Pith. sign in

REVIEW 3 major objections 5 minor 280 references

A frugal primal-dual splitting with minimal lifting over arbitrary rooted trees

T0 review · 3 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

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

Pith's one-line read A frugal primal-dual splitting on arbitrary rooted trees solves general structured monotone inclusions with minimal lifting.

desk verdict New tree-structured splitting that unifies a lot of prior work, but the minimal-lifting claim leans on under-verified support. read the letter →

arxiv 2607.18932 v1 pith:EXRHOCI5 submitted 2026-07-21 math.OC

Feng Xue , Hui Zhang This is my paper · ORCID
classification math.OC MSC 47H0549M2949M2790C25
keywords frugalsplittingminimalliftingprimal-dualtree-structuredgraphsmonotoneinclusionsparallelsumscocoerciveoperatorsDouglas-Rachford
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 a very general class of structured monotone inclusions — sums of monotone operators, linear compositions with parallel sums, and cocoercive operators — can be solved by a single frugal splitting algorithm that evaluates each resolvent once per iteration and uses the minimal number of auxiliary variables. The algorithm is organized by an arbitrary rooted tree: primal nodes carry the monotone operators, dual variables and cocoercive terms are assigned to nodes arbitrarily, and edges enforce consensus. The authors prove weak convergence to a fixed point whose image solves the inclusion, under explicit metric conditions on the edge weights, and show that the scheme recovers Douglas–Rachford, forward-DRS, and parallel Chambolle–Pock as special cases. For convex minimization it yields an O(1/k) ergodic rate on the primal-dual gap and an o(1/k) asymptotic regularity rate. The significance is that one flexible, level-synchronous, distributed template covers many previously separate algorithms.

What carries the argument

The tree-structured consensus-ADMM encoding: each primal node updates against its parent and children via weighted equality constraints, dual variables s_j are attached to non-leaf nodes and updated in Chambolle–Pock style through B_j^{-1} with a forward step of D_j^{-1}, and cocoercive operators C_l are evaluated at the parent node. The analysis hinges on the product-space block operator A (monotone plus skew) and the degenerate preconditioner Q; their interplay yields the metric inequalities of Proposition 4.3 that drive the convergence proof.

What would settle it

Work through the smallest nontrivial case — n = 2, m = 1, p = 1 with a star tree — and directly compute A from Lemma 4.1 to check that it is maximally monotone. More generally, search for any tree topology, dual assignment, or cocoercive loading where the block operator A violates maximal monotonicity or the metric conditions of Corollary 4.4; even one such instance would falsify the unconditional convergence claim.

Watch

Extended reading notes

Core claim

The central discovery is that the splitting scheme (3.3) — built from a tree whose nodes are primal variables and whose edges carry multipliers and metrics — defines a frugal parametrized resolvent splitting with (n−1,m)-fold minimal lifting for the general inclusion (1.2). The proof recasts the algorithm as a fixed-point iteration in a product Hilbert space, 0 ∈ A x̃ + Q(x̃−x) + r, where A is a maximally monotone block operator of monotone-plus-skew structure and Q is a degenerate preconditioner. Under metric conditions linking the edge weights to the cocoercivity constants and the linear-operator norms, the fixed-point operator is nonexpansive and demiclosed, implying weak convergence, and

Load-bearing premise

The convergence proof relies on the block operator A defined in Lemma 4.1 being maximally monotone, which the paper justifies by citing a lemma from the first author's earlier work; if that cited lemma fails for the full block structure, the convergence argument collapses.

Editorial extensions

If this is right

  • If correct, a single algorithm template solves any instance of (1.2) with n ≥ 2 and arbitrary m, p using the minimum (n−1,m) auxiliary dimensions.
  • It provides a unified distributed computing framework: the same scheme implements centralized (star tree) and decentralized (chain tree) variants, with arbitrary intermediate topologies.
  • It recovers Douglas–Rachford, forward-DRS, and (parallel) Chambolle–Pock as special cases, unifying their convergence proofs.
  • For convex minimization, the ergodic primal-dual gap bound O(1/k) requires only boundedness of the feasible sets, not strong convexity.
  • The o(1/k) asymptotic regularity rate is the first such rate for this class of problems with parallel sums and arbitrary trees.

Reading between the lines

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

  • If the maximal monotonicity of the product-space operator A holds, the same reformulation technique could extend to other graph structures (e.g., graphs with cycles) by treating A as a graph Laplacian-like operator; the rank of Q would then determine the lifting number.
  • The metric conditions suggest a practical recipe for step-size selection: choose M_i and N_j proportional to the cocoercivity constants and inverse operator norms, with relaxation parameters in (0,2); automatic step-size rules could be derived.
  • The correction mechanism used to compensate for dual-variable assignments on parent nodes is a novel device that might be reused in asynchronous or stochastic variants of the scheme.
  • The algorithm's dependence on the tree topology offers a way to tune convergence speed by tree-shape optimization, a direction the paper explicitly leaves open.
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

3 major / 5 minor

Summary. The paper proposes a frugal primal-dual splitting algorithm over arbitrary rooted trees for structured monotone inclusions that combine sums of maximally monotone operators, linear compositions with parallel sums, and cocoercive operators. The algorithm is written in a fixed-point form (3.3) with a reduced iteration on (z_i, s_j), and the paper claims it is a fixed-point encoding with (n−1,m)-fold minimal lifting. Convergence is analyzed through a product-space reformulation (Lemma 4.1): under metric conditions, weak convergence to a fixed point whose image solves (1.2) is proved (Theorem 4.5), and o(1/k) asymptotic regularity is derived (Proposition 4.6). Applications include distributed forward-DRS, pure resolvent splitting, recovery of parallel Chambolle–Pock, and O(1/k) ergodic primal-dual gap for convex minimization. The central advertised contributions are the general problem class, the frugal/minimal-lifting guarantee, and the flexible tree-based distributed implementation.

Significance. If the claims are valid, this is a substantial contribution: it is the first splitting method for the full class (1.2) with parallel sums, cocoercive operators, and linear compositions, and the tree formulation enables flexible level-synchronous distributed computation. The paper also provides concrete recoveries of Douglas–Rachford and Chambolle–Pock variants, and nontrivial rate statements (o(1/k) asymptotic regularity, O(1/k) ergodic gap). The product-space reformulation is a useful analysis device. However, two load-bearing facts are not derived in the manuscript: the minimality of the lifting for the enlarged class with parallel sums, and the maximal monotonicity of the block operator A. Both are imported from external or self-cited results without verification. These gaps need to be closed before the central claims can be accepted.

major comments (3)
  1. [Theorem 3.3(ii), Sect. 1.2(i)] The minimal-lifting claim is not derived in this paper. The proof says minimality follows from [4, Theorem 5] and [29]. But [4, Theorem 5] treats (1.1) with p=0 and no parallel-sum terms, and the paper itself states in Sect. 1.2(i) that [4,29] do not cover the parallel-sum type. Thus the (n−1,m) lower bound for the general class (1.2) with B_j□D_j and C_l is unsupported. Since minimal lifting is a central advertised contribution, the authors must either prove the lower bound for the full class or restrict the claim.
  2. [Appendix B, Lemma 4.1] The convergence analysis — Proposition 4.3, Corollary 4.4, Theorem 4.5, and Theorem 5.4 — all rely on A being maximally monotone. The only justification is "by [43, Lemma 2.6]" (Appendix B), with no verification that the skew-coupled block operator with zero z-diagonal blocks, multiple children, and dual assignments satisfies that lemma's hypotheses. A is not simply a diagonal monotone operator plus a standard skew-adjoint operator; the z-part has zero diagonal, so the cited result may not apply directly. A complete proof, or a precise statement with all hypotheses explicitly checked, is required.
  3. [Fact 4(ii), Theorem 5.4] Fact 4(ii) asserts T_uz=(A+Q)^{-1}Q and Theorem 5.4 uses zer A = Fix T_uz and closedness of gra A. In Lemma 4.1, Q is degenerate with zero blocks on u-components, so invertibility of A+Q is not automatic from maximal monotonicity of A. Fact 1 shows the individual resolvent steps are well defined, but the operator identity and the domain/range conditions under which (A+Q)^{-1}Q is single-valued and defined on the whole space are not established. This step is load-bearing for identifying the weak limit as a solution.
minor comments (5)
  1. [Fact 3(i)] The display "1Pp l=1 β−1 l" should read 1/(∑_{l=1}^p β_l^{-1}); the current typesetting is unclear.
  2. [Table 1.1] The entry "/reve" appears to be a placeholder or typo. It should be replaced with the intended symbols (e.g., checkmarks or em-dashes) to make the table readable.
  3. [Section 5.3, Proposition 5.5] The use of a "virtual" primal node A_1 ≡ 0 to recover Chambolle–Pock is valid, but the phrase "virtual" should be defined explicitly (e.g., A_1 = 0 as a maximal monotone operator) to avoid confusion in the counting of lifting.
  4. [Notation, Remark 4] The block-operator notation in Remark 4 is dense; a small concrete example for a tree with two children would greatly improve readability.
  5. [References] Reference [29] is a private communication and not publicly verifiable. If it is used for a load-bearing result, its statement should be reproduced or cited to an archival version.

Circularity Check

0 steps flagged · score 1.0 of 10

No circularity: self-citations are to general published lemmas, not fits of the target result.

full rationale

The algorithm and its fixed-point encoding (Prop 3.2, Thm 3.3) are constructed algebraically from the tree splitting, and the (n-1,m) lifting count is read off the reduced variables (z_i,s_j); minimality is imported from the external lower bound [4, Thm 5], not from the paper's own assumptions. The convergence proof (Lemma 4.1, Prop 4.3, Cor 4.4, Thm 4.5) follows a standard Fejér/demiclosedness template: Lemma 4.1 is an algebraic reformulation of the prediction step, and Proposition 4.3 uses the actual monotonicity of the block operator A together with Young-type residual estimates in Lemma 4.2; no parameter is fitted from the desired limit and no inequality is assumed to equal the target result. The paper does lean on prior work by the same authors for general monotone-operator facts — [43, Lem. 2.6] for maximal monotonicity of A, [46, Lem. 2.4] for demiclosedness, and [44] for the ergodic gap estimate — but these are published, parameter-free statements about general monotone operators (monotone+skew maximality and Browder demiclosedness), not specializations of the theorem being proved. Thus the reliance is a citation burden, not a reduction of the conclusion to its inputs. The omitted proof of Prop 5.8 by reference to [44] is a presentation gap, not circularity.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The central algorithmic construction is self-contained, but the convergence theory borrows several load-bearing lemmas from the authors' own prior papers, and the minimal-lifting lower bound relies on published [4] plus unpublished [29]. No new physical or mathematical entities are postulated.

free parameters (4)
  • tau_j > 0
    Arbitrary positive constants introduced in Lemma 4.2 to bound cross terms involving L_j and the residuals. They enter the sufficient conditions on M_i and N_j but are not fitted to data.
  • tau_i = sum_{j in J_{c,i}} tau_j
    Derived from the arbitrary tau_j; used in the metric conditions for convergence.
  • beta_i = 1 / sum_{l in P_i} beta_l^{-1}
    Derived from the cocoercivity constants of the operators loaded on node i. It is a property of the problem, not a fitted value.
  • Metrics M_i, N_j and relaxation parameters theta_i, zeta_j
    User-selected step-size/metric parameters constrained by Corollary 4.4. They are not fitted to data but are hand-chosen by the practitioner.
assumptions (6)
  • standard math Standard monotone operator theory holds: Minty's theorem, Browder's demiclosedness principle, Moreau's resolvent identity, and maximal monotonicity of monotone+skew sums.
    Used throughout Sections 3–4 and the appendices; see [7,15,24,43,46].
  • domain assumption Problem data satisfy the stated regularity: A_i and B_j maximal monotone, D_j strongly monotone, C_l cocoercive, L_j bounded linear.
    This is the problem class (1.2) itself; Assumption 1 additionally requires M_i,N_j positive definite with closed ranges.
  • domain assumption Existence of a solution and a qualification condition for strong duality in the convex minimization setting of Section 5.4.
    Invoked before defining the primal-dual gap and applying Fermat's rule; the paper cites [20, Remark 2.1] and does not prove it.
  • ad hoc to paper The minimal lifting bound (n−1,m) for the enlarged class with parallel sums is inherited from [4, Theorem 5] and, for cocoercive terms, from the unpublished [29].
    Theorem 3.3 does not derive the lower bound; it says 'minimality follow from [4, Definitions 9, 10, Theorem 5]' and relies on [29] for p∈N, which is only a private communication.
  • standard math The block operator A in Lemma 4.1 is maximally monotone.
    The proof states this follows from [43, Lemma 2.6], a published result by the first author; the lemma is not reproduced in the paper.
  • standard math The extended demiclosedness principle for metric nonexpansive operators holds as stated in [46, Lemma 2.4].
    Used in Corollary 4.4 to pass from nonexpansiveness to demiclosedness of I−T_{zs,θ,ζ}.

how reviews work

0 comments
Cite this review

Pith. "Pith review of A frugal primal-dual splitting with minimal lifting over arbitrary rooted trees." pith.science (2026). https://pith.science/paper/EXRHOCI5

@misc{pith2026260718932,
  author       = {Pith},
  title        = {Pith review of: A frugal primal-dual splitting with minimal lifting over arbitrary rooted trees},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EXRHOCI5}},
  note         = {Machine review of arXiv:2607.18932}
}
read the original abstract

We develop a frugal primal-dual splitting with minimal lifting for solving structured monotone inclusions, involving cocoercive operators, linear compositions and parallel sums. This is established by defining hierarchical nodes and edges between them over a tree-structured graph, with arbitrary assignments of dual variables and cocoercive elements to primal nodes. This arbitrariness allows a great flexibility in terms of level-synchronous distributed computing, such as centralized or decentralized. The particular instances naturally extend the Douglas--Rachford splitting on various graphs, recover the parallel Chambolle--Pock, and solve a class of structured convex minimization problems with O(1/k) ergodic rate for its primal-dual gap function. Furthermore, we introduce a reformulation technique in a product Hilbert space to facilitate the convergence analysis, specifically to derive the o(1/k)-rate of asymptotic regularity.

Figures

Figures reproduced from arXiv: 2607.18932 by the authors.

Figure 2.1
Figure 2.1. An ancestral line of a tree with dual assignments and cocoercive loading. [PITH_FULL_IMAGE:figures/full_fig_p006_2_1.png] view at source ↗
Figure 2.2
Figure 2.2. A star-shaped tree for centralized computing. [PITH_FULL_IMAGE:figures/full_fig_p010_2_2.png] view at source ↗
Figure 2.3
Figure 2.3. A fully sequential tree for decentralized computing. [PITH_FULL_IMAGE:figures/full_fig_p010_2_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

280 extracted references · 3 canonical work pages

  1. [29]

    Computational Optimization and Applications , volume=

    A product space reformulation with reduced dimension for splitting algorithms , author=. Computational Optimization and Applications , volume=

  2. [1]

    A Projected Primal–Dual Method for Solving Constrained Monotone Inclusions , author=. J. Optim. Theory Appl. , volume =

  3. [2]

    Arxiv preprint: arXiv:2111.09757 , year=

    Resolvent splitting with minimal lifting for composite monotone inclusions , author=. Arxiv preprint: arXiv:2111.09757 , year=

    arXiv

  4. [4]

    Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR) , month =

    Gu, Shuhang and Zhang, Lei and Zuo, Wangmeng and Feng, Xiangchu , title =. Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR) , month =

  5. [5]

    , title =

    Combettes, Patrick L. , title =. SIAM Journal on Optimization , volume =. 2013 , doi =

    2013

  6. [6]

    A primal-dual splitting algorithm for monotone inclusions with applications , author=

  7. [7]

    Set-Valued Var

    Solving monotone inclusions involving the sum of three maximally monotone operators and a cocoercive operator with applications , author=. Set-Valued Var. Anal. , volume=

  8. [8]

    and Yin, W

    Davis, D. and Yin, W. , title=. Splitting Methods in Communication, Imaging, Science, and Engineering , year=

Show all 280 references
  1. [9]

    Journal of Mathematical Analysis and Applications , volume=

    Warped proximal iterations for monotone inclusions , author=. Journal of Mathematical Analysis and Applications , volume=

  2. [10]

    Optimization , volume=

    Variable metric forward–backward splitting with applications to monotone inclusions in duality , author=. Optimization , volume=. 2014 , publisher=

    2014

  3. [12]

    Alcantara, J. H. and Dao, M. N. and Takeda, A. , journal=

  4. [13]

    and Hong, M

    Chang, T.H. and Hong, M. and Wang, X. , journal=. Multi-agent distributed optimization via inexact consensus

  5. [14]

    Inverse Problems , volume=

    Aprimal–dual fixed point algorithm for convex separable minimization with applications to image restoration , author=. Inverse Problems , volume=

  6. [15]

    Fixed Point Theory Appl , volume=

    A primal–dual fixed point algorithm for minimization of the sum of three convex separable functions , author=. Fixed Point Theory Appl , volume=

  7. [16]

    SIAM Journal on Imaging Sciences , volume=

    Preconditioning of a generalized forward-backward splitting and application to optimization on graphs , author=. SIAM Journal on Imaging Sciences , volume=

  8. [17]

    Optimization Letters , volume=

    Frugal and decentralised resolvent splittings defined by nonexpansive operators , author=. Optimization Letters , volume=

  9. [18]

    SIAM Journal on Optimization , volume=

    Frugal splitting operators: representation, min imal lifting, and convergence , author=. SIAM Journal on Optimization , volume=

  10. [19]

    Applied Mathematics and Computation , volume=

    Backward-forward-reflected-backward splitting for three operator monotone inclusions , author=. Applied Mathematics and Computation , volume=

  11. [20]

    Ryu, E.K. and V\. Finding the forward-. Journal of Optimization Theory and Applications , volume=

  12. [21]

    Set-Valued and Variational Analysis , volume=

    A reflected forward-backward splitting method for monotone inclusions involving Lipschitzian operators , author=. Set-Valued and Variational Analysis , volume=

  13. [22]

    SIAM Journal on Optimization , volume=

    A forward-backward splitting method for monotone inclusions without cocoercivity , author=. SIAM Journal on Optimization , volume=

  14. [23]

    Optimization , pages=

    Linear convergence of resolvent splitting with minimal lifting and its application to a primal-dual algorithm , author=. Optimization , pages=

  15. [24]

    Abstract and Applied Analysis , number=

    On the convergence analysis of the alternating direction method of multipliers with three blocks , author=. Abstract and Applied Analysis , number=

  16. [25]

    Computational Optimization and Applications , volume=

    Distributed forward-backward methods for ring networks , author=. Computational Optimization and Applications , volume=

  17. [26]

    SIAM Journal on Imaging Sciences , volume=

    A generalized forward-backward splitting , author=. SIAM Journal on Imaging Sciences , volume=

  18. [27]

    Numerical Algorithms , volume=

    A primal-dual splitting algorithm for composite monotone inclusions with minimal lifting , author=. Numerical Algorithms , volume=

  19. [28]

    SIAM Journal on Optimization , volume=

    Forward-backward algorithms devised by graphs , author=. SIAM Journal on Optimization , volume=

  20. [30]

    Forward-backward-half forward algorithm for solving monotone inclusions , author=. SIAM J. Optim. , volume=

  21. [32]

    Degenerate preconditioned proximal point algorithms , author=. SIAM J. Optim. , volume=

  22. [33]

    Primal-dual splittings as fixed point iterations in the range of linear operators , author=. J. Glob. Optim. , year=

  23. [34]

    Journal of the Operations Research Society of China , volume=

    On the metric resolvent: nonexpansiveness, convergence rates and applications , author =. Journal of the Operations Research Society of China , volume=. 2025 , publisher=

    2025

  24. [35]

    Some extensions of the operator splitting schemes based on

    Xue, Feng , journal=. Some extensions of the operator splitting schemes based on. 2023 , publisher=

    2023

  25. [36]

    Xue, Feng , journal=. On. 2024 , publisher=

    2024

  26. [37]

    Equivalent resolvents of

    Xue, Feng , journal=. Equivalent resolvents of. 2024 , publisher=

    2024

  27. [38]

    Optimization Letters , volume=

    The operator splitting schemes revisited: Primal-dual gap and degeneracy reduction by a unified analysis , author =. Optimization Letters , volume=. 2024 , publisher=

    2024

  28. [39]

    Journal of Optimization Theory and Applications,

    On Degenerate Preconditioned Proximal Point Methods under Restricted Monotonicity , author =. Journal of Optimization Theory and Applications,

  29. [40]

    Computational and Applied Mathematics,

    A generalized forward--backward splitting operator: Degenerate analysis and applications , author =. Computational and Applied Mathematics,

  30. [41]

    Results in Mathematics,

    On the nonexpansive operators based on arbitrary metric: A degenerate analysis , author =. Results in Mathematics,

  31. [42]

    A generalized proximal point algorithm and its convergence rate , author=. SIAM J. Optim. , volume=

  32. [43]

    On the convergence of the proximal point algorithm for convex minimization , author=. SIAM J. Control Optim. , volume=

  33. [44]

    Graph and distributed extensions of the

    Bredies, Kristian and Chenchene, Enis and Naldi, Emanuele , journal=. Graph and distributed extensions of the. 2024 , volume=

    2024

  34. [45]

    2021 , publisher=

    Large-Scale Convex Optimization via Monotone Operators , author=. 2021 , publisher=

    2021

  35. [46]

    and Pesquet, Jean-Christophe , journal=

    Combettes, Patrick L. and Pesquet, Jean-Christophe , journal=. Fixed Point Strategies in Data Science , year=

  36. [47]

    SIAM Review , volume=

    Proximal Splitting Algorithms for Convex Optimization: A Tour of Recent Advances, with New Twists , author=. SIAM Review , volume=

  37. [48]

    Combettes, Patrick L. and V\. Variable metric quasi-. Nonlinear Analysis: Theory, Methods & Applications , volume=. 2013 , publisher=

    2013

  38. [49]

    , journal=

    Ryu, Ernest K. , journal=. Uniqueness of. 2020 , publisher=

    2020

  39. [50]

    Mathematical Programming , volume=

    Resolvent splitting for sums of monotone operators with minimal lifting , author=. Mathematical Programming , volume=. 2023 , publisher=

    2023

  40. [51]

    Mathematical Programming , volume=

    Convergence rates with inexact non-expansive operators , author=. Mathematical Programming , volume=

  41. [52]

    Augmented

    Tai, Xue-Cheng and Wu, Chunlin , booktitle=. Augmented. 2009

    2009

  42. [53]

    Augmented

    Wu, Chunlin and Tai, Xue-Cheng , journal=. Augmented. 2010 , publisher=

    2010

  43. [54]

    Multiplier and gradient methods , author=. J. Optimization Theory and Applications , volume=. 1969 , publisher=

    1969

  44. [55]

    Resolvent of the parallel composition and the proximity operator of the infimal postcomposition , author=. Optim. Lett. , volume=

  45. [56]

    Pacific J

    Understanding the convergence of the alternating direction method of multipliers: Theoretical and computational perspectives , author=. Pacific J. Optim. , volume=

  46. [57]

    , editor=

    Glowinski, R. , editor=. On alternating direction methods of multipliers: A historical perspective , bookTitle=. 2014 , publisher=

    2014

  47. [58]

    and Pan, T.-W

    Glowinski, R. and Pan, T.-W. and Tai, X.-C. , editor=. Some Facts About Operator-Splitting and Alternating Direction Methods , bookTitle=. 2016 , publisher=

    2016

  48. [59]

    Rockafellar, R. T. , journal=. Augmented. 1976 , publisher=

    1976

  49. [60]

    A new Primal-Dual algorithm for minimizing the sum of three functions with a linear operator , author=. J. Sci. Comput. , volume=. 2018 , publisher=

    2018

  50. [61]

    Self Equivalence of the Alternating Direction Method of Multipliers , bookTitle=

    Yan, Ming and Yin, Wotao , editor=. Self Equivalence of the Alternating Direction Method of Multipliers , bookTitle=. 2016 , publisher=

    2016

  51. [62]

    Svaiter, B. F. , journal=. On Weak Convergence of the. 2011 , publisher=

    2011

  52. [63]

    and Xu, M

    He, B. and Xu, M. and Yuan, X. , journal=. Block-Wise. 2018 , publisher=

    2018

  53. [64]

    and Yuan, X

    Yang, J. and Yuan, X. , journal=. Linearized augmented. 2013 , publisher=

    2013

  54. [65]

    Splitting Methods for Monotone Operators with Applications to Parallel Optimization, , author=. Ph.D. Thesis, M.I.T. , year=

  55. [66]

    Douglas-

    Patrinos, Panagiotis and Stella, Lorenzo and Bemporad, Alberto , booktitle=. Douglas-. 2014

    2014

  56. [67]

    , editor=

    Gabay, D. , editor=. Applications of the method of multipliers to variational inequalities , bookTitle=. 2018 , publisher=

    2018

  57. [68]

    Cevher, V. and V\. Stochastic Forward Douglas-Rachford Splitting Method for Monotone Inclusions , bookTitle=. 2018 , publisher=

    2018

  58. [69]

    Forward-

    Brice\. Forward-. Optimization , year=

  59. [70]

    Pure Appl

    Stochastic approximations and perturbations in forward–backward splitting for monotone operators , author=. Pure Appl. Funct. Anal. , year=

  60. [71]

    Linear Convergence and Metric Selection for

    Giselsson, Pontus and Boyd, Stephen , journal=. Linear Convergence and Metric Selection for. 2017 , volume=

    2017

  61. [72]

    IEEE Transactions on Signal Processing , year=

    Majorization-Minimization Algorithms in Signal Processing, Communications, and Machine Learning , author=. IEEE Transactions on Signal Processing , year=

  62. [73]

    SIAM Journal on Control and Optimization , year=

    A Variable Metric Proximal Point Algorithm for Monotone Operators , author=. SIAM Journal on Control and Optimization , year=

  63. [74]

    SIAM Journal on Optimization , year=

    A Class of Inexact Variable Metric Proximal Point Algorithms , author=. SIAM Journal on Optimization , year=

  64. [75]

    and Liu, H

    He, B. and Liu, H. and Wang, Z. and Yuan, X. , journal=. A Strictly Contractive. 2014 , volume=

    2014

  65. [76]

    and Marrocco, A

    Glowinski, R. and Marrocco, A. , journal=. Sur l'approximation par \'. 1975 , volume=

    1975

  66. [77]

    1984 , publisher=

    Numerical Methods for Nonlinear Variational Problems , author=. 1984 , publisher=

    1984

  67. [78]

    Journal of Scientific Computing , year=

    Convergence Analysis of Primal–Dual Based Methods for Total Variation Minimization with Finite Element Approximation , author=. Journal of Scientific Computing , year=

  68. [79]

    Generalized alternating direction method of multipliers: New theoretical insights and applications , author=. Math. Program. Comput. , year=

  69. [80]

    SIAM Journal on Numerical Analysis , year=

    Local Linear Convergence of the Alternating Direction Method of Multipliers for Quadratic Programs , author=. SIAM Journal on Numerical Analysis , year=

  70. [81]

    Journal of Computational & Applied Mathematics , year=

    A generalized proximal-point-based prediction–correction method for variational inequality problems , author=. Journal of Computational & Applied Mathematics , year=

  71. [82]

    A parameterized proximal point algorithm for separable convex optimization , author=. Optim. Lett. , year=

  72. [83]

    and Li, J

    Bai, J. and Li, J. and Xu, F. and Zhang, H. , journal=. Generalized symmetric. 2018 , volume=

    2018

  73. [84]

    Journal of Optimization Theory and Applications , year=

    On Glowinski’s Open Question on the Alternating Direction Method of Multipliers , author=. Journal of Optimization Theory and Applications , year=

  74. [85]

    Journal of Optimization Theory and Applications , year=

    Linear Convergence Rates for Variants of the Alternating Direction Method of Multipliers in Smooth Cases , author=. Journal of Optimization Theory and Applications , year=

  75. [86]

    SIAM Journal on Optimization , year=

    On the O(1/t) Convergence Rate of Alternating Direction Method with Logarithmic--Quadratic Proximal Regularization , author=. SIAM Journal on Optimization , year=

  76. [87]

    Computational Optimization and Applications , year=

    The generalized proximal point algorithm with step size 2 is not necessarily convergent , author=. Computational Optimization and Applications , year=

  77. [88]

    SIAM Journal on Imaging Science , year=

    Constrained total variational deblurring models and fast algorithms based on alternating direction method of multipliers , author=. SIAM Journal on Imaging Science , year=

  78. [89]

    ICCV-2011 , year=

    Diagonal preconditioning for first order primal-dual algorithms in convex optimization , author=. ICCV-2011 , year=

    2011

  79. [90]

    ECCV-2004 , year=

    A _1 -Unified Variational Framework for Image Restoration , author=. ECCV-2004 , year=

    2004

  80. [91]

    On the equivalence of the primal-dual hybrid gradient method and

    O'Connor, Daniel and Vandenberghe, Lieven , journal=. On the equivalence of the primal-dual hybrid gradient method and. 2020 , pages=

    2020

  81. [92]

    Primal-dual decomposition by operator splitting and applications to image deblurring , author=. SIAM J. Imaging Sciences , volume=. 2014 , publisher=

    2014

  82. [93]

    and Osher, S

    Cai, J. and Osher, S. and Shen, Z. , journal=. Linearized

  83. [94]

    and Osher, S

    Cai, J. and Osher, S. and Shen, Z. , journal=. Convergence of the linearized

  84. [95]

    , journal=

    Passty, G.B. , journal=. Ergodic convergence to a zero of the sum of monotone operators in

  85. [96]

    Journal of Mathematical Imaging and Vision , volume=

    Accelerated Alternating Descent Methods for Dykstra-Like Problems , author=. Journal of Mathematical Imaging and Vision , volume=. 2017 , publisher=

    2017

  86. [97]

    Pacific Journal of Optimization , volume=

    A Parallel Inertial Proximal Optimization Method , author=. Pacific Journal of Optimization , volume=

  87. [98]

    Local Linear Convergence of

    Tao, Shaozhe and Boley, Daniel and Zhang, Shuzhong , journal=. Local Linear Convergence of. 2016 , publisher=

    2016

  88. [99]

    SIAM Journal on Optimization , volume=

    A Dynamical Approach to an Inertial Forward-Backward Algorithm for Convex Minimization , author=. SIAM Journal on Optimization , volume=. 2014 , publisher=

    2014

  89. [100]

    Convergence Rate Analysis of Inertial

    Shehu, Yekini , journal=. Convergence Rate Analysis of Inertial. 2018 , publisher=

    2018

  90. [101]

    and Brox, T

    Ochs, P. and Brox, T. and Pock, T. , journal=. i. 2015 , publisher=

    2015

  91. [102]

    Advances in Computational Mathematics , volume=

    Bo. Advances in Computational Mathematics , volume=

  92. [103]

    SIAM Journal on Optimization , volume=

    Weak Convergence of a Relaxed and Inertial Hybrid Projection-Proximal Point Algorithm for Maximal Monotone Operators in Hilbert Space , author=. SIAM Journal on Optimization , volume=. 2004 , publisher=

    2004

  93. [104]

    and Hendrich, C

    Bot, Radu Ioan and Csetnek, E.R. and Hendrich, C. , journal=. Inertial. 2015 , publisher=

    2015

  94. [105]

    Numerical Algorithms , volume=

    An inertial forward-backward-forward primal-dual splitting algorithm for solving monotone inclusion problems , author=. Numerical Algorithms , volume=. 2014 , publisher=

    2014

  95. [106]

    Advances in Neural Information Processing Systems , year=

    Local Linear Convergence of Forward--Backward under Partial Smoothness , author=. Advances in Neural Information Processing Systems , year=

  96. [107]

    Advances in Neural Information Processing Systems , year=

    Trajectory of Alternating Direction Method of Multipliers and Adaptive Acceleration , author=. Advances in Neural Information Processing Systems , year=

  97. [108]

    Activity Identification and Local Linear Convergence of

    Liang, Jingwei and Fadili, Jalal and Peyr\'. Activity Identification and Local Linear Convergence of. International Conference on Scale Space and Variational Methods in Computer Vision , pages=. 2015

    2015

  98. [109]

    Computational Optimization and Applications , volume=

    Local and global convergence of a general inertial proximal splitting scheme for minimizing composite functions , author=. Computational Optimization and Applications , volume=

  99. [110]

    Linear Algebra & Its Applications , volume=

    On the proximal minimization algorithm with D -Functions , author=. Linear Algebra & Its Applications , volume=

  100. [111]

    On the ergodic convergence rates of a first-order primal-dual algorithm , author=. Math. Program., Ser. A , volume=. 2016 , publisher=

    2016

  101. [112]

    An inertial forward-backward algorithm for monotone inclusions , author=. J. Math. Imaging Vis. , volume=. 2015 , publisher=

    2015

  102. [113]

    Bioucas-Dias, Jos. A new. IEEE Transactions on Image Processing , volume=. 2007 , publisher=

    2007

  103. [114]

    Multiscale Modeling & Simulation , volume=

    An Iterative Regularization Method for Total Variation-Based Image Restoration , author=. Multiscale Modeling & Simulation , volume=. 2005 , publisher=

    2005

  104. [115]

    Bregman iterative algorithms for _1 -minimization with applications to compressed sensing , author=. SIAM J. Imaging Sceiences , volume=. 2008 , publisher=

    2008

  105. [116]

    A simplified view of first order methods for optimization , author=. Math. Program., Ser. B , volume=. 2018 , publisher=

    2018

  106. [117]

    Convergence analysis of a proximal-like minimization algorithm using Bregman functions , author=. SIAM J. Optim. , volume=. 1993 , publisher=

    1993

  107. [118]

    The split

    Goldstein, Tom and Osher, Stanley , journal=. The split. 2009 , publisher=

    2009

  108. [119]

    SIAM Journal on Scientific Computing , volume=

    Alternating direction algorithms for L1-problems in compressive sensing , author=. SIAM Journal on Scientific Computing , volume=. 2011 , publisher=

    2011

  109. [120]

    SIAM Journal on Scientific Computing , volume=

    An Efficient TVL1 Algorithm for Deblurring Multichannel Images Corrupted by Impulsive Noise , author=. SIAM Journal on Scientific Computing , volume=. 2009 , publisher=

    2009

  110. [121]

    SIAM Journal on Imaging Sciences , volume=

    A New Alternating Minimization Algorithm for Total Variation Image Reconstruction , author=. SIAM Journal on Imaging Sciences , volume=. 2008 , publisher=

    2008

  111. [122]

    Communications on Pure and Applied Mathematics , volume=

    An iterative thresholding algorithm for linear inverse problems with a sparsity constraint , author=. Communications on Pure and Applied Mathematics , volume=. 2004 , publisher=

    2004

  112. [123]

    Foundations and Trends in Machine Learning , volume=

    Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers , author=. Foundations and Trends in Machine Learning , volume=

  113. [124]

    Foundations and Trends in Optimization , volume=

    Proximal Algorithms , author=. Foundations and Trends in Optimization , volume=

  114. [125]

    Journal of Mathematical Analysis and Applications , volume=

    Backward–forward algorithms for structured monotone inclusions in Hilbert spaces , author=. Journal of Mathematical Analysis and Applications , volume=

  115. [126]

    Set-valued Analysis , volume=

    An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping , author=. Set-valued Analysis , volume=

  116. [127]

    A monotone+skew splitting model for composite monotone inclusions in duality , author=. SIAM J. Control Optim. , volume=. 2011 , publisher=

    2011

  117. [128]

    Convergence analysis for a primal-dual monotone+skew splitting algorithm with applications to total variation minimization , author=. J. Math. Imaging Vis. , volume=. 2014 , publisher=

    2014

  118. [129]

    On the convergence rate of

    He, Bingsheng and Yuan, Xiaoming , journal=. On the convergence rate of

  119. [130]

    Customized proximal point algorithms for linearly constrained convex minimization and saddle-point problems: a unified approach , author=. Comp. Optim. Appl. , volume=

  120. [131]

    A customized proximal point algorithm for convex minimization with linear constraints , author=. Comp. Optim. Appl. , volume=

  121. [132]

    Convergence Analysis of the Generalized Alternating Direction Method of Multipliers with Logarithmic–Quadratic Proximal Regularization , author=. J. Optim. Theorey Appl. , volume=. 2015 , publisher=

    2015

  122. [133]

    and Yuan, X

    He, B. and Yuan, X. , journal=. On non-ergodic convergence rate of. 2015 , publisher=

    2015

  123. [134]

    and Xu, H.K

    He, B. and Xu, H.K. and Yuan, X. , journal=. On the Proximal. 2016 , publisher=

    2016

  124. [135]

    and Yuan, X

    He, B. and Yuan, X. , journal=. A class of. 2018 , publisher=

    2018

  125. [136]

    and He, B

    Chen, C. and He, B. and Yuan, X. , journal=. The direct extension of. 2016 , publisher=

    2016

  126. [137]

    A class of customized proximal point algorithms for linearly constrained convex optimization , author=. Comp. Appl. Math. , volume=. 2018 , publisher=

    2018

  127. [138]

    and Yuan, X

    Tao, M. and Yuan, X. , journal=. On the. 2012 , publisher=

    2012

  128. [139]

    He, Bingsheng and Yuan, Xiaoming , journal=. On the. 2012 , publisher=

    2012

  129. [140]

    Convergence Analysis of Primal-Dual Algorithms for a Saddle-Point Problem: From Contraction Perspective , author=. SIAM J. Imaging Sciences , volume=. 2012 , publisher=

    2012

  130. [141]

    Journal of Mathematical Imaging and Vision , volume=

    An Algorithmic Framework of Generalized Primal-Dual Hybrid Gradient Methods for Saddle Point Problems , author=. Journal of Mathematical Imaging and Vision , volume=. 2017 , publisher=

    2017

  131. [142]

    Journal of Nonlinear and Convex Analysis , volume=

    An Algorithm for Splitting Parallel Sums of Linearly Composed Monotone Operators, with Applications to Signal Recovery , author=. Journal of Nonlinear and Convex Analysis , volume=

  132. [143]

    and Osher, S

    Rudin, L.I. and Osher, S. and Fatemi, E. , journal=. 1992 , publisher=

    1992

  133. [144]

    2017 , publisher=

    Convex Analysis and Monotone Operator Theory in Hilbert Spaces , author=. 2017 , publisher=

    2017

  134. [145]

    2006 , publisher=

    Numerical Optimization (Second Edition) , author=. 2006 , publisher=

    2006

  135. [146]

    2004 , publisher=

    Convex Optimization , author=. 2004 , publisher=

    2004

  136. [147]

    2009 , publisher=

    Convex Optimization Theory , author=. 2009 , publisher=

    2009

  137. [148]

    2015 , publisher=

    Convex Optimization Algorithms , author=. 2015 , publisher=

    2015

  138. [149]

    2017 , publisher=

    First-Order Methods in Optimization , author=. 2017 , publisher=

    2017

  139. [150]

    2020 , publisher=

    Accelerated Optimization for Machine Learning: First-Order Algorithms , author=. 2020 , publisher=

    2020

  140. [151]

    2008 , publisher=

    A Wavelet Tour of Signal Processing: The Sparse Way , author=. 2008 , publisher=

    2008

  141. [152]

    Set-Valued and Variational Analysis , volume=

    A Three-Operator Splitting Scheme and its Optimization Applications , author=. Set-Valued and Variational Analysis , volume=

  142. [153]

    , journal=

    Moreau, J.-J. , journal=. Proximit\'

  143. [154]

    and Osher, S

    Goldstein, T. and Osher, S. , journal=. The split

  144. [155]

    Efficient schemes for total variation minimization under constraints in image processing , author=. SIAM J. Sci. Comput. , volume=

  145. [156]

    Inverse Problems , volume=

    On a generalization of the iterative soft-thresholding algorithm for the case of non-separable penalty , author=. Inverse Problems , volume=. 2011 , publisher=

    2011

  146. [157]

    International Journal on Geomathematics , volume=

    Iterative algorithms for total variation-like reconstructions in seismic tomography , author=. International Journal on Geomathematics , volume=

  147. [158]

    Annals of Statistics , volume=

    Robust estimation of a location parameter , author=. Annals of Statistics , volume=

  148. [159]

    Compressive Sensing for 3D Data Processing Tasks: Applications, Models and Algorithms , author=. Ph.D. Thesis, CAAM, Rice University, Houston, Texas , year=

  149. [160]

    IEEE Transactions on Image Processing , volume=

    Fast image recovery using variable splitting and constrained optimization , author=. IEEE Transactions on Image Processing , volume=. 2010 , publisher=

    2010

  150. [161]

    and Bioucas-Dias, J.M

    Afonso, M.V. and Bioucas-Dias, J.M. and Figueiredo M.A.T. , journal=. An agumented. 2011 , publisher=

    2011

  151. [162]

    IEEE Signal Processing Magazine , volume=

    Playing with Duality: An overview of recent primal?dual approaches for solving large-scale optimization problems , author=. IEEE Signal Processing Magazine , volume=

  152. [163]

    44 , author=

    Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Science, vol. 44 , author=. 1983 , publisher=

    1983

  153. [164]

    Duality methods for solving variational inequalities , author=. Comput. Math. Appl. , volume=. 1981 , publisher=

    1981

  154. [165]

    2004 , publisher=

    Introductory lectures on convex optimization: a basic course , author=. 2004 , publisher=

    2004

  155. [166]

    Core Discussion Papers , volume=

    Gradient methods for minimizing composite objective function , author=. Core Discussion Papers , volume=

  156. [167]

    , journal=

    Nesterov, Y. , journal=. A method for solving the convex programming problem with convergence rate

  157. [168]

    Smooth minimization of non-smooth functions , author=. Math. Program. (A) , volume=

  158. [169]

    SIAM Journal on Numerical Analysis , volume=

    Splitting Algorithms for the Sum of Two Nonlinear Operators , author=. SIAM Journal on Numerical Analysis , volume=. 1979 , publisher=

    1979

  159. [170]

    , journal=

    Eckstein, Jonathan and Bertsekas, Dimitri P. , journal=. On the. 1992 , publisher=

    1992

  160. [171]

    On the Numerical Solution of Heat Conduction Problems in Two and Three Space Variables , author=. Trans. Amer. Math. Soc. , volume=. 1956 , publisher=

    1956

  161. [172]

    Journal of the Society for Industrial and Applied Mathematics , volume=

    The Numerical Solution of parabolic and elliptic differential equations , author=. Journal of the Society for Industrial and Applied Mathematics , volume=. 1955 , publisher=

    1955

  162. [173]

    A modified forward-backward splitting method for maximal monotone mappings , author=. SIAM J. Control Optim. , volume=. 2000 , publisher=

    2000

  163. [174]

    Applications of a Splitting Algorithm to Decomposition in Convex Programming and Variational Inequalities , author=. SIAM J. Control Optim. , volume=. 1991 , publisher=

    1991

  164. [175]

    IEEE Trans

    Signal recovery with certain involved convex data-fidelity constraints , author=. IEEE Trans. on Signal Processing , volume=. 2015 , publisher=

    2015

  165. [176]

    SIAM Journal on Imaging Sciences , volume=

    A fast iterative shrinkage-thresholding algorithm for linear inverse problems , author=. SIAM Journal on Imaging Sciences , volume=. 2009 , publisher=

    2009

  166. [177]

    IEEE Transactions on Image Processing , volume=

    Fast Gradient-Based Algorithms for Constrained Total Variation Image Denoising and Deblurring Problems , author=. IEEE Transactions on Image Processing , volume=. 2009 , publisher=

    2009

  167. [178]

    Journal of Mathematical Imaging and Vision , volume=

    An algorithm for total variation minimization and applications , author=. Journal of Mathematical Imaging and Vision , volume=. 2004 , publisher=

    2004

  168. [179]

    , journal=

    Chambolle, A. , journal=. Total Variation Minimization and a Class of Binary. 2005 , publisher=

    2005

  169. [180]

    A first-order primal-dual algorithm for convex problems with applications to imaging , author=. J. Math. Imag. Vis. , volume=. 2011 , publisher=

    2011

  170. [181]

    Weak convergence of the sequence of successive approximations for nonexpansive mappings , author=. Bull. Amer. Math. Soc. , volume=

  171. [182]

    Journal of Computational and Applied Mathematics , volume=

    Convergence of a splitting inertial proximal method for monotone operators , author=. Journal of Computational and Applied Mathematics , volume=

  172. [183]

    Optimization , volume=

    Solving monotone inclusions via compositions of nonexpansie averaed operators , author=. Optimization , volume=

  173. [184]

    Multiscale Modeling & Simulation , volume=

    Signal recovery by proximal forward-backward splitting , author=. Multiscale Modeling & Simulation , volume=. 2005 , publisher=

    2005

  174. [185]

    Inverse Problems , volume=

    A proximal decomposition method for solving convex variational inverse problems , author=. Inverse Problems , volume=. 2008 , publisher=

    2008

  175. [186]

    and Pesquet, J.C

    Combettes, P.L. and Pesquet, J.C. , journal=. A. 2007 , publisher=

    2007

  176. [187]

    Some first-order algorithms for total variation based image restoration , author=. J. Math. Imag. Vis. , volume=. 2009 , publisher=

    2009

  177. [188]

    Inverse problems with

    Dup\'. Inverse problems with. IEEE Int. Conf. on Image Proc. , pages=. 2011

    2011

  178. [189]

    Diagonal scaling in

    Giselsson, Pontus and Boyd, Stephen , booktitle=. Diagonal scaling in. 2014

    2014

  179. [190]

    and Cremers, D

    Pock, T. and Cremers, D. and Bischof, H. and Chambolle, A. , booktitle=. An algorithm fo rminimizing the. 2009

    2009

  180. [191]

    IEEE Int

    Monotone operator splitting for optimization problems in sparse recovery , author=. IEEE Int. Conf. on Image Proc. , pages=. 2009

    2009

  181. [192]

    and Chartrand, R

    Le, T. and Chartrand, R. and Asaki, T.J. , journal=. A Variational Approach to Reconstructing Images Corrupted by. 2007 , publisher=

    2007

  182. [193]

    , journal=

    Setzer, S. , journal=. Split. 2009 , publisher=

    2009

  183. [194]

    and Steidl, G

    Setzer, S. and Steidl, G. and Teuber, T. , journal=. Deblurring. 2010 , publisher=

    2010

  184. [195]

    and Steidl, G

    Setzer, S. and Steidl, G. , journal=. Split. 2008 , publisher=

    2008

  185. [196]

    , journal=

    Setzer, S. , journal=. Operator Splittings,. 2011 , publisher=

    2011

  186. [197]

    Splitting methods in image processing , author=. Ph.D. thesis, University of Mannheim , year=

  187. [198]

    Primal dual algorithms for convex models and applications to image restoration, registration and nonlocal inpainting , author=. Ph.D. thesis, University of California, Los Angeles , year=

  188. [199]

    SIAM Journal on Imaging Sciences , volume=

    A General Framework for a Class of First Order Primal-Dual Algorithms for Convex Optimization in Imaging Science , author=. SIAM Journal on Imaging Sciences , volume=. 2010 , publisher=

    2010

  189. [200]

    , journal=

    Esser, E. , journal=. Applications of Lagrangian-Based Alternating Direction Methods and Connections to Split. 2009 , publisher=

    2009

  190. [201]

    CAM Report 08-34, UCLA , year=

    An efficient primal-dual hybrid gradient algorithm for total variation image restoration , author=. CAM Report 08-34, UCLA , year=

  191. [202]

    Thchnical Report 2790, Optimization online , year=

    Convergence analysis of primal-dual algorithms for total variation image restoration , author=. Thchnical Report 2790, Optimization online , year=

  192. [203]

    Stanford Mathematical Studies in the Social Sciences, vol.II

    Studies in linear and nonllinear programming , author=. Stanford Mathematical Studies in the Social Sciences, vol.II. , year=

  193. [204]

    Computational Optimization and Applications , volume=

    Duality-based algorithms for total-variation-regularized image restoration , author=. Computational Optimization and Applications , volume=. 2010 , publisher=

    2010

  194. [205]

    Monotone operators and the proximal point algorithm , author=. SIAM J. Control Optim. , volume=. 1976 , publisher=

    1976

  195. [206]

    , journal=

    Martinet, B. , journal=. R\'

  196. [207]

    1996 , publisher=

    Convex analysis , author=. 1996 , publisher=

    1996

  197. [208]

    2004 , publisher=

    Variational Analysis , author=. 2004 , publisher=

    2004

  198. [209]

    Journal of Mathematical Imaging and Vision , volume=

    On the Convergence of Primal–Dual Hybrid Gradient Algorithms for Total Variation Image Restoration , author=. Journal of Mathematical Imaging and Vision , volume=. 2012 , publisher=

    2012

  199. [210]

    Splitting algorithms for the sum of two nonlinear operators , author=. SIAM J. Numer. Anal. , volume=. 1976 , publisher=

    1976

  200. [211]

    Operations Research Letters , volume=

    A fast dual proximal gradient algorithm for convex minimization and applications , author=. Operations Research Letters , volume=. 2014 , publisher=

    2014

  201. [212]

    and Burger, M

    Zhang, X. and Burger, M. and Osher, S. , journal=. A Unified Primal-Dual Algorithm Framework Based on. 2011 , publisher=

    2011

  202. [213]

    SIAM Journal on Imaging Sciences , volume=

    Bregmanized nonlocal regularization for deconvolution and sparse reconstruction , author=. SIAM Journal on Imaging Sciences , volume=. 2010 , publisher=

    2010

  203. [214]

    and Benfenati, A

    Bonettini, S. and Benfenati, A. and Ruggiero, V. , booktitle=. Primal-dual first order methods for total variation image restoration in presence of. 2008

    2008

  204. [215]

    1993 , publisher=

    Convex analysis and minimization algorithms, Volume I , author=. 1993 , publisher=

    1993

  205. [216]

    A splitting algorithm for dual monotone inclusions involving cocoercive operators , author=. Adv. Comput. Math. , volume=

  206. [217]

    Set-Valued Anal

    Dualization of signal recovery problems , author=. Set-Valued Anal. , volume=

  207. [218]

    and Pesquet, J.C

    Combettes, P.L. and Pesquet, J.C. , journal=. Primal-dual splitting algorithm for solving inclusions with mixtures of composite,

  208. [219]

    A proximal-based decomposition method for convex minimization problems , author=. Math. Program. , volume=

  209. [220]

    Monotone operator theory in convex optimization , author=. Math. Program., Ser. B , volume=. 2018 , publisher=

    2018

  210. [221]

    , journal=

    Condat, L. , journal=. A primal-dual splitting method for convex optimization involving

  211. [222]

    and Sawatzky, A

    Burger, M. and Sawatzky, A. and Steidl, G. , title =. Splitting Methods in Communication, Imaging, Science, and Engineering , year =

  212. [223]

    and Sawatzky, A

    Burger, M. and Sawatzky, A. and Steidl, G. , editor=. First Order Algorithms in Variational Image Processing , bookTitle=. 2017 , publisher=

    2017

  213. [224]

    and Pesquet, J.C

    Combettes, P.L. and Pesquet, J.C. , editor=. Proximal Splitting Methods in Signal Processing , bookTitle=. 2011 , publisher=

    2011

  214. [225]

    , editor=

    Powell, M.J. , editor=. A method for non-linear constraints in minimization problems , year=

  215. [226]

    Graph signal recovery via primal-dual algorithms for total variation minimization , author=. IEEE J. of Selected Topics in Signal Processing , volume=. 2017 , publisher=

    2017

  216. [227]

    Optimization , volume=

    On the convergence rate of a forward-backward type primal-dual primal-dual splitting algorithm for convex optimization problems , author=. Optimization , volume=

  217. [228]

    On the convergence rate improvement of a primal-dual splitting algorithm for solving monotone inclusion problems , author=. Math. Program., Ser. A , volume=

  218. [229]

    A primal-dual splitting algorithm for finding zeros of sums of maximally monotone operators , author=. SIAM J. Optim. , volume=

  219. [230]

    Bo. A. SIAM J. Optim. , volume=. 2013 , publisher=

    2013

  220. [231]

    A proximal point analysis of the preconditioned alternating direction method of multipliers , author=. J. Optim. Theory Appl. , volume=

  221. [232]

    and Sun, H.P

    Bredies, K. and Sun, H.P. , journal=. Preconditioned. 2015 , publisher=

    2015

  222. [233]

    Alternating Direction Algorithms for Total Variation Deconvolution in Image Deconstruction , author=

  223. [234]

    On the optimal linear convergence rate of a generalized proximal point algorithm , author=. J. Sci. Comput. , volume=. 2018 , publisher=

    2018

  224. [235]

    Constrained Total Variation Deblurring Models and Fast Algorithms Based on Alternating Direction Method of Multipliers , author=. SIAM J. Imaging Sciences , volume=. 2013 , publisher=

    2013

  225. [236]

    On the global and linear convergence of the generalized alternating direction method of multipliers , author=. J. Sci. Comput. , volume=. 2016 , publisher=

    2016

  226. [237]

    and Lai, M

    Deng, W. and Lai, M. and Peng, Z. and Yin, W. , journal=. Parallel Multi-Block. 2017 , publisher=

    2017

  227. [238]

    kerman, P

    A. kerman, P. Chenchene, E. an Giselsson, and E. Naldi , Splitting the forward-backward algorithm: A full characterization , Arxiv preprint: arXiv:2504.10999, (2025)

    2025 arXiv

  228. [239]

    Afonso, J.M

    M.V. Afonso, J.M. Bioucas-Dias, and Figueiredo M.A.T. , Fast image recovery using variable splitting and constrained optimization , IEEE Transactions on Image Processing, 19 (2010), pp. 2345--2356

    2010

  229. [240]

    681--695

    height 2pt depth -1.6pt width 23pt, An agumented L agrangian approach to the constrained optimization formulation of imaging inverse problems , IEEE Transactions on Image Processing, 20 (2011), pp. 681--695

    2011

  230. [241]

    F. J. Arag\' o n-Artacho, R.I. Bo t , and D. Torregrosa-Bel\' e n , A primal-dual splitting algorithm for composite monotone inclusions with minimal lifting , Numerical Algorithms, 93 (2023), pp. 103--130

    2023

  231. [242]

    F. J. Arag\' o n-Artacho, R. Campoy, and C. L\' o pez-Pastor , Forward-backward algorithms devised by graphs , SIAM Journal on Optimization, 35 (2025), pp. 2423--2451

    2025

  232. [243]

    F. J. Arag\' o n-Artacho, Y. Malitsky, M.K. Tam, and D. Torregrosa-Bel\' e n , Distributed forward-backward methods for ring networks , Computational Optimization and Applications, 86 (2023), pp. 845--870

    2023

  233. [244]

    Bauschke and Patrick L

    Heinz H. Bauschke and Patrick L. Combettes , Convex Analysis and Monotone Operator Theory in Hilbert Spaces , Second Edition, CMS Books in Mathematics, Springer, New York, NY, 2017

    2017

  234. [245]

    Bo t , E

    R.I. Bo t , E. Chenchene, and J.M. Fadili , Fast K rasnoselskii-- M ann method with overrelaxation and preconditioners , Arxiv preprint: arXiv:2411.18574, (2024)

    2024

  235. [246]

    Radu Ioan Bot and Christopher Hendrich , Convergence analysis for a primal-dual monotone+skew splitting algorithm with applications to total variation minimization , J. Math. Imaging Vis., 49 (2014), pp. 551--568

    2014

  236. [247]

    Bredies, E

    K. Bredies, E. Chenchene, D.A. Lorenz, and E. Naldi , Degenerate preconditioned proximal point algorithms , SIAM J. Optim., 32 (2022), pp. 2376--2401

    2022

  237. [248]

    1569--1594

    Kristian Bredies, Enis Chenchene, and Emanuele Naldi , Graph and distributed extensions of the Douglas-Rachford method , SIAM Journal on Optimization, 34 (2024), pp. 1569--1594

    2024

  238. [249]

    Brice\ n o Arias and P.L

    L.M. Brice\ n o Arias and P.L. Combettes , A monotone+skew splitting model for composite monotone inclusions in duality , SIAM J. Control Optim., 21 (2011), pp. 1230--1250

    2011

  239. [250]

    Brice\ n o Arias and D

    L.M. Brice\ n o Arias and D. Davis , Forward-backward-half forward algorithm for solving monotone inclusions , SIAM J. Optim., 28 (2018), pp. 2839--2871

    2018

  240. [251]

    Brice\ n o Arias and S

    L.M. Brice\ n o Arias and S. L\' o pez Rivera , A projected primal–dual method for solving constrained monotone inclusions , J. Optim. Theory Appl., 180 (2019), pp. 907--924

    2019

  241. [252]

    B\` u i and Patrick L

    Minh N. B\` u i and Patrick L. Combettes , Warped proximal iterations for monotone inclusions , Journal of Mathematical Analysis and Applications, 491 (2020), p. 124315

    2020

  242. [253]

    Campoy , A product space reformulation with reduced dimension for splitting algorithms , Computational Optimization and Applications, 83 (2022), pp

    R. Campoy , A product space reformulation with reduced dimension for splitting algorithms , Computational Optimization and Applications, 83 (2022), pp. 319--348

    2022

  243. [254]

    Chambolle and T

    A. Chambolle and T. Pock , A first-order primal-dual algorithm for convex problems with applications to imaging , J. Math. Imag. Vis., 40 (2011), pp. 120--145

    2011

  244. [255]

    Chang, M

    T.H. Chang, M. Hong, and X. Wang , Multi-agent distributed optimization via inexact consensus ADMM , IEEE Transactions on Signal Processing, 63 (2014), pp. 482--497

    2014

  245. [256]

    Combettes , Monotone operator theory in convex optimization , Math

    P.L. Combettes , Monotone operator theory in convex optimization , Math. Program., Ser. B, 170 (2018), pp. 177--206

    2018

  246. [257]

    Combettes, D

    P.L. Combettes, D. D\ u ng, and B.C. V\ u , Dualization of signal recovery problems , Set-Valued Anal., 18 (2010), pp. 373--404

    2010

  247. [258]

    Combettes and J.C

    P.L. Combettes and J.C. Pesquet , Primal-dual splitting algorithm for solving inclusions with mixtures of composite, L ipschitzian, and parallel-sum type monotone operators , Set-Valued Var. Anal., 20 (2012), pp. 307--330

    2012

  248. [259]

    Combettes and V.R

    P.L. Combettes and V.R. Wajs , Signal recovery by proximal forward-backward splitting , Multiscale Modeling & Simulation, 4 (2005), pp. 1168--1200

    2005

  249. [260]

    Combettes , Systems of structured monotone inclusions: Duality, algorithms, and applications , SIAM Journal on Optimization, 23 (2013), pp

    Patrick L. Combettes , Systems of structured monotone inclusions: Duality, algorithms, and applications , SIAM Journal on Optimization, 23 (2013), pp. 2420--2447

    2013

  250. [261]

    P. L. Combettes and B.C. V\ u , Variable metric forward–backward splitting with applications to monotone inclusions in duality , Optimization, 63 (2014), pp. 1289--1318

    2014

  251. [262]

    375--435

    Laurent Condat, Daichi Kitahara, Andr\' e s Contreras, and Akira Hirabayashi , Proximal splitting algorithms for convex optimization: A tour of recent advances, with new twists , SIAM Review, 65 (2023), pp. 375--435

    2023

  252. [263]

    Dao, Matthew K

    Minh N. Dao, Matthew K. Tam, and Thang D. Truong , A general approach to distributed operator splitting , Arxiv preprint: arXiv:2504.14987v3, (2026)

    2026 arXiv

  253. [264]

    Davis and W

    D. Davis and W. Yin , A three-operator splitting scheme and its optimization applications , Set-Valued and Variational Analysis, 25 (2017), pp. 829--858

    2017

  254. [265]

    He and X

    B. He and X. Yuan , On non-ergodic convergence rate of Douglas - Rachford alternating direction method of multipliers , Numerische Mathematik, 130 (2015), pp. 567--577

    2015

  255. [266]

    Changchi Huang, Jigen Peng, Liqian Qin, and Yuchao Tang , A primal-dual splitting algorithm for monotone inclusions with applications , private communication

  256. [267]

    Lions and B

    P.L. Lions and B. Mercier , Splitting algorithms for the sum of two nonlinear operators , SIAM Journal on Numerical Analysis, 16 (1979), pp. 964--979

    1979

  257. [268]

    Tam , Resolvent splitting for sums of monotone operators with minimal lifting , Mathematical Programming, 201 (2023), pp

    Yura Malitsky and Matthew K. Tam , Resolvent splitting for sums of monotone operators with minimal lifting , Mathematical Programming, 201 (2023), pp. 231--262

    2023

  258. [269]

    Morin, S

    M. Morin, S. Banert, and P. Giselsson , Frugal splitting operators: representation, min imal lifting, and convergence , SIAM Journal on Optimization, 34 (2024), pp. 1595--1621

    2024

  259. [270]

    R. T. Rockafellar , Augmented L agrangians and applications of the proximal point algorithm in convex programming , Math. Oper. Res., 1 (1976), pp. 97--116

    1976

  260. [271]

    height 2pt depth -1.6pt width 23pt, Convex analysis , Princeton Landmarks in Mathematics and Physics, Princeton University Press, 1996

    1996

  261. [272]

    Tyrrell Rockafellar and Roger J-B Wets , Variational Analysis , Springer, Grundlehren der Mathematischen Wissenschaft, vol

    R. Tyrrell Rockafellar and Roger J-B Wets , Variational Analysis , Springer, Grundlehren der Mathematischen Wissenschaft, vol. 317, 2004

    2004

  262. [273]

    Ryu and B.C

    E.K. Ryu and B.C. V\ u , Finding the forward- D ouglas- R achford-forward method , Journal of Optimization Theory and Applications, 184 (2020), pp. 858--876

    2020

  263. [274]

    Ryu , Uniqueness of DRS as the 2 operator resolvent-splitting and impossibility of 3 operator resolvent-splitting , Mathematical Programming, 182 (2020), pp

    Ernest K. Ryu , Uniqueness of DRS as the 2 operator resolvent-splitting and impossibility of 3 operator resolvent-splitting , Mathematical Programming, 182 (2020), pp. 233--273

    2020

  264. [275]

    B. F. Svaiter , On weak convergence of the D ouglas- R achford method , SIAM Journal on Control and Optimization, 49 (2011), pp. 280--287

    2011

  265. [276]

    Tam , Frugal and decentralised resolvent splittings defined by nonexpansive operators , Optimization Letters, 18 (2024), pp

    M.K. Tam , Frugal and decentralised resolvent splittings defined by nonexpansive operators , Optimization Letters, 18 (2024), pp. 1541--1559

    2024

  266. [277]

    V\ u , A splitting algorithm for dual monotone inclusions involving cocoercive operators , Adv

    B.C. V\ u , A splitting algorithm for dual monotone inclusions involving cocoercive operators , Adv. Comput. Math., 38 (2013), pp. 667--681

    2013

  267. [278]

    , 77 (2022)

    Feng Xue , On the nonexpansive operators based on arbitrary metric: A degenerate analysis , Results in Mathematics, https://doi.org/10.1007/s00025-022-01766-6. , 77 (2022)

    2022 doi

  268. [279]

    , 42 (2023)

    height 2pt depth -1.6pt width 23pt, A generalized forward--backward splitting operator: Degenerate analysis and applications , Computational and Applied Mathematics, https://doi.org/10.1007/s40314-022-02143-3. , 42 (2023)

    2023 doi

  269. [280]

    1981--1992

    height 2pt depth -1.6pt width 23pt, On D ouglas-- R achford splitting that generally fails to be a proximal mapping: A degenerate proximal point analysis , Optimization, 73 (2024), pp. 1981--1992

    2024

  270. [281]

    155--194

    height 2pt depth -1.6pt width 23pt, The operator splitting schemes revisited: Primal-dual gap and degeneracy reduction by a unified analysis , Optimization Letters, 18 (2024), pp. 155--194

    2024

  271. [282]

    966--988

    height 2pt depth -1.6pt width 23pt, On the metric resolvent: nonexpansiveness, convergence rates and applications , Journal of the Operations Research Society of China, 13 (2025), pp. 966--988

    2025

  272. [283]

    , 208 (2026)

    Feng Xue and Hui Zhang , On degenerate preconditioned proximal point methods under restricted monotonicity , Journal of Optimization Theory and Applications, https://doi.org/10.1007/s10957-025-02915-x. , 208 (2026)

    2026 doi

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.