Pith. sign in

REVIEW 2 major objections 3 minor 38 references

Contractive difference-of-convex algorithms

T0 review · 2 major / 3 minor · reviewed 2026-08-15 · deepseek-v4-flash

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

Pith's one-line read The paper proves that each linearized proximal subproblem in a difference-of-convex method is a fixed point problem for a contraction, and uses that reformulation to build cDCA, an algorithm with an adaptive inner stopping rule and global…

desk verdict A clean contraction-based subproblem solver with a genuinely adaptive stopping rule; the whole-sequence convergence claim overreaches for the ℓ1−2 experiments. read the letter →

arxiv 2505.10800 v1 pith:RCDNEFE7 submitted 2025-05-16 math.OC

Songnian He , Qiao-Li Dong , Michael Th. Rassias This is my paper · ORCID
classification math.OC MSC 90C3065K0590C26
keywords difference-of-convexoptimizationlinearizedproximalmethodcontractivemappingPicarditerationadaptiveterminationruleKurdyka-Lojasiewiczpropertyfixedpointconvergenceanalysis
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 targets difference-of-convex problems minimizing $F = f + g - h$, where standard DCA iterations reduce to convex subproblems that usually lack closed-form solutions and are solved by slow iterative schemes. The core claim is that the linearized proximal method subproblem (10) is equivalent to the fixed point problem of a contraction, $\operatorname{Prox}_{\mu g}[T_k^{\lambda,\mu}]$, for $\mu\in(0,2/(2\lambda+L_f)]$. Based on this, the paper proposes cDCA, which approximates each subproblem by Picard iteration and stops the inner loop adaptively when the inner displacement drops below a fixed fraction of the previous outer displacement. The convergence results say every cluster point is critical, the objective values converge, and under a standard asymptotic regularity condition the whole sequence converges to a critical point without any pre-tuned decreasing inner tolerance schedule.

What carries the argument

The central object is the composite map $\operatorname{Prox}_{\mu g}[T_k^{\lambda,\mu}] = \operatorname{Prox}_{\mu g}[(1-\mu\lambda)(I-\frac{\mu}{1-\mu\lambda}\nabla f)(\cdot)+\mu\lambda x_k+\mu\eta_k]$. It is a contraction because $\nabla f$ is $1/L_f$-cocoercive, so $I-\nu\nabla f$ is nonexpansive for $\nu\le2/L_f$, and $\operatorname{Prox}_{\mu g}$ is firmly nonexpansive; the affine shift preserves the contraction constant $1-\mu\lambda$. This object carries the argument: it turns the convex subproblem into a Banach fixed point problem, justifies Picard iteration, and makes the adaptive stopping rule (16) natural, because inner contraction error can be controlled by outer displacement. The auxiliary function $E$ then supplies the descent framework that drives both subsequential and whole-sequence convergence.

What would settle it

Run cDCA on a one-dimensional problem with $f(x)=x^2/2$, $g=0$, and $h(x)=|x|^{3/2}$, so $\nabla h$ is convex but not Lipschitz on the convex hull of any sequence passing through $0$, while the other hypotheses of Theorem 4.1 hold. If the outer sequence still converges to a critical point, the Lipschitz-gradient assumption in Theorem 4.2 is not needed; if it fails or stalls, that assumption is load-bearing. Also record whether the inner termination rule (16) is ever satisfied when two consecutive outer iterates coincide exactly.

Watch

Extended reading notes

Core claim

Lemma 3.1 is the hinge: for $x_k$ and $\eta_k\in\partial h(x_k)$, the unique minimizer of (10) is exactly the unique fixed point of $\operatorname{Prox}_{\mu g}[T_k^{\lambda,\mu}]$, where $T_k^{\lambda,\mu}(x)=(1-\mu\lambda)x-\mu\nabla f(x)+\mu\lambda x_k+\mu\eta_k$. For $\mu\in(0,2/(2\lambda+L_f)]$ the composed operator is a contraction with coefficient $1-\mu\lambda$, and the best coefficient $L_f/(2\lambda+L_f)$ is attained at $\mu=2/(2\lambda+L_f)$. The paper therefore replaces the usual inner proximal-gradient loop by Picard iteration of this contraction, terminated by (16): $\|x_{m_k}^{k+1}-x_{m_k-1}^{k+1}\|\le\delta\|x_{k-1}-x_k\|$ with $\delta\in(0,2\lambda/L_f)$. The convergence analysis shows the auxiliary function $E(x,y)=F(x)+\tau\|x-y\|^2$ decreases along the pairs $(x_k,x_{k-1})$, that outer step lengths are square-summable, and that all cluster points satisfy $0\in\nabla f(x)+\partial g(x)-\partial h(x)$. Theorem 4.2 then upgrades subsequential convergence to whole-sequence convergence when $E$ has the KL property at the limit and $\nabla h$ is Lipschitz on the convex hull of the iterates.

Load-bearing premise

For the whole sequence to converge (not just along subsequences), the proof needs the slope of the convex function $h$ to change at a bounded rate all along the path the iterates travel, and the auxiliary function $E$ to satisfy a standard asymptotic flatness condition; neither follows from the basic assumptions and neither is verified in the experiments.

Editorial extensions

If this is right

  • Every outer iteration of cDCA needs only finitely many inner Picard steps when consecutive outer iterates differ, so no user-supplied inner tolerance sequence is required.
  • All cluster points of the cDCA sequence are critical points of $F$, and the sequence of objective values converges.
  • Adding the KL property and Lipschitz differentiability of $h$ along the iterates upgrades this to convergence of the whole sequence to a critical point, with rates when the KL exponent is known.
  • In the reported experiments on $\ell_1\text{-}2$ and logarithmic regularized least squares, cDCA uses fewer total iterations and less CPU time than ADCA and pDCAe at comparable final objective values.

Reading between the lines

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

  • Beyond the paper, the contraction constant $L_f/(2\lambda+L_f)$ gives a direct bound on the inner iteration count: aiming for inner error $\varepsilon$ costs about $\log(\varepsilon)/\log(L_f/(2\lambda+L_f))$ Picard steps, so the adaptive rule can be converted into an explicit worst-case complexity estimate.
  • Beyond the paper, the fixed-point equivalence depends on $g$ only through $\operatorname{Prox}_{\mu g}$, so the same contraction argument should apply to any convex prox-friendly regularizer without changing the inner-loop analysis.
  • Beyond the paper, a natural test of Theorem 4.2's necessity is to run cDCA with $h(x)=|x|^{3/2}$ in one dimension; if the whole sequence converges anyway, the $L_h$-Lipschitz assumption could be removed or weakened.
  • Beyond the paper, the adaptive stopping criterion (16) is portable: any outer method that produces a stabilizing quadratic term could adopt a relative inner-error rule keyed to the previous outer displacement instead of an absolute tolerance schedule.
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

2 major / 3 minor

Summary. The paper studies the difference-of-convex minimization problem min F(x)=f(x)+g(x)-h(x), where f is smooth convex with Lipschitz gradient, g is prox-friendly convex, and h is convex. Its first contribution is Lemma 3.1, which reformulates the linearized proximal method (LPM) subproblem as a fixed-point problem of a contraction, Prox_{μg}[T_k^{λ,μ}], with contraction coefficient 1-μλ when μ∈(0,2/(2λ+L_f)]. Building on this, the authors propose cDCA, which solves each LPM subproblem approximately by Picard iteration with the adaptive inner stopping rule (16), and an outer stopping rule in Step 2. The main theoretical results are Theorem 4.1, establishing subsequential convergence to a critical point and summability of squared successive displacements, and Theorem 4.2, establishing whole-sequence convergence under a Kurdyka-Łojasiewicz condition on an auxiliary function E and Lipschitz differentiability of ∇h on the convex hull of the generated sequence. Numerical experiments compare cDCA with ADCA and pDCAe on ℓ1-2 and logarithmic regularized least-squares problems.

Significance. The contraction reformulation of the LPM subproblem is clean, and the adaptive inner termination rule is a useful idea: if the theory holds, each outer iteration is computed by finitely many inner steps whose error is controlled by the previous outer displacement, avoiding a fixed inner tolerance schedule. The descent estimate (17) in Theorem 4.1 is carefully derived, and the subsequential convergence proof is largely sound. The main limitation is that the whole-sequence convergence theorem's smoothness assumption is not verified for the two numerical test problems, so the advertised global convergence is broader than what is actually established for those examples.

major comments (2)
  1. [Section 5, Eqs. (32)–(33) and Theorem 4.2] Theorem 4.2 requires ∇h to be L_h-Lipschitz continuous on co({x_k}). For the ℓ1−2 test problem (32), h(x)=γ‖x‖, which is not differentiable at 0; for the logarithmic problem (33), the chosen h(x)=∑γ[|x_i|/ε − log(|x_i|+ε)+log ε] is also nondifferentiable on coordinate hyperplanes. The sentence after (32) asserting that γ<1/(2‖A^T b‖∞) makes the assumptions of Theorem 4.2 satisfied is not justified: the cited passage in [34] concerns a different algorithm, and no argument is given that the cDCA iterates, or their convex hull, avoid the nondifferentiability set. Consequently the abstract's whole-sequence convergence claim is not established for either numerical test case; only the subsequential Theorem 4.1 is guaranteed there. Please either prove the needed avoidance/differentiability property for cDCA or explicitly state that Theorem 4.2 applies only to problems whose h is differentiable on the generated convex hull.
  2. [Theorem 4.1(ii), around Eq. (27)] In Step 2 of Algorithm 3.2, when the first inner tolerance test fails, the algorithm sets x_{k+1}=x_{m_k+1}^{k+1}; the stationarity equation (19) and its limit version (27) are written only for the iterate x_{m_k}^{k+1}. The sentence following (24) says the extra case follows 'by a similar derivation', but the analogous equation for x_{m_k+1}^{k+1} is not displayed, and the proof of (ii) invokes (19) directly for the final outer iterate. The gap is easily repaired because the contraction property gives ||x_{m_k+1}^{k+1}−x_{m_k}^{k+1}|| ≤ (1−μλ)||x_{m_k}^{k+1}−x_{m_k−1}^{k+1}|| ≤ δ||x_k−x_{k−1}||, so the same bound (16) holds for the extra iterate; please add this argument.
minor comments (3)
  1. [Section 3.2, Lemma 3.2] The stopping criterion is only shown to detect a critical point in the exact equality case x_{m_k}^{k+1}=x_k=x_{m_k+1}^{k+1}. In finite precision the algorithm stops when the relative distance is below tol, but no quantitative relation between tol and a subgradient residual of F is given; the text after the lemma calls the output an 'approximate critical point', which is stronger than what is proven. A bound along the lines of Lemma 4.1 would make the termination rule more defensible.
  2. [Sections 5.1 and 5.2] The parameter λ is tuned in Section 5.1 on the same distribution of random instances used for the comparisons in Table 3 and Table 4, while ADCA and pDCAe use literature defaults; the reported advantages in iterations and CPU time should be interpreted with this selection bias in mind.
  3. [Throughout] Typographical issues include 'ffrstly' and 'ffxed' in the abstract, the corrupted string 'Kurdyka—/suppress Lojasiewicz' in Definition 2.1 and Remark 4.2, and minor grammar issues in the abstract; these should be cleaned up.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity in the derivation; the fixed-point equivalence is proved from the optimality conditions, and the convergence results rest on explicit assumptions rather than on the authors' own prior results.

full rationale

The paper's central derivation is self-contained. Lemma 3.1 derives the equivalence between the LPM subproblem (10) and the fixed-point equation (12) directly from the first-order optimality condition, displayed in the chain (13); the operator Prox_{μg}[T_k^{λ,μ}] is shown to be a contraction using standard facts (Lemma 2.5 cocoercivity of ∇f, Lemma 2.1 on I−ν∇f, Lemma 2.4 firm nonexpansiveness of Prox), with μ/(1−μλ) ≤ 2/Lf verified from the choice μ ≤ 2/(2λ+Lf). Algorithm 3.2 then approximates that fixed point by Picard iteration with the adaptive stopping rule (16); Theorem 4.1 obtains summability of successive differences from (24), boundedness of the subgradients {η_k} from Lemma 2.2, and subdifferential closure from Lemma 2.3, without invoking any prior result of the authors. Theorem 4.2 uses the external result [22, Thm 3.7] only after Lemma 4.1 establishes the needed subgradient bound under the explicitly stated assumption that ∇h is Lh-Lipschitz on co({x_k}); this is an independent criterion, not an input smuggled in by self-citation. The only self-citation, [9] in Remark 3.1, merely motivates a heuristic choice of the inner-loop initial point, which the theory explicitly permits to be arbitrary, so it is not load-bearing. Two non-circular limitations should be noted separately: tuning λ in Section 5.1 on the same random-instance distribution used in Section 5.2 weakens the experimental comparison but does not enter the derivation chain; and the assertion after Eq. (32) that γ < 1/(2||A^T b||∞) makes the assumptions of Theorem 4.2 satisfied, citing [34], is a scope/correctness question because h = γ||x|| is nondifferentiable at 0, so Theorem 4.2 may not cover the ℓ1−2 test case as stated. Neither issue is a circular reduction of the paper's mathematical claims.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The theoretical contribution relies on standard convex analysis facts plus the problem's standing assumptions. The only added assumptions are the KL property and Lipschitz differentiability of h, needed for whole-sequence convergence. The algorithm hyperparameters lambda and delta are free parameters, and lambda is tuned on the same data in the numerical section. No new physical or mathematical entity beyond the auxiliary function E is introduced.

free parameters (4)
  • lambda = 0.1 L_f (chosen by sweep in Section 5.1)
    The convergence proof allows any lambda > 0, but the numerical comparison uses lambda = 0.1 L_f selected as the best value on the same problem classes in Section 5.1. Thus the empirical speedup claim is tuned.
  • delta = 1.99 lambda / L_f
    Hand chosen inside the theoretical interval (0, 2 lambda / L_f). It controls the inner stopping threshold and is fixed experimentally, not fitted to achieve the convergence theorem.
  • tol = varies by regularizer and problem size (e.g., 1e-6 for L1-2, 6.5e-5 or 2e-5 for logarithmic)
    Algorithm tolerance chosen per experiment; it affects how many outer and inner iterations run. The convergence theory assumes a nonzero tolerance and no finite termination.
  • alpha_k and beta_k = 0.6 for all k in experiments
    Inertial coefficients in the inner initial point x_{k+1}^0. The theory allows arbitrary inner initialization, so these are experimental hyperparameters.
assumptions (4)
  • domain assumption f is convex with L_f-Lipschitz gradient; g is proper, lsc, convex, and prox-friendly; h is convex and finite; F = f+g-h is level-bounded.
    These are the standing assumptions stated in the introduction and used throughout: they guarantee subgradients exist, the proximal operator is well-defined, and the solution set is nonempty.
  • standard math The gradient of a convex function with L-Lipschitz gradient is 1/L-cocoercive (Lemma 2.5) and I - nu T is nonexpansive for cocoercive T (Lemma 2.1).
    These standard facts from monotone operator theory are the basis for proving Prox_{mu g}[T_k] is a contraction.
  • standard math Prox_{mu g} is firmly nonexpansive and the proximal characterization p = Prox_{alpha phi}(x) iff (x-p)/alpha in partial phi(p).
    Used in Lemma 3.1 and in the derivation of the subgradient relation (19).
  • domain assumption For Theorem 4.2, E has the Kurdyka-Lojasiewicz property at (x_hat, x_hat) and grad h is L_h-Lipschitz on co({x_k}).
    These are extra assumptions introduced in Lemma 4.1 and Theorem 4.2 solely to obtain convergence of the whole sequence. They are not implied by the base assumptions and are not verified numerically.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Contractive difference-of-convex algorithms." pith.science (2026). https://pith.science/paper/RCDNEFE7

@misc{pith2026250510800,
  author       = {Pith},
  title        = {Pith review of: Contractive difference-of-convex algorithms},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RCDNEFE7}},
  note         = {Machine review of arXiv:2505.10800}
}
read the original abstract

The difference-of-convex algorithm (DCA) and its variants are the most popular methods to solve the difference-of-convex optimization problem. Each iteration of them is reduced to a convex optimization problem, which generally needs to be solved by iterative methods such as proximal gradient algorithm. However, these algorithms essentially belong to some iterative methods of fixed point problems of averaged mappings, and their convergence speed is generally slow. Furthermore, there is seldom research on the termination rule of these iterative algorithms solving the subproblem of DCA. To overcome these defects, we ffrstly show that the subproblem of the linearized proximal method (LPM) in each iteration is equal to the ffxed point problem of a contraction. Secondly, by using Picard iteration to approximately solve the subproblem of LPM in each iteration, we propose a contractive difference-ofconvex algorithm (cDCA) where an adaptive termination rule is presented. Both global subsequential convergence and global convergence of the whole sequence of cDCA are established. Finally, preliminary results from numerical experiments are promising.

Figures

Figures reproduced from arXiv: 2505.10800 by the authors.

Figure 1
Figure 1. The true solution and the solution by cDCA for [PITH_FULL_IMAGE:figures/full_fig_p017_1.png] view at source ↗
Figure 2
Figure 2. The decay of objective values of cDCA, ADCA and pDCA [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

38 extracted references · 37 canonical work pages

  1. [34]

    Wen, B., Chen, X., Pong, T.K.: A proximal difference-of-conex alg orithm with extrapolation. Comput. Optim. Appl. 69, 297–324 (2018)

    work page 2018

  2. [1]

    IEEE Trans

    Alvarado, A., Scutari, G., Pang, J.S.: A new decomposition method f or mul- tiuser DC-programming and its applications. IEEE Trans. Signal Process. 62, 2984–2998 (2014)

    work page 2014

  3. [2]

    An, L.T.H., Tao, P.D.: The DC (difference of convex functions) prog ramming and DCA revisited with DC models of real world nonconvex optimization prob- lems. Ann. Oper. Res. 133, 23–46 (2005)

    work page 2005

  4. [3]

    Arag´ on Artacho, F.J., Vuong, P.T.: The boosted difference of convex functions algorithm for nonsmooth functions, SIAM J. Optim. 30, 980–1006 (2020)

    work page 2020

  5. [4]

    programming

    Banert, S., Bot, R.I.: A general double-proximal gradient algorit hm for d.c. programming. Math. Program. 178, 301–326 (2019)

    work page 2019

  6. [5]

    Springer, New York, 2017

    Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces , 2nd ed. Springer, New York, 2017. 18

    work page 2017

  7. [6]

    Beck, A., Teboulle, M.: A fast iterative shrinkage-thresholding alg orithm for linear inverse problems. SIAM J. Imaging Sci. 2, 183–202 (2009)

    work page 2009

  8. [7]

    Gotoh, J., Takeda, A., Tono, K.: DC formulations and algorithms fo r sparse optimization problems. Math. Program. 169, 141–176 (2018)

    work page 2018

Show all 38 references
  1. [8]

    de Oliveira, W.: The ABC of DC Programming, Set-Valued Var. Anal. 28, 679–706 (2020)

    2020

  2. [9]

    et al.: MiKM: multi-step inertial Kras- nosel’skiˇ ı–Mann algorithm and its applications

    Dong, Q.L., Huang, J.Z., Li, X.H. et al.: MiKM: multi-step inertial Kras- nosel’skiˇ ı–Mann algorithm and its applications. J. Glob. Optim. 73, 801–824 (2019)

    2019

  3. [10]

    Le Thi, H.A., Le, H.M., Phan, D.N., Tran, B.: Stochastic DCA for minimiz ing a large sum of DC functions with application to multi-class logistic regre ssion, Neural Networks, 132, 220–231 (2020)

    2020

  4. [11]

    4th International Conference on Advanced Data Mining and Applications

    Le Thi, H.A., Nguyen, V.V., Ouchani, S.: Gene selection for cancer c lassifica- tion using DCA. 4th International Conference on Advanced Data Mining and Applications. 5139, 62–72 (2008)

    2008

  5. [12]

    Le Thi, H.A., Pham Dinh, T.: Open issues and recent advances in DC p ro- gramming and DCA. J. Glob. Optim. (2023). https://doi.org/10.1007/s10898- 023-01272-1

    2023 doi

  6. [13]

    Le Thi, H.A., Pham, D.T.: DC programming and DCA: thirty years of d evel- opments. Math. Program. 169, 5–68 (2018)

    2018

  7. [14]

    Lin, D., Liu, C.: The modified second APG method for DC optimization prob- lems. Optim. Lett. 13, 805–824 (2019)

    2019

  8. [15]

    Liu, T., Pong, T.K., Takeda, A.: A refined convergence analysis of pDCAe with applications to simultaneous sparse recovery and outlier detection . Comput. Optim. Appl. 73, 69–100 (2019)

    2019

  9. [16]

    Liu, T., Pong, T.K., Takeda, A.: A successive difference-of-conv ex approxima- tion method for a class of nonconvex nonsmooth optimization proble ms. Math. Program. 176, 339–367 (2019)

    2019

  10. [17]

    Lou, Y., Yan, M.: Fast L1-L2 Minimization via a proximal operator. J. Sci. Comput. 74, 767–785 (2018)

    2018

  11. [18]

    Lu, Z., Zhou, Z.: Nonmonotone enhanced proximal DC algorithms for a class of structured nonsmooth DC programming. SIAM J. Optim. 29(4), 2725–2752 (2019) 19

    2019

  12. [19]

    Lu, Z., Zhou, Z., Sun, Z.: Enhanced proximal DC algorithms with extrapolation for a class of structured nonsmooth DC minimization. Math. Program. 176, 369–401 (2019)

    2019

  13. [20]

    Nesterov, Y.E.: A method for solving the convex programming pr oblem with convergence rate O(1/k2), Dokl. Akad. Nauk SSSR , 269, 543–547 (1983) (in Russian)

    1983

  14. [21]

    Oblomskaja, L.: Methods of successive approximation for linear equations in Banach spaces, USSR Compt. Math. and Math. Phys. 8, 239–253 (1968)

    1968

  15. [22]

    Imaging Sci

    Ochs, P., Chen, Y., Brox, T., Pock, T.: iPiano: Inertial proximal a lgorithm for nonconvex optimization, SIAM J. Imaging Sci. 7(2), 1388–1419 (2014)

    2014

  16. [23]

    Pang, J.S., Razaviyayn, M., Alvarado, A.: Computing B-stationar y points of nonsmooth DC programs. Math. Oper. Res. 42(1), 95–118 (2017)

    2017

  17. [24]

    Pham, T.N., Dao, M.N., Amjady, N., Shah, R.: A proximal splitting algo - rithm for generalized DC programming with applications in signal recov ery, 2024 https://arxiv.org/abs/2409.01535

    2024 arXiv

  18. [25]

    Acta Math

    Pham, D.T., Le Thi, H.A.: Convex analysis approach to DC programm ing: theory, algorithms and applications. Acta Math. Vietnam. 22, 289–355 (1997)

    1997

  19. [26]

    optimization algorithm for solving the trust-region subproblem

    Pham, D.T., Le Thi, H.A.: A D.C. optimization algorithm for solving the trust-region subproblem. SIAM J. Optim. 8, 476–505 (1998)

    1998

  20. [27]

    Fermat Day 85: Mathematics for Optimization, North Holland 1986

    Pham Dinh, T., Souad, E.B.: Algorithms for solving a class of nonconvex opti- mizations problems: methods of subgradient . Fermat Day 85: Mathematics for Optimization, North Holland 1986

    1986

  21. [28]

    In: Pro- ceedings of the 27th International Joint Conference on Artificial Intelligence, IJCAI-18, pp

    Phan, D.N., Le, H.M., Le Thi, H.A.: Accelerated difference of convex func- tions algorithm and its application to sparse binary logistic regression . In: Pro- ceedings of the 27th International Joint Conference on Artificial Intelligence, IJCAI-18, pp. 1369–1375. International J...

    2018

  22. [29]

    Liang, J.: Geometry of first-order methods and adap tive acceleration, arXiv:2003 .03910

    Poon, C.. Liang, J.: Geometry of first-order methods and adap tive acceleration, arXiv:2003 .03910

    2003

  23. [30]

    Rockafellar, R.T.: Convex Analysis , Princeton University Press, Princeton, 1970

    1970

  24. [31]

    Sun, W.Y., Sampaio, R.J.B., Candido, M.A.B.: Proximal point algorithm f or minimization of DC Functions. J. Comput. Math. 21(4), 451–462 (2003) 20

    2003

  25. [32]

    Syrtseva, K., de Oliveira, W., Demassey, S., van Ackooij, W.: Minimiz ing the difference of convex and weakly convex functions via bundle method . Pac. J. Optim. 20(4), 499-741, (2024)

    2024

  26. [33]

    Springer, Berlin 2016

    Tuy, H.: Convex Analysis and Global Optimization, 2nd edn. Springer, Berlin 2016

    2016

  27. [35]

    Xu, H.K.: Averaged mappings and the gradient-projection algor ithm. J. Optim. Theory Appl. 150, 360–378 (2011)

    2011

  28. [36]

    Ye, J.J., Yuan, X., Zeng, S., Zhang, J.: Difference of convex algor ithms for bilevel programs with applications in hyperparameter selection.Math. Program. 198, 1583–1616 (2023)

    2023

  29. [37]

    Yin, P., Lou, Y., He, Q., Xin, J.: Minimization of ℓ1−2 for compressed sensing. SIAM J. Sci. Comput. 37(1), A536–A563 (2015)

    2015

  30. [38]

    Yu, P., Pong, T.K., Lu, Z.: Convergence rate analysis of a sequen tial convex programming method with line search for a class of constrained differ ence-of- convex optimization problems, SIAM J. Optim. 31, 2024–2054 (2021) 21

    2021

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.