarly in the discrete case), which appears to be a typo for the expression coming from Lemma 3.6. Since these bounds are the quantitative statements of convergence to a neighbourhood of u⋆, the constant should be cleaned up and checked. This is a presentation issue rather than a structural one, but it matters for the paper's explicit-rate claims.
minor comments (4)
  1. [References [64] and [65]] References [64] and [65] list the same paper (same title, journal, volume, and pages); one entry should be removed and the citations renumbered.
  2. [Section 6, after Eq. (6.6)] The sentence "For simplicity, we also let M = M(1)" is ambiguous; it should state explicitly that M(1) denotes the weighted mass matrix evaluated at the constant function 1, or at σ_h = 1 in the vector representation.
  3. [Lemma 3.5] The proof invokes Brouwer's fixed point theorem after establishing that φ is a contraction; in the Hilbert-space setting the Banach fixed point theorem is the applicable result and also gives uniqueness directly.
  4. [Table 1] The table header "Ave. Wcycles" and the text's "Wcycle" would benefit from a definition of the multigrid cycle used (presumably a W-cycle) and from a statement of how the average is taken over outer iterations.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the convergence rates are derived from explicit assumptions via Lyapunov estimates, with constants displayed and no fitted parameters; self-citations are background only.

full rationale

The paper's derivation chain is self-contained and non-circular. The central rates in Theorems 4.5, 4.6, 5.3 and 5.4 follow from Lemmas 4.3, 4.4, 5.1 and 5.2, whose constants are explicit functions of the convexity and Lipschitz parameters, the metric-deviation functions, the inexactness levels, and the stated assumptions H1-H4. No quantity reported as a prediction is obtained by fitting a parameter to the numerical experiments; the experiments in Section 6 are used for validation only. The fixed point u*_phi is introduced as the equilibrium of the ODE and is used as a Lyapunov reference point, which is standard practice; its distance to the true minimizer u* is bounded independently in Lemma 3.6. Assumption H4 is a regularity/coupling condition on the metric family; it is load-bearing but it is an assumption stated before the theorems, not a rewording of the convergence conclusion. The self-citations in the manuscript appear only as related-work remarks and are not used to justify the main estimates. An apparent bookkeeping mismatch in Lemma 4.3 between E^(1)=D_f(u,u*) and derivative expressions written for D_f(u,u*_phi) is a correctness concern, not a circularity: it does not make the claimed rates equivalent to the assumptions by construction.

Assumptions & free parameters 2 free parameters · 7 assumptions · 1 invented entities

The central claim rests on a standard convexity/smoothness framework plus three assumptions that are specific to this paper: the Loewner-order closeness of the metric sequence (H1), the quantified inexactness (H2), and the Lipschitz-type conditions (H3)-(H4). There are no free parameters fitted to the data in the convergence theory itself, but the numerical experiments introduce hand-chosen hyperparameters (τ and the n_mg schedule). The inexact projection operator is a new construct, though it is fully defined and its properties are verified.

free parameters (2)
  • τ (relaxation parameter in IPPGDv-τ) = 0.5 for case (a0,a1,a2)=(1,1,5), 0.2 for (1,6,5)
    The numerical experiments choose τ by hand and the authors state that smaller τ can improve convergence. The theory allows τ in a range, but the specific values are selected to obtain faster practical performance, not derived from the data.
  • Number of inner multigrid W-cycles (n_mg) = Starts at 1, increases to max 6 dynamically
    The dynamic schedule controls the inexactness level δ_k. The paper does not quantify the resulting δ_k or verify that it satisfies the smallness assumptions (5.10), so this schedule acts as a free implementation parameter.
assumptions (7)
  • domain assumption f is μ-strongly convex and L-smooth with respect to the M-inner product (2.10a), uniformly for all M in the admissible family.
    Assumed at the start of Section 2.3 and used throughout the Lyapunov analysis to bound Bregman divergences and gradients.
  • domain assumption eS ≽ S so that ⟨·, eP_M·⟩_M is an inner product (Section 3.1, Remark 3.1).
    This condition is needed for inequalities (3.3c), (3.4b) and for the positivity of the Lyapunov-function components.
  • ad hoc to paper Assumption 4.1 (H1): M(t) and eS(t) are uniformly comparable to M* and eS* with deviation functions Θ(t), eΘ(t).
    Introduced by the authors to handle the variable metric; it bounds the distance between the current and limiting operators in the Loewner order.
  • domain assumption Assumption 4.1 (H2): the inexactness level satisfies (1−δ(t))eS(t) ≼ S(t) ≼ eS(t).
    This is a standard way to quantify how close the approximate Schur complement is to the exact one; it is the main smallness parameter in the rates.
  • ad hoc to paper Assumption 4.1 (H3): the operator difference (eS^{-1} − S^{-1}) is Lipschitz in the sense that its deviation from the limit is bounded by K_S Θ_m δ.
    This smoothness of the inexactness residual is used in Lemma 4.1 to bound the difference between the inexact projection at time t and at the limit. It is not verified for the PDE application.
  • ad hoc to paper Assumption 4.1 (H4): Θ_m(t) ≤ μ^{1/2}_{f,M*}/K_θ · ||u(t) − u*_φ||_{M*}.
    This is the most fragile assumption: it ties the metric deviation to the distance to the fixed point, which is exactly the kind of quantity the convergence proof seeks to control. It is a strong regularity condition on the metric construction.
  • standard math For the PDE application in Section 6, the coefficient function ν satisfies conditions (1)-(3) and the mixed finite element spaces are used with the standard assumptions of the Thomas-Raviart and piecewise-constant spaces.
    These are standard regularity conditions for quasilinear elliptic problems and standard finite element assumptions, not specific to this paper.
invented entities (1)
  • Inexact projection operator eP_M = I − M^{-1}B^T eS^{-1}B independent evidence
    purpose: Approximates the exact projection onto ker(B) by replacing the exact Schur complement S with a cheaper operator eS, usually a multigrid approximation.
    This is a new mathematical operator introduced by the paper, but its properties (idempotence up to factorization, inner-product inequalities) are proven in Lemma 3.2 and it is used in the numerical solver, so it has independent computational support.

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Pith. "Pith review of Inexact projected preconditioned gradient methods with variable metrics: a Lyapunov convergence theory (extended version)." pith.science (2026). https://pith.science/paper/ZMYT3WRN

@misc{pith2026250603671,
  author       = {Pith},
  title        = {Pith review of: Inexact projected preconditioned gradient methods with variable metrics: a Lyapunov convergence theory (extended version)},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZMYT3WRN}},
  note         = {Machine review of arXiv:2506.03671}
}
read the original abstract

Projected gradient methods are widely used for constrained optimization. A key application is for partial differential equations (PDEs), where the objective functional represents physical energy and the linear constraints enforce conservation laws. However, computing the projections onto constraint sets generally requires solving large-scale ill-conditioned systems. A common strategy is to relax projection accuracy and apply preconditioners, which leads to inexact preconditioned projected gradient descent (IPPGD) methods studied here. Furthermore, variable preconditioners dynamically incorporating updated nonlinear information often enhance convergence rates. However, due to the complex interplay between inexactness and adaptive preconditioners, the theoretical analysis and the dynamic behavior of the IPPGD methods still remain quite open. We propose an effective strategy for constructing the inexact projection operator and develop a gradient-type flow to model the resulting IPPGD methods. Discretization of this flow not only recovers the original IPPGD method but also yields a potentially faster novel method. Furthermore, we apply Lyapunov analysis, designing a delicate Lyapunov function, to prove the exponential convergence at the continuous level and linear convergence at the discrete level under certain assumptions. Finally, we validate our approach through numerical experiments, demonstrating robust performance and computational efficiency.

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Works this paper leans on

70 extracted references · 69 canonical work pages

  1. [1]

    A. A. Aguiar, O. P. Ferreira, and L. F. Prudente , Inexact gradient projection method with relative error tolerance, Comput. Optim. Appl., 84 (2023), pp. 363–395. 3

    work page 2023

  2. [2]

    Amaya, A differential equations approach to function minimization , Rev

    J. Amaya, A differential equations approach to function minimization , Rev. Mat. Apl., 4 (1987), pp. 1–7. 3

    work page 1987

  3. [3]

    Andreani, E

    R. Andreani, E. G. Birgin, J. M. Mart ´ınez, and J. Yuan , Spectral projected gradient and variable metric methods for optimization with linear inequalities , IMA J. Numer. Anal., 25 (2005), pp. 221–252. 2, 3

    work page 2005

  4. [4]

    A. Antipin, Continuous and iterative processes with projection and projection-type operators , Problems in Cyber- netics: Computational problems in the analysis of large systems, (1989), pp. 5–43. 4

    work page 1989

  5. [5]

    A. S. Antipin , Minimization of convex functions on convex sets by means of differential equations , Differ. Equ., 30 (1994). 4

    work page 1994

  6. [6]

    Bacuta, A unified approach for Uzawa algorithms , SIAM J

    C. Bacuta, A unified approach for Uzawa algorithms , SIAM J. Numer. Anal., 44 (2006), pp. 2633–2649. 2

    work page 2006

  7. [7]

    D. P. Bertsekas, On the Goldstein-Levitin-Polyak gradient projection method , IEEE Trans. Autom. Control., 21 (1976), pp. 174–184. 2

    work page 1976

  8. [8]

    E. G. Birgin, J. Mart´ınez, and M. Raydan, Spectral projected gradient methods, Encyclopedia of Optimization, 2 (2009). 2

    work page 2009

Show all 70 references
  1. [9]

    E. G. Birgin, J. M. Mart ´ınez, and M. Raydan , Nonmonotone spectral projected gradient methods on convex sets, SIAM J. Optim., 10 (2000), pp. 1196–1211. 2

    2000

  2. [10]

    E. G. Birgin, J. M. Mart´ınez, and M. Raydan, Inexact spectral projected gradient methods on convex sets, IMA J. Numer. Anal., 23 (2003), pp. 539–559. 2, 3

    2003

  3. [11]

    E. G. Birgin, J. M. Mart ´ınez, and M. Raydan , Spectral projected gradient methods: Review and perspectives , J. Stat. Softw., 60 (2014), pp. 1–21. 2

    2014

  4. [12]

    Boffi, F

    D. Boffi, F. Brezzi, and M. Fortin , Mixed finite element methods and applications , vol. 44 of Springer Series in Computational Mathematics, Springer, Heidelberg, 2013. 19

    2013

  5. [13]

    Botsaris, Differential gradient methods, J

    C. Botsaris, Differential gradient methods, J. Math. Anal. Appl., 63 (1978), pp. 177–198. 3

    1978

  6. [14]

    , A class of differential descent methods for constrained optimization , J. Math. Anal. Appl., 79 (1981), pp. 96–112. 1, 4

    1981

  7. [15]

    J. H. Bramble, J. E. Pasciak, and A. T. V assilev , Analysis of the inexact Uzawa algorithm for saddle point problems, SIAM J. Numer. Anal., 34 (1997), pp. 1072–1092. 2

    1997

  8. [16]

    Comput., 69 (2000), pp

    , Uzawa type algorithms for nonsymmetric saddle point problems , Math. Comput., 69 (2000), pp. 667–689. 2

    2000

  9. [17]

    J. H. Bramble, J. E. Pasciak, and J. Xu , Parallel multilevel preconditioners, Math. Comp., 55 (1990). 5

    1990

  10. [18]

    A. A. Brown and M. C. Bartholomew-Biggs, ODE versus SQP methods for constrained optimization, J. Optim. Theory Appl., 62 (1989), pp. 371–386. 4

    1989

  11. [19]

    L. Chen, R. Guo, and J. Wei, Transformed primal-dual methods with variable-preconditioners, arXiv:2312.12355, (2024). 3

    2024 arXiv

  12. [20]

    Chen and J

    L. Chen and J. Wei , Transformed primal-dual methods for nonlinear saddle point systems , J. Numer. Math., (2023). 3

    2023

  13. [21]

    Chen, Global and superlinear convergence of inexact Uzawa methods for saddle point problems with nondiffer- entiable mappings , SIAM J

    X. Chen, Global and superlinear convergence of inexact Uzawa methods for saddle point problems with nondiffer- entiable mappings , SIAM J. Numer. Anal., 35 (1998), pp. 1130–1148. 2

    1998

  14. [22]

    Chen, BDM mixed methods for a nonlinear elliptic problem , J

    Z. Chen, BDM mixed methods for a nonlinear elliptic problem , J. Comput. Appl. Math., 53 (1994), pp. 207–223. 19

    1994

  15. [23]

    Cheng and J

    X.-L. Cheng and J. Zou , An inexact Uzawa-type iterative method for solving saddle point problems , Int. J. Comput. Math., 80 (2003), pp. 55–64. 2

    2003

  16. [24]

    Devolder, F

    O. Devolder, F. Glineur, and Y. Nesterov , First-order methods of smooth convex optimization with inexact oracle, Math. Program., 146 (2014), pp. 37–75. 2

    2014

  17. [25]

    Dobson, J

    P. Dobson, J. M. Sanz-Serna, and K. C. Zygalakis , On the connections between optimization algorithms, lyapunov functions, and differential equations: Theory and insights , SIAM J. Optim., 35 (2025), pp. 537–566. 3

    2025

  18. [26]

    J. C. Dunn , Global and asymptotic convergence rate estimates for a class of projected gradient processes , SIAM J. Control Optim., 19 (1981), pp. 368–400. 2

    1981

  19. [27]

    R. L. Dykstra, An algorithm for restricted least squares regression , J. Am. Stat. Assoc., 78 (1983), pp. 837–842. 2

    1983

  20. [28]

    Y. G. Evtushenko and V. G. Zhadan , Stable barrier-projection and barrier-newton methods in linear program- ming, Comput. Optim. Appl., 3 (1994), pp. 289–303. 4

    1994

  21. [29]

    O. P. Ferreira, M. Lemes, and L. F. Prudente , On the inexact scaled gradient projection method , Comput. Optim. Appl., 81 (2022), pp. 91–125. 3

    2022

  22. [30]

    B. Gao, N. T. Son, P.-A. Absil, and T. Stykel , Riemannian optimization on the symplectic stiefel manifold , SIAM J. Optim., 31 (2021), pp. 1546–1575. 1, 4

    2021

  23. [31]

    Goldstein, On gradient projection, in Proc

    A. Goldstein, On gradient projection, in Proc. 12th Ann. Allerton Conference and Circuits and Systems, Allerton Park, IL, 1974, pp. 38–40. 2

    1974

  24. [32]

    A. A. Goldstein , Convex programming in Hilbert space , Bull. Amer. Math. Soc., 70 (1964). 2

    1964

  25. [33]

    M. A. Gomes-Ruggiero, J. M. Mart ´ınez, and S. A. Santos , Spectral projected gradient method with inexact restoration for minimization with nonconvex constraints , SIAM J. Sci. Comput., 31 (2009), pp. 1628–1652. 2

    2009

  26. [34]

    D. S. Gonc ¸alves, M. L. N. Gonc ¸alves, and T. C. Menezes , Inexact variable metric method for convex- constrained optimization problems, Optimization, 71 (2022), pp. 145–163. 2, 3

    2022

  27. [35]

    Han, Nonlinear Elliptic Equations of the Second Order , Amer

    Q. Han, Nonlinear Elliptic Equations of the Second Order , Amer. Math. Soc., 2016. 19 24

    2016

  28. [36]

    Henning and D

    P. Henning and D. Peterseim , Sobolev gradient flow for the gross–pitaevskii eigenvalue problem: Global conver- gence and computational efficiency , SIAM J. Numer. Anal., 58 (2020), pp. 1744–1772. 1, 4

    2020

  29. [37]

    Hiptmair, G

    R. Hiptmair, G. Widmer, and J. Zou, Auxiliary space preconditioning in H0(curl; Ω), Numer. Math., 103 (2006), pp. 435–459. 5

    2006

  30. [38]

    Hu and J

    Q. Hu and J. Zou , An iterative method with variable relaxation parameters for saddle-point problems , SIAM J. Matrix Anal. Appl., 23 (2001), pp. 317–338. 2

    2001

  31. [39]

    Hu and J

    Q. Hu and J. Zou , Two new variants of nonlinear inexact Uzawa algorithms for saddle-point problems , Numer. Math., 93 (2002), pp. 333–359. 2

    2002

  32. [40]

    Optim., 16 (2006), pp

    , Nonlinear inexact Uzawa algorithms for linear and nonlinear saddle-point problems , SIAM J. Optim., 16 (2006), pp. 798–825. 2

    2006

  33. [41]

    Huang, L

    J. Huang, L. Chen, and H. Rui , Multigrid methods for a mixed finite element method of the Darcy–Forchheimer model, J. Sci. Comput., 74 (2018), pp. 396–411. 19

    2018

  34. [42]

    Jiang, D

    K. Jiang, D. Sun, and K.-C. Toh , An inexact accelerated proximal gradient method for large scale linearly constrained convex sdp, SIAM J. Optim., 22 (2012), pp. 1042–1064. 2, 3

    2012

  35. [43]

    KAZEMI and M

    P. KAZEMI and M. ECKART , Minimizing the Gross-Pitaevskii energy functional with the sobolev gradient — analytical and numerical results , Int. J. Comput. Methods, 07 (2010), pp. 453–475. 1, 4

    2010

  36. [44]

    Levitin and B

    E. Levitin and B. Polyak , Constrained minimization methods , USSR Computational Mathematics and Mathe- matical Physics, 6 (1966), pp. 1–50. 2

    1966

  37. [45]

    Luo, Accelerated differential inclusion for convex optimization , Optimization, 72 (2023), pp

    H. Luo, Accelerated differential inclusion for convex optimization , Optimization, 72 (2023), pp. 1139–1170. 3

    2023

  38. [46]

    Luo and L

    H. Luo and L. Chen, From differential equation solvers to accelerated first-order methods for convex optimization, Math. Program., 195 (2022), pp. 735–781. 3

    2022

  39. [47]

    May, On the convergence of the continuous gradient projection method, Optimization, 68 (2019), pp

    R. May, On the convergence of the continuous gradient projection method, Optimization, 68 (2019), pp. 1791–1806. 4

    2019

  40. [48]

    Nesterov, Introductory lectures on convex optimization: A basic course , vol

    Y. Nesterov, Introductory lectures on convex optimization: A basic course , vol. 87, Springer Science & Business Media, 2003. 5, 6

    2003

  41. [49]

    Patrascu and I

    A. Patrascu and I. Necoara , On the convergence of inexact projection primal first-order methods for convex minimization, IEEE Trans. Autom. Control., 63 (2018), pp. 3317–3329. 3

    2018

  42. [50]

    Polyak, A general method for solving extremum problems , Dokl

    B. Polyak, A general method for solving extremum problems , Dokl. Akad. Nauk SSSR, 174 (1967). 2

    1967

  43. [51]

    Polyak and P

    B. Polyak and P. Shcherbakov , Lyapunov functions: An optimization theory perspective**this work was sup- ported by the russian scientific foundation, project no. 16-11-10015 , IF AC-PapersOnLine, 50 (2017), pp. 7456–

    2017

  44. [52]

    J. B. Rosen, The gradient projection method for nonlinear programming. part i. linear constraints , Journal of the Society for Industrial and Applied Mathematics, 8 (1960), pp. 181–217. 2

    1960

  45. [53]

    , The gradient projection method for nonlinear programming. part ii. nonlinear constraints , Journal of the Society for Industrial and Applied Mathematics, 9 (1961), pp. 514–532. 2

    1961

  46. [54]

    J. M. Sanz Serna and K. C. Zygalakis , The connections between lyapunov functions for some optimization algorithms and differential equations , SIAM J. Numer. Anal., 59 (2021), pp. 1542–1565. 3

    2021

  47. [55]

    Scheurer , Existence et approximation de points selles pour certains probl` emes non lin´ eaires, RAIRO

    B. Scheurer , Existence et approximation de points selles pour certains probl` emes non lin´ eaires, RAIRO. Anal. num´ er., 11 (1977), pp. 369–400. 19

    1977

  48. [56]

    Schropp and I

    J. Schropp and I. Singer , A dynamical systems approach to constrained minimization , Numer. Funct. Anal., 21 (2000), pp. 537–551. 4

    2000

  49. [57]

    M. ´O. Searc´oid, Metric Spaces, Springer-Verlag, 2006. 5

    2006

  50. [58]

    B. Shi, S. S. Du, M. I. Jordan, and W. J. Su , Understanding the acceleration phenomenon via high-resolution differential equations, Math. Program., 195 (2022), pp. 79–148. 3

    2022

  51. [59]

    Shikhman and O

    V. Shikhman and O. Stein , Constrained optimization: Projected gradient flows , J. Optim. Theory Appl., 140 (2009), pp. 117–130. 1, 4

    2009

  52. [60]

    W. Su, S. Boyd, and E. J. Cand‘es , A differential equation for modeling Nesterov’s accelerated gradient method: Theory and insights , J. Mach. Learn. Res., (2016). 3

    2016

  53. [61]

    TANABE, An algorithm for the constrained maximization in nonlinear programming , J

    K. TANABE, An algorithm for the constrained maximization in nonlinear programming , J. Operations Research Soc. of Japan, 17 (1973). 4

    1973

  54. [62]

    Tanabe, A geometric method in nonlinear programming , J

    K. Tanabe, A geometric method in nonlinear programming , J. Optim. Theory Appl., 30 (1980), pp. 181–210. 4

    1980

  55. [63]

    Uzawa, Iterative methods for concave programming , Studies in linear and nonlinear programming, 6 (1958), pp

    H. Uzawa, Iterative methods for concave programming , Studies in linear and nonlinear programming, 6 (1958), pp. 154–165. 2

    1958

  56. [64]

    W ang and Q

    C. W ang and Q. Liu , Convergence properties of inexact projected gradient methods , Optimization, 55 (2006), pp. 301–310. 2

    2006

  57. [65]

    , Convergence properties of inexact projected gradient methods , Optimization, 55 (2006), pp. 301–310. 3

    2006

  58. [66]

    A. C. Wilson, B. Recht, and M. I. Jordan , A lyapunov analysis of accelerated methods in optimization , J. Mach. Learn. Res., 22 (2021), pp. 1–34. 3

    2021

  59. [67]

    Xu and L

    J. Xu and L. Zikatanov, Algebraic multigrid methods, Acta Numerica, 26 (2017), pp. 591–721. 5

    2017

  60. [68]

    Y. Xu, I. Yousept, and J. Zou, An adaptive edge element approximation of a quasilinear H(curl)-elliptic problem , Math. Models Methods Appl. Sci., 30 (2020), pp. 2799–2826. 19

    2020

  61. [69]

    Yamashita, A differential equation approach to nonlinear programming , Math

    H. Yamashita, A differential equation approach to nonlinear programming , Math. Program., 18 (1980), pp. 155–

    1980

  62. [7461]

    20th IF AC World Congress. 3

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