Pith. sign in

REVIEW 4 major objections 6 minor 44 references

Continuous and discrete-time accelerated methods for an inequality constrained convex optimization problem

T0 review · 4 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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

Pith's one-line read A Bregman-Lagrangian dynamical system is proved to converge exponentially to the solution of an inequality-constrained convex program, and its discretizations give accelerated gradient methods.

desk verdict The Bregman-Lagrangian machinery is applied cleanly to a fixed log-barrier, but the paper's central theorem silently treats the barrier as fixed while Assumption 3.1 makes it time-dependent, so the exponential convergence claim is not actually proved. read the letter →

arxiv 2411.14828 v1 pith:PEATNWIQ submitted 2024-11-22 math.OC

Juan Liu , Nan-Jing Huang , Xian-Jun Long , Xue-song Li This is my paper · ORCID
classification math.OC MSC 90C2590C3065K0537N40
keywords inequalityconstrainedconvexoptimizationlogarithmicbarrierBregmanLagrangiancontinuous-timedynamicalsystemLyapunovfunctionacceleratedgradientmethodEulerdiscretizationconvergencerate
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 tries to show that an inequality-constrained convex optimization problem can be solved by a smooth dynamical system that tracks the constrained optimum exponentially fast. The strategy is to replace the hard constraints with a logarithmic barrier whose strength and a slack variable are scheduled by time, turning the problem into an unconstrained one, and then to derive a Bregman-Lagrangian flow from Hamilton's principle. The main result states that, when the barrier parameter grows as $e^{\beta t}$ and the slack decays as $e^{-\beta t}$, the trajectory $X_t$ of the flow satisfies $\|X_t - x^*\|_2 \le C_0 e^{-\beta t/2}$. If the proof is right, it gives a continuous-time explanation of acceleration for constrained problems and a principled route to discrete accelerated gradient methods with provable rates.

What carries the argument

The load-bearing object is the Bregman Lagrangian $L(X_t,\dot X_t,t)=e^{\alpha t+\gamma t}(V_h(X_t+e^{-\alpha t}\dot X_t,X_t)-e^{\beta t}\Phi(X_t))$, where $V_h$ is the Bregman divergence of a strongly convex distance-generating function $h$. Under the ideal scaling conditions $\dot\beta_t \le e^{\alpha t}$ and $\dot\gamma_t = e^{\alpha t}$, its Euler-Lagrange equation collapses to the differential inclusion $d\nabla h(X_t+e^{-\alpha t}\dot X_t) = -e^{\alpha t+\beta t}\nabla\Phi(X_t)\,dt$. The convergence argument is carried by the Lyapunov function $E_t=e^{\beta t}(\Phi(X_t)-\Phi(\hat{x}^*)) + V_h(\hat{x}^*, X_t+e^{-\alpha t}\dot X_t)$, whose monotone decrease transfers to distance to the approximate optimum, and by Lemma 2.1, which transfers that to distance to the true optimum. For the discrete half, the same Lyapunov structure is discretized: implicit Euler gives Algorithm 1, and a hybrid Euler step with an extra sequence $y_k$ gives Algorithm 2.

What would settle it

Numerically integrate (3.4) for the quadratic example (5.1) with $\beta_t = 2t$ and measure $\log\|X_t - x^*\|_2$; Theorem 3.1 predicts a slope approaching $-1$, meaning $\|X_t - x^*\|_2 \sim e^{-t}$. A slope that flattens, or any iteration at which $s - g_i(X_t)$ reaches zero and the log-barrier gradient blows up, would falsify the claim.

Watch

Extended reading notes

Core claim

The paper's central claim is Theorem 3.1: under Assumption 3.1, the solution of the continuous-time system (3.4) converges to the unique optimal solution $x^*$ of the original problem, with the explicit rate $\|X_t - x^*\|_2 = O(e^{-\beta t/2})$ when the objective is smooth. The mechanism is a two-level approximation: the log-barrier $\Phi(x)=f(x)-\frac{1}{c}\sum_i\log(s-g_i(x))$ has a unique minimizer $\hat{x}^*$, Lemma 2.1 bounds the gap between $\hat{x}^*$ and $x^*$ by $m/c + s\|\lambda^*\|_1$, and Lemma 3.1 shows the flow converges to $\hat{x}^*$ with the same exponential factor. Making $c(t)=e^{\beta t}$ and $s(t)=e^{-\beta t}$ balances the two errors, so the trajectory homes in on the true optimum at rate $e^{-\beta t/2}$. The paper further derives discrete algorithms by Euler discretization: Algorithm 1 matches the continuous rate $O(1/\sqrt{A_k})$, while Algorithm 2, a hybrid implicit-explicit scheme with an additional mirror-descent sequence, is claimed to achieve $O(1/k)$.

Load-bearing premise

The whole argument depends on being able to schedule the barrier parameter and slack as $c(t)=e^{\beta t}$ and $s(t)=e^{-\beta t}$ while the trajectory remains inside the open set $\hat K$; if that scheduling cannot be maintained, the exponential tracking bound collapses.

Editorial extensions

If this is right

  • With $\beta_t = 2p\log t$, the continuous system converges polynomially at rate $O(1/t^p)$, and with $\beta_t = 2t^p$ it converges as $O(e^{-t^p})$, giving a tunable family of rates from the same Lyapunov argument.
  • Algorithm 1 inherits the continuous rate: with $A_k = (\delta k)^{2p}$ it achieves $O(1/(\delta k)^p)$, and with $A_k = e^{2\delta k p}$ it achieves $O(e^{-\delta k p})$.
  • Algorithm 2 achieves $O(1/k)$ convergence to $x^*$ for the smoothed problem, and the numerical experiments show it reaches a fixed accuracy with fewer iterations than the plain gradient scheme (Algorithm 3).
  • When all constraint functions are identically zero, the framework reduces to unconstrained accelerated methods, recovering the standard setting as a special case.

Reading between the lines

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

  • A natural extension is to make the barrier schedule adaptive, choosing $c$ from the current duality gap $\|\lambda^*\|_1$ and using Lemma 2.1 as a stopping criterion; the paper's proof structure suggests the same Lyapunov machinery would carry over if $\dot c/c$ stays comparable to $\dot\beta$.
  • The proof of Theorem 4.2 uses step sizes $\eta_k \le 1/(kL_\Phi)$, while Algorithm 2 is stated with a fixed $\eta$; whether the $O(1/k)$ rate holds under the fixed-step version is an open testable question.
  • Because the slack variable starts large and decays, the method can be started from infeasible points; this suggests a direct comparison with interior-point path-following, where the barrier parameter plays the same role and the path $\hat{x}^*(c,s)$ is the central path.
Share X Bluesky LinkedIn Reddit HN

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. The paper proposes continuous-time and discrete-time accelerated methods for inequality-constrained convex optimization (1.1). It approximates the constrained problem by a logarithmic-barrier objective (2.7), derives a Bregman-Lagrangian dynamical system (3.4) via Hamilton's principle, and proves exponential convergence of the continuous-time trajectory to the true optimum under Assumption 3.1. It then discretizes the dynamics to obtain three algorithms and gives convergence rates, together with numerical experiments on a quadratic program.

Significance. If the main results were correct, the paper would be a useful contribution to the variational/continuous-time understanding of accelerated methods for inequality-constrained problems, a setting that has received less attention than equality-constrained or unconstrained problems. The Bregman-Lagrangian derivation (Proposition 3.1) is standard and, for a fixed barrier, the Lyapunov argument in Lemma 3.1 is essentially sound. The paper also makes a concrete computational proposal, Algorithm 2, and documents numerical behavior. However, as shown below, the central time-varying barrier argument and the discrete-time step-size conditions contain load-bearing gaps, so the claims are not established in their present form.

major comments (4)
  1. [§3.2, Lemma 3.1 and Theorem 3.1] Lemma 3.1 is proved for a fixed objective Φ and a fixed minimizer x̂*, but Assumption 3.1 makes c(t)=e^{β_t} and s(t)=e^{-β_t}, so Φ_t and x̂*_t are time-dependent. The Lyapunov function in (3.6) is E_t = e^{β_t}(Φ_t(X_t)-Φ_t(x̂*_t)) + V_h(x̂*_t, X_t+e^{-α_t}Ẋ_t), and its derivative contains no ∂Φ_t/∂t or dx̂*_t/dt terms. The proof of Lemma 3.1 therefore does not imply non-increase of E_t, and inequality (3.5) does not follow for the time-varying barrier. Extra terms such as e^{β_t}(∂Φ_t/∂t(X_t)-∂Φ_t/∂t(x̂*_t)) and <∇h(x̂*_t)-∇h(Y_t), dx̂*_t/dt> must be estimated; no such estimate is given. This gap affects Theorem 3.1 and also the discrete-time Theorems 4.1 and 4.2, which invoke Assumption 3.1 while using Lyapunov functions with an implicitly fixed x̂*.
  2. [§4.2, Theorem 4.2 and Algorithm 2] Theorem 4.2 states that if A_k=C k^2, 0<C≤μ/(4L_Φ), and 0<η≤1/(kL_Φ), then ε_{k+1}≤0 and ‖x_k-x*‖=O(1/k). However, Algorithm 2 fixes η>0 in its initialization and uses the same η in every iteration. Since 1/(kL_Φ) tends to 0, no fixed positive η can satisfy the theorem's condition for all k. The stated convergence proof therefore does not apply to Algorithm 2 as written. The numerical section (Section 5) uses the constant values η=1/4 and η=1/2 and asserts that all conditions of Theorem 4.2 are satisfied, which is inconsistent with the theorem's own step-size requirement.
  3. [§4.2, proof of Theorem 4.2] The derivation of Φ(y_{k+1})-Φ(x_{k+1}) ≤ -1/(2L_Φ)‖∇Φ(x_{k+1})‖² is not justified. From smoothness and y_{k+1}-x_{k+1}=-η∇Φ(x_{k+1}) one obtains Φ(y_{k+1})-Φ(x_{k+1}) ≤ -(η - L_Φ η²/2)‖∇Φ(x_{k+1})‖². For η=1/(kL_Φ), the coefficient is (2k-1)/(2k²L_Φ), which equals 1/(2L_Φ) only at k=1 and is smaller for k>1. Consequently, the subsequent inequality δα_k²/(2μ) - A_{k+1}/(2δL_Φ) ≤ 0 does not imply ε_{k+1}≤0; with the correct k-dependent coefficient, the term δα_k²/(2μ) grows like k² and cannot be controlled for all k by the stated choice of C. This gap affects the claimed O(1/k) rate even if η were allowed to be time-varying.
  4. [§4.3, Algorithm 3 and Corollary 4.1] Corollary 4.1 is stated under Assumption 3.1, but Algorithm 3 sets A_k = Σ_{i=0}^k 1/(i+1)², which is bounded above by π²/6. Hence s_k=1/A_k does not tend to 0, violating Assumption 3.1(ii). The use of Lemma 2.1 to pass from convergence of Φ(x_k)-Φ(x̂*) to convergence of x_k to the true optimum x* is therefore not justified in this case, since the approximation error |f(x̂*)-f(x*)| need not vanish. Corollary 4.1's conclusion ‖x_k-x*‖=O(1/√k) requires an additional argument that is not supplied.
minor comments (6)
  1. [§3.1, Proposition 3.1 proof] In the partial derivative ∂L/∂X_t, the expression should be e^{γ_t}⟨∇h(X_t), ·⟩ rather than an undefined product; there is also a missing gradient symbol in the second line. These typos make the derivation harder to follow, although the fixed-barrier calculation is plausible.
  2. [§3.2, Lemma 3.1 proof] The derivative computation contains the term e^{β_t}⟨Φ(X_t), Ẋ_t⟩, which should be e^{β_t}⟨∇Φ(X_t), Ẋ_t⟩. The same typo appears in the displayed formula before substitution of (3.4).
  3. [§4.2, equation (4.4)] The second line of (4.4) reads ‖y_{k+1}-x_{k+1}‖₂ = -η∇Φ(x_{k+1}), which equates a scalar norm to a vector. This should be y_{k+1}-x_{k+1} = -η∇Φ(x_{k+1}).
  4. [§2, Lemma 4.2 proof] In the proof of Lemma 4.2, the expression for V_h(x, x̂*) uses ∇h(y) with an undefined y; it should be ∇h(x̂*).
  5. [§3.2, Theorem 3.1 proof] The final bound contains the typo f(x̂* - f(x*)); the intended expression is f(x̂*) - f(x*). Also, the constant C₀ is defined with a parenthesis mismatch in the displayed formula.
  6. [General notation] The symbol T is used both as the time interval in (3.1) and as the transpose operator in (2.4) and elsewhere. This is confusing; a different symbol for the time horizon would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the derivation is built from explicit external assumptions and a priori barrier schedules; the flagged concerns are internal proof gaps, not circular reductions.

full rationale

The paper's derivation chain is not circular. The Bregman Lagrangian (3.1) and the ideal scaling conditions (3.2) are stated as assumptions and explicitly credited to Wibisono et al. [23,26]; Proposition 3.1 derives the continuous-time system (3.4) from the Euler-Lagrange equation under those stated conditions. The barrier approximation error in Lemma 2.1 is quoted from Boyd and Vandenberghe's standard sensitivity and barrier analysis [3, Sec.5.6 and Sec.11], not from the authors' own prior work. Lemma 3.1 constructs the Lyapunov function (3.6) and proves non-increase using only the stated scaling conditions, strong convexity, and the Bregman divergence properties; the rate O(e^{-beta t/2}) is a direct consequence of the a priori chosen schedule beta_t, not of any fitted parameter. Theorem 3.1 combines these ingredients with c(t)=e^{beta_t} and s(t)=e^{-beta_t}, which are set in Assumption 3.1 as user-chosen time scalings, not calibrated to the solution or to numerical data. The discrete-time proofs similarly use the same explicit assumptions and standard convexity inequalities. There is no self-citation chain, no fitted quantity renamed as a prediction, and no target result embedded in the definition of the Lyapunov function or the algorithm parameters. The manuscript's genuine weaknesses, such as applying fixed-barrier lemmas to the time-varying barrier required by Assumption 3.1 and the mismatch between Theorem 4.2's condition eta <= 1/(k L_Phi) and Algorithm 2's fixed positive eta, are correctness or proof gaps rather than circular reductions; they do not make the claimed result equivalent to its own inputs by construction.

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

The central results rest on the time-varying barrier and slack schedules (c=e^{beta t}, s=e^{-beta t}, with user-chosen beta_t), the standard convex-analysis assumptions (strong convexity, smoothness, Slater), and the unproven existence of a solution inside the enlarged feasible set. The discrete accelerated method additionally depends on a step-size condition whose consistency with Algorithm 2 is dubious.

free parameters (5)
  • beta_t (time-scaling exponent) = 2p log t or 2t^p (user-chosen)
    Controls the exponential convergence rate O(e^{-beta t/2}) and the barrier/slack schedules; not fitted to data.
  • barrier parameter c(t) = e^{beta t} = e^{beta t}
    Assumption 3.1(i); makes the log-barrier approximation tighten at rate e^{-beta t}.
  • slack s(t) = e^{-beta t} = e^{-beta t}
    Assumption 3.1(ii); keeps iterates in the enlarged feasible set while shrinking.
  • eta (gradient step size in Algorithm 2) = constant, e.g., 1/4 or 1/2
    Chosen by hand; Theorem 4.2's condition eta <= 1/(k L_Phi) contradicts a constant eta.
  • C in A_k = C k^2 (Algorithm 2) = C = 0.1 (Euclidean), 2/3 (entropy)
    Determines the Lyapunov decrease rate; must satisfy C <= mu/(4 L_Phi).
assumptions (5)
  • domain assumption Slater's condition and strict feasibility of the constrained problem
    Used to guarantee existence of KKT multipliers and the error bound in Lemma 2.1.
  • domain assumption f is m_f-strongly convex and L_f-smooth; g_i convex; X convex compact
    Theorems 3.1, 4.1, 4.2 rely on strong convexity for distance bounds and smoothness for descent steps.
  • domain assumption Assumption 3.1: c(t)=e^{beta t}, s(t)=e^{-beta t}, and bounded optimal dual norm
    Central to Theorem 3.1; makes the approximation error vanish at the same exponential rate as the optimization error.
  • ad hoc to paper Solution X_t remains inside K_hat for all t
    Remark 3.1 argues the gradient blows up at the boundary, but existence of a global solution in K_hat is not proven rigorously.
  • standard math Existence of L_Phi such that Hessian of Phi is bounded above on the sublevel set (Lemma 4.1)
    Standard consequence of twice continuous differentiability on a compact sublevel set.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Continuous and discrete-time accelerated methods for an inequality constrained convex optimization problem." pith.science (2026). https://pith.science/paper/PEATNWIQ

@misc{pith2026241114828,
  author       = {Pith},
  title        = {Pith review of: Continuous and discrete-time accelerated methods for an inequality constrained convex optimization problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PEATNWIQ}},
  note         = {Machine review of arXiv:2411.14828}
}
read the original abstract

This paper is devoted to the study of acceleration methods for an inequality constrained convex optimization problem by using Lyapunov functions. We first approximate such a problem as an unconstrained optimization problem by employing the logarithmic barrier function. Using the Hamiltonian principle, we propose a continuous-time dynamical system associated with a Bregman Lagrangian for solving the unconstrained optimization problem. Under certain conditions, we demonstrate that this continuous-time dynamical system exponentially converges to the optimal solution of the inequality constrained convex optimization problem. Moreover, we derive several discrete-time algorithms from this continuous-time framework and obtain their optimal convergence rates. Finally, we present numerical experiments to validate the effectiveness of the proposed algorithms.

Figures

Figures reproduced from arXiv: 2411.14828 by the authors.

Figure 2.1
Figure 2.1. The dashed lines show the function I−(u), and the solid curves show ˆI−(u) = −(1/c) log(−u), for c = 0.5, 1, 2. The curve for c = 2 gives the best approximation. Substituting ˆI− for I− in (2.5), we obtain the following approximation: min x∈K f(x) − 1 c X i∈I log(−gi(x)), (2.6) where K := {x ∈ X : gi(x) < 0, ∀i ∈ I}, and the function φ(x) = (1/c) P i∈I log(−gi(x)) with dom(φ) = K is known as the logarithmic barrier … view at source ↗
Figure 5.1
Figure 5.1. the coordinates of the optimal trajectory [PITH_FULL_IMAGE:figures/full_fig_p016_5_1.png] view at source ↗
Figure 5.2
Figure 5.2. the coordinates of the constrain function [PITH_FULL_IMAGE:figures/full_fig_p016_5_2.png] view at source ↗
Figures from the paper (2 more)
Figure 5.3
Figure 5.3. Figure 5.3: the performance of Algorithm 2 applied to (5.1) under diffe [PITH_FULL_IMAGE:figures/full_fig_p017_5_3.png]
Figure 5.4
Figure 5.4. Figure 5.4: The objective function value gap obtained by Algorithm 2 ( [PITH_FULL_IMAGE:figures/full_fig_p017_5_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

44 extracted references · 40 canonical work pages

  1. [1]

    Beck, A. (2023). Introduction to nonlinear optimization: theor y, algorithms, and applications with python and MATLAB (2nd ed.). Society for Industrial and Applied Ma thematics, Philadelphia

    work page 2023

  2. [2]

    Boyd, S., Parikh, N., Chu, E., Peleato, B., Eckstein, J. (2011). Dis tributed optimization and statisti- cal learning via the alternating direction method of multipliers. Found ations and Trends in Machine Learning, 3(1), 1-122

    work page 2011

  3. [3]

    Boyd, S., Vandenberghe, L. (2004). Convex optimization, Camb ridge University Press, Cambridge

    work page 2004

  4. [4]

    Nocedal, J., Wright, S. (2006). Numerical optimization (2nd ed.). Springer Series in Operations Research and Financial Engineering. New York

    work page 2006

  5. [5]

    Bertsekas, D.P., Nedi´c, A., Ozdaglar, A.E. (2003). Convex analysis and optimization. Athen a Scien- tific, Belmont

    work page 2003

  6. [6]

    K.C., Ruszczynski

    Shor, N.Z., Kiwiel. K.C., Ruszczynski. A. (2012). Minimization method s for non-differentiable func- tions. Springer Series in Computational Mathematics, New York

    work page 2012

  7. [7]

    Esmaeili, H., Kimiaei, M. (2014). A new adaptive trust-region metho d for system of nonlinear equations. Applied Mathematical Modelling, 38(11-12), 3003-3015

    work page 2014

  8. [8]

    Chen, Y., Lan, G., Ouyang, Y. (2014). Optimal primal-dual metho ds for a class of saddle point problems. SIAM Journal on Optimization, 24(4), 1779-1814

    work page 2014

Show all 44 references
  1. [9]

    Nesterov, Y.E. (1983). A method of solving a convex programmin g problem with convergence rate O(1/k2). Soviet Mathematics Doklady, 27(2), 372-376

    1983

  2. [10]

    Zhang, J., Luo, Z.Q. (2020). A proximal alternating direction me thod of multipliers for linearly constrained nonconvex minimization. SIAM Journal on Optimization, 30(3), 2272-2302

    2020

  3. [11]

    Kong, W., Monteiro, R.D.C. (2023). An accelerated inexact damp ened augmented Lagrangian method for linearly-constrained nonconvex composite optimization problems. Computational Op- timization and Applications, 85, 509-545

    2023

  4. [12]

    Nesterov, Y. (2005). Smooth minimization of non-smooth func tions. Mathematical Programming, 103(1), 127-152

    2005

  5. [13]

    Krichene, W., Bayen, A.M., Bartlett, P.L. (2015). Accelerated m irror descent in continuous and discrete time. Proceedings of the 28th International Conferenc e on Neural Information Processing Systems, 2, pages 2845-2853

    2015

  6. [14]

    Lan, G., Lu, Z., Monteiro, R.D.C. (2011). Primal-dual first-orde r methods with O(1/ǫ) iteration- complexity for cone programming. Mathematical Programming, 126 (1), 1-29

    2011

  7. [15]

    Beck, A., Teboulle, M. (2009). A fast iterative shrinkage-thre sholding algorithm for linear inverse problems. SIAM Journal on Imaging Sciences, 2(1), 183-202

    2009

  8. [16]

    Lan, G. (2012). An optimal method for stochastic composite o ptimization. Mathematical Program- ming, 133(1-2), 365-397. 18

    2012

  9. [17]

    Liu, Y.F., Liu, X., Ma, S. (2019). On the nonergodic convergence rate of an inexact augmented Lagrangian framework for composite convex programming. Mathe matics of Operations Research, 44(2), 632-650

    2019

  10. [18]

    Ghadimi, S., Lan, G. (2012). Optimal stochastic approximation a lgorithms for strongly convex stochastic composite optimization I: A generic algorithmic framewor k. SIAM Journal on Optimiza- tion, 22(4), 1469-1492

    2012

  11. [19]

    Ghadimi, S., Lan, G. (2015). Accelerated gradient methods for nonconvex nonlinear and stochastic programming. Mathematical Programming, 156(1), 59-99

    2015

  12. [20]

    Su, W., Boyd, S., Cand` es, E.J. (2016). A differential equation f or modeling Nesterov’s accelerated gradient method: Theory and insights. Journal of Machine Learnin g Research, 17, 1-43

    2016

  13. [21]

    Shi, B., Du, S.S., Jordan, M.I., Su, W.J. (2021). Understanding th e acceleration phenomenon via high-resolution differential equations. Mathematical Programming , 195, 1-70

    2021

  14. [22]

    Nemirovski, A., Yudin, D.B. (1983). Problem complexity and metho d efficiency in optimization. Wiley & Sons, New York

    1983

  15. [23]

    Wibisono, A., Wilson, A.C., Jordan, M.I. (2016). A variational pers pective on accelerated methods in optimization. Proceedings of the National Academy of Sciences, 1 13(47), E7351-E7358

    2016

  16. [24]

    Betancourt, M., Jordan, M.I., Wilson, A. (2018). On symplectic o ptimization. Arxiv preprint arXiv1802.03653

    2018 arXiv

  17. [25]

    Muehlebach, M., Jordan, M.I. (2021). Optimization with momentu m: dynamical, control-theoretic, and symplectic perspectives. Journal of Machine Learning Resear ch, 22, 1-50

    2021

  18. [26]

    Wilson, A.C., Recht, B., Jordan, M.I. (2021). A Lyapunov analysis of accelerated methods in opti- mization. Journal of Machine Learning Research, 22, 1-34

    2021

  19. [27]

    Chen, S., Shi, B., Yuan, Y.X. (2022). Revisiting the acceleration p henomenon via high-resolution differential equations. arXiv:2212.05700

    2022 arXiv

  20. [28]

    Wang, Y., Jia, Z., Wen, Z. (2021). Search direction correction w ith normalized gradient makes first-order methods faster. SIAM Journal on Scientific Computin g. 43(5), A3184-A3211

    2021

  21. [29]

    Luo, H., Chen, L. (2022). From differential equation solvers to accelerated first-order methods for convex optimization. Mathematical Programming. 195, 735-781

    2022

  22. [30]

    Chen, L., Luo, H. (2019). First order optimization methods bas ed on Hessian-driven Nesterov accel- erated gradient flow. arXiv:1912.09276

    2019 arXiv

  23. [31]

    Bao, C.L., Chen, L., Li, J.H. (2023). The global R-linear converge nce of nesterov’s accelerated gradient method with unknown strongly convex parameter. arXiv:2 308.14080

    2023

  24. [32]

    Bao, C.L., Chen, L., Li, J.H., Shen, Z.W. (2024). Accelerated grad ient methods with gradient restart: Global linear convergence. ArXiv: 2401.07672. 19

    2024 arXiv

  25. [33]

    Fazlyab, M., Koppel, A., Ribeiro, A., Preciado, V.M. (2017). A varia tional approach to dual methods for constrained convex optimization. In Proceedings of the Americ an Control Conference (pp. 5269- 5275)

    2017

  26. [34]

    Zeng, X., Lei, J., Chen, J. (2022). Dynamical primal-dual accele rated method with applications to network optimization. arXiv:1912.03690

    2022 arXiv

  27. [35]

    R., Tits, A

    Panier, E. R., Tits, A. L., Herskovits, J. N. (1988). A QP-free, globally convergent, locally superlin- early convergent algorithm for inequality constrained optimization. SIAM Journal on Control and Optimization. 26(4), 788-811

    1988

  28. [36]

    B., Pan, H

    Jian, J. B., Pan, H. Q., Tang, C. M., Li, J. L. (2015). A strongly su b-feasible primal-dual quasi interiorpoint algorithm for nonlinear inequality constrained optimizat ion. Applied Mathematics and Computation. 266, 560-578

    2015

  29. [37]

    Su, K., Ren, L. (2023). A modified nonmonotone filter QP-free m ethod. Miskolc Mathematical Notes. 24(1), 457-471

    2023

  30. [38]

    He, X., Hu, R., Fang, Y.P. (2021). Convergence rates of inertia l primal-dual dynamical methods for separable convex optimization problems. SIAM Journal on Optimizat ion. 59(5), 3278-3301

    2021

  31. [39]

    Attouch, H., Chbani, Z., Fadili, J., Riahi, H. (2022). Fast converg ence of dynamical ADMM via time scaling of damped inertial dynamics. Journal of Optimization The ory and Applications. 193(5), 704-736

    2022

  32. [40]

    Fazlyab, M., Paternain, S., Preciado, V.M., Ribeiro, A. (2018). Pr ediction-correction interior-point method for time-varying convex optimization. IEEE Transactions o n Automatic Control, 63(7), 1973-1986

    2018

  33. [41]

    Bailey, C. (1982). Hamilton’s principle and the calculus of variation s. Acta Mechanica, 44, 49-57

    1982

  34. [42]

    Lyapunov, A.M. (1992). General problem of the stability of mot ion. International Journal of Control, 55(3), 531-773

    1992

  35. [43]

    Kloeden, P.E., Platen, E. (1992). Higher-order implicit strong nu merical schemes for stochastic differential equations. Journal of Statistical Physics, 66(3-4), 283-314

    1992

  36. [44]

    Nesterov, Y., Shikhman, V. (2015). Quasi-monotone subgrad ient methods for nonsmooth convex minimization. Journal of Optimization Theory and Applications, 165(3 ), 917-940. 20

    2015

Pith tools

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