pith. sign in

arxiv: 2502.21202 · v1 · submitted 2025-02-28 · 📡 eess.IV · eess.SP· math.OC

An Adaptive Multiparameter Penalty Selection Method for Multiconstraint and Multiblock ADMM

Luke Lozenski , Michael T. McCann , Brendt Wohlberg This is my paper

Pith reviewed 2026-05-23 02:04 UTC · model grok-4.3

classification 📡 eess.IV eess.SPmath.OC
keywords ADMMpenalty parametersmulticonstraint optimizationadaptive selectionsignal processingimage processingconvergence
0
0 comments X

The pith

A new adaptive method selects multiple penalty parameters for ADMM to account for scale differences in multiconstraint problems.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper introduces an online method for selecting multiple penalty parameters in the alternating direction method of multipliers (ADMM) for optimization problems with multiple constraints or block matrix components. Standard ADMM uses a single penalty parameter that requires tuning, but scale differences between constraints can cause slow convergence. The new method adaptively adjusts for these scale differences, making it robust to problem transformations and initial parameter choices while remaining simple to implement. Numerical experiments indicate it performs better than several existing penalty selection approaches.

Core claim

The proposed method for online selection of multiple penalty parameters in ADMM applied to problems with multiple constraints or functionals with block matrix components is able to adaptively account for differences in scale between constraints. This provides robustness with respect to problem transformations and initial selection of penalty parameters. The method is simple to understand and implement, and numerical experiments show it performs favorably compared to a variety of existing penalty parameter selection methods.

What carries the argument

The adaptive multiparameter penalty selection method that dynamically adjusts individual penalty parameters for each constraint based on observed scale differences.

Load-bearing premise

That the adaptive adjustment based on scale differences will reliably lead to better convergence in most practical cases, as supported by experiments without a general theoretical proof.

What would settle it

Finding a specific multiconstraint optimization problem where the proposed adaptive method converges slower or less reliably than a carefully tuned single-penalty ADMM despite the presence of scale differences.

Figures

Figures reproduced from arXiv: 2502.21202 by Brendt Wohlberg, Luke Lozenski, Michael T. McCann.

Figure 4
Figure 4. Figure 4: B. Scaled Constraint Sum of Squares Consider the constrained optimization problem argmin x,z 1 2 x T Qx + q T x + 1 2 z T Rz + r T x s.t. j m(a T j x + b T j z − cj ) = 0, j = 1, . . . , J, (25) where each constraint is scaled by it index j raised to a chosen power m, x ∈ RM, z ∈ R N , aj and q are randomly sampled from an M-dimensional standard normal distribution, bj and r are randomly sampled from an N-… view at source ↗
Figure 1
Figure 1. Figure 1: Magnitude and angle (radians) of maximum eigenvalues of complex iteration matrix [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Magnitude and angle (radians) of maximum eigenvalues of complex iteration matrix [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Relative residual of ADMM solutions corresponding to an iteration matrix with complex eigenvalues for fixed penalty [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Relative residual of ADMM solutions corresponding to an iteration matrix with complex eigenvalues for single-parameter [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Relative residual after 50 iterations for ADMM solutions of [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Relative residual of ADMM solutions for ℓ1 fidelity, total variation regularization sparse CT reconstruction problem utilizing fixed, multiparameter BBS, and multiparameter SRA methods after 50 iterations plotted on a surface as a function of initial ρ1 and ρ2. Note that the fixed method converges only in a region off of the ρ1 = ρ2 diagonal. The multiparameter BSS method demonstrates very poor performance… view at source ↗
Figure 7
Figure 7. Figure 7: Relative residual of ADMM solutions for ℓ1 fidelity, total variation regularization sparse CT reconstruction problem for single-parameter and multiparameter adaptive penalty pa￾rameter rules after 50 iterations plotted as a function of initial ρ = ρ1 = ρ2. Note that none of the single-parameter methods converge, corresponding to the optimal ρ being off-diagonal in [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Example reconstructions from each penalty parameter method initialized at [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗
read the original abstract

This work presents a new method for online selection of multiple penalty parameters for the alternating direction method of multipliers (ADMM) algorithm applied to optimization problems with multiple constraints or functionals with block matrix components. ADMM is widely used for solving constrained optimization problems in a variety of fields, including signal and image processing. Implementations of ADMM often utilize a single hyperparameter, referred to as the penalty parameter, which needs to be tuned to control the rate of convergence. However, in problems with multiple constraints, ADMM may demonstrate slow convergence regardless of penalty parameter selection due to scale differences between constraints. Accounting for scale differences between constraints to improve convergence in these cases requires introducing a penalty parameter for each constraint. The proposed method is able to adaptively account for differences in scale between constraints, providing robustness with respect to problem transformations and initial selection of penalty parameters. It is also simple to understand and implement. Our numerical experiments demonstrate that the proposed method performs favorably compared to a variety of existing penalty parameter selection methods.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The paper proposes an adaptive multiparameter penalty selection method for the alternating direction method of multipliers (ADMM) applied to optimization problems with multiple constraints or multiblock structures. The method is designed to automatically account for scale differences between constraints, yielding robustness to problem transformations and to the choice of initial penalty parameters. The approach is presented as simple to implement, and the authors report that numerical experiments show favorable performance relative to existing penalty-parameter selection heuristics.

Significance. If the empirical gains hold under broader testing, the method could offer a practical, low-effort tuning strategy for ADMM users in signal and image processing. The absence of any convergence-rate bound, fixed-point analysis, or scale-invariance proof, however, confines the contribution to an algorithmic heuristic whose reliability remains problem-dependent.

major comments (2)
  1. The manuscript contains no convergence analysis, invariance proof, or fixed-point argument establishing that the adaptive multiparameter rule preserves or improves convergence when constraint scales differ by orders of magnitude (see the skeptic note and the abstract's claim of robustness). This is load-bearing for the central assertion that the method “adaptively account[s] for differences in scale.”
  2. [Abstract] The abstract states that the method “performs favorably compared to a variety of existing penalty parameter selection methods in numerical experiments,” yet supplies no information on test problems, data sets, stopping criteria, or quantitative metrics. Without these details the performance claim cannot be assessed.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript. We address each major comment below. We agree that the abstract requires expansion with experimental details and will revise it. Regarding convergence analysis, we clarify the heuristic nature of the contribution without adding theoretical results.

read point-by-point responses

  1. Referee: The manuscript contains no convergence analysis, invariance proof, or fixed-point argument establishing that the adaptive multiparameter rule preserves or improves convergence when constraint scales differ by orders of magnitude (see the skeptic note and the abstract's claim of robustness). This is load-bearing for the central assertion that the method “adaptively account[s] for differences in scale.”

    Authors: We acknowledge that the manuscript provides no convergence analysis, invariance proof, or fixed-point argument. The method is presented as a practical heuristic for selecting multiple penalty parameters in multiconstraint or multiblock ADMM, with robustness to scale differences demonstrated empirically in the numerical experiments. We do not assert theoretical guarantees, so the central claim rests on observed performance rather than proof. We will add explicit language in the introduction and conclusion stating that the approach is a heuristic without convergence-rate or invariance guarantees. revision: partial

  2. Referee: The abstract states that the method “performs favorably compared to a variety of existing penalty parameter selection methods in numerical experiments,” yet supplies no information on test problems, data sets, stopping criteria, or quantitative metrics. Without these details the performance claim cannot be assessed.

    Authors: We agree that the abstract lacks sufficient detail on the experiments. In the revised manuscript we will expand the abstract to briefly specify the classes of test problems, the problem instances or data sets employed, the stopping criteria, and the quantitative metrics used to evaluate performance against existing methods. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper introduces an algorithmic procedure for adaptive multiparameter selection in ADMM to address scale differences among constraints. No derivation, theorem, or prediction is presented that reduces by construction to its own fitted inputs or to a self-citation chain. The central claim rests on the algorithmic description plus numerical comparisons, which constitute independent empirical content rather than a closed loop. No self-definitional steps, fitted-input predictions, or load-bearing self-citations appear in the provided text.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, which does not detail any free parameters, axioms, or new entities beyond the proposed method itself.

axioms (1)
  • domain assumption ADMM can be extended with multiple penalty parameters to handle scale differences between constraints
    The method builds on the standard ADMM framework assuming multiple penalties can address the issue.

pith-pipeline@v0.9.0 · 5717 in / 1072 out tokens · 56398 ms · 2026-05-23T02:04:52.202914+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

51 extracted references · 51 canonical work pages · 2 internal anchors

  1. [1]

    Proximal splitting methods in signal processing,

    P. L. Combettes and J.-C. Pesquet, “Proximal splitting methods in signal processing,” Fixed-point algorithms for inverse problems in science and engineering, pp. 185–212, 2011

    work page 2011

  2. [2]

    Sur l’approximation, par ´el´ements finis d’ordre un, et la r ´esolution, par p ´enalisation-dualit´e d’une classe de probl `emes de dirichlet non lin ´eaires,

    R. Glowinski and A. Marroco, “Sur l’approximation, par ´el´ements finis d’ordre un, et la r ´esolution, par p ´enalisation-dualit´e d’une classe de probl `emes de dirichlet non lin ´eaires,” ESAIM: Mathematical Mod- elling and Numerical Analysis - Mod ´elisation Math´ematique et Analyse Num´erique, vol. 9, no. R2, pp. 41–76, Aug. 1975

    work page 1975

  3. [3]

    A dual algorithm for the solution of nonlin- ear variational problems via finite element approximation,

    D. Gabay and B. Mercier, “A dual algorithm for the solution of nonlin- ear variational problems via finite element approximation,” Computers & Mathematics with Applications , vol. 2, no. 1, pp. 17–40, 1976. doi:10.1016/0898-1221(76)90003-1

    work page doi:10.1016/0898-1221(76)90003-1 1976

  4. [4]

    Distributed optimization and statistical learning via the alternating direction method of multipliers,

    S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Foundations and Trends in Machine Learning , vol. 3, no. 1, pp. 1–122, Jan. 2011. doi:10.1561/2200000016

    work page doi:10.1561/2200000016 2011

  5. [5]

    On the global and linear convergence of the gen- eralized alternating direction method of multipliers,

    W. Deng and W. Yin, “On the global and linear convergence of the gen- eralized alternating direction method of multipliers,”Journal of Scientific Computing, vol. 66, no. 3, pp. 889–916, May 2015. doi:10.1007/s10915- 015-0048-x

    work page doi:10.1007/s10915- 2015

  6. [6]

    Linear Convergence and Metric Selection for Douglas-Rachford Splitting and ADMM

    P. Giselsson and S. Boyd, “Linear convergence and metric selection for Douglas-Rachford splitting and ADMM,”arXiv, 2014, arXiv:1410.8479

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  7. [7]

    Global convergence of ADMM in nonconvex nonsmooth optimization,

    Y . Wang, W. Yin, and J. Zeng, “Global convergence of ADMM in nonconvex nonsmooth optimization,” Journal of Scientific Computing , vol. 78, pp. 29–63, 2019. doi:10.1007/s10915-018-0757-z 13

    work page doi:10.1007/s10915-018-0757-z 2019

  8. [8]

    ADMM for monotone operators: con- vergence analysis and rates,

    R. I. Bot , and E. R. Csetnek, “ADMM for monotone operators: con- vergence analysis and rates,” Advances in Computational Mathematics , vol. 45, pp. 327–359, 2019

    work page 2019

  9. [9]

    Consensus ADMM for inverse problems governed by multiple PDE models,

    L. Lozenski and U. Villa, “Consensus ADMM for inverse problems governed by multiple PDE models,” arXiv preprint arXiv:2104.13899 , 2021

    work page arXiv 2021

  10. [10]

    ADMM-type methods for generalized Nash equilibrium problems in Hilbert spaces,

    E. B ¨orgens and C. Kanzow, “ADMM-type methods for generalized Nash equilibrium problems in Hilbert spaces,” SIAM Journal on Optimization, vol. 31, no. 1, pp. 377–403, 2021

    work page 2021

  11. [11]

    Parallel multi-block ADMM with O(1/k) convergence,

    W. Deng, M.-J. Lai, Z. Peng, and W. Yin, “Parallel multi-block ADMM with O(1/k) convergence,” Journal of Scientific Computing , vol. 71, pp. 712–736, 2017

    work page 2017

  12. [12]

    On the sublinear convergence rate of multi-block ADMM,

    T.-Y . Lin, S.-Q. Ma, and S.-Z. Zhang, “On the sublinear convergence rate of multi-block ADMM,”Journal of the Operations Research Society of China, vol. 3, pp. 251–274, 2015

    work page 2015

  13. [13]

    A splitting-based iterative algorithm for accelerated statistical x-ray ct reconstruction,

    S. Ramani and J. A. Fessler, “A splitting-based iterative algorithm for accelerated statistical x-ray ct reconstruction,” IEEE Transac- tions on Medical Imaging , vol. 31, no. 3, pp. 677–688, 2012. doi:10.1109/TMI.2011.2175233

    work page doi:10.1109/tmi.2011.2175233 2012

  14. [14]

    First m87 event horizon telescope results. iv. imaging the central supermassive black hole,

    A. et. al., “First m87 event horizon telescope results. iv. imaging the central supermassive black hole,”The Astrophysical Journal Letters, vol. 875, no. 1, p. L4, 2019

    work page 2019

  15. [15]

    An admm algorithm for a class of total variation regularized estimation problems,

    B. Wahlberg, S. Boyd, M. Annergren, and Y . Wang, “An admm algorithm for a class of total variation regularized estimation problems,” IFAC Proceedings Volumes, vol. 45, no. 16, pp. 83–88, 2012

    work page 2012

  16. [16]

    Plug-and-play priors for model based reconstruction,

    S. V . Venkatakrishnan, C. A. Bouman, and B. Wohlberg, “Plug-and-play priors for model based reconstruction,” in 2013 IEEE global conference on signal and information processing . IEEE, 2013, pp. 945–948

    work page 2013

  17. [17]

    Asynchronous distributed ADMM for consensus optimization,

    R. Zhang and J. Kwok, “Asynchronous distributed ADMM for consensus optimization,” inInternational conference on machine learning. PMLR, 2014, pp. 1701–1709

    work page 2014

  18. [18]

    Convergence rate of distributed ADMM over networks,

    A. Makhdoumi and A. Ozdaglar, “Convergence rate of distributed ADMM over networks,” IEEE Transactions on Automatic Control , vol. 62, no. 10, pp. 5082–5095, 2017

    work page 2017

  19. [19]

    Asynchronous distributed ADMM for large-scale optimization - part I: Algorithm and convergence analysis,

    T.-H. Chang, M. Hong, W.-C. Liao, and X. Wang, “Asynchronous distributed ADMM for large-scale optimization - part I: Algorithm and convergence analysis,”IEEE Transactions on Signal Processing, vol. 64, no. 12, pp. 3118–3130, 2016

    work page 2016

  20. [20]

    D-ADMM: A communication-efficient distributed algorithm for separable optimiza- tion,

    J. F. Mota, J. M. Xavier, P. M. Aguiar, and M. P ¨uschel, “D-ADMM: A communication-efficient distributed algorithm for separable optimiza- tion,”IEEE Transactions on Signal processing, vol. 61, no. 10, pp. 2718– 2723, 2013

    work page 2013

  21. [21]

    Multi-agent distributed opti- mization via inexact consensus ADMM,

    T.-H. Chang, M. Hong, and X. Wang, “Multi-agent distributed opti- mization via inexact consensus ADMM,” IEEE Transactions on Signal Processing, vol. 63, no. 2, pp. 482–497, 2014

    work page 2014

  22. [22]

    On the linear convergence of the ADMM in decentralized consensus optimization,

    W. Shi, Q. Ling, K. Yuan, G. Wu, and W. Yin, “On the linear convergence of the ADMM in decentralized consensus optimization,” IEEE Transactions on Signal Processing, vol. 62, no. 7, pp. 1750–1761, 2014

    work page 2014

  23. [23]

    A general analysis of the convergence of ADMM,

    R. Nishihara, L. Lessard, B. Recht, A. Packard, and M. Jordan, “A general analysis of the convergence of ADMM,” in Proceedings of the 32nd International Conference on Machine Learning , ser. Proceedings of Machine Learning Research, vol. 37, Lille, France, 07–09 Jul 2015, pp. 343–352

    work page 2015

  24. [24]

    Diagonal scaling in douglas-rachford splitting and admm,

    P. Giselsson and S. Boyd, “Diagonal scaling in douglas-rachford splitting and admm,” in 53rd IEEE Conference on Decision and Control , Dec. 2014, pp. 5033–5039. doi:10.1109/cdc.2014.7040175

    work page doi:10.1109/cdc.2014.7040175 2014

  25. [25]

    Alternating direction method with self-adaptive penalty parameters for monotone variational inequalities,

    B. He, H. Yang, and S. Wang, “Alternating direction method with self-adaptive penalty parameters for monotone variational inequalities,” Journal of Optimization Theory and Applications, vol. 106, pp. 337–356,

    work page

  26. [26]

    doi:10.1023/a:1004603514434

    work page doi:10.1023/a:1004603514434

  27. [27]

    ADMM Penalty Parameter Selection by Residual Balancing

    B. Wohlberg, “ADMM penalty parameter selection by residual balanc- ing,” 2017,” arXiv:1704.06209

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  28. [28]

    Adaptive ADMM with spectral penalty parameter selection,

    Z. Xu, M. Figueiredo, and T. Goldstein, “Adaptive ADMM with spectral penalty parameter selection,” in Proceedings of the 20th International Conference on Artificial Intelligence and Statistics , ser. Proceedings of Machine Learning Research, vol. 54, Apr. 2017, pp. 718–727

    work page 2017

  29. [29]

    Non-stationary Douglas-Rachford and alternating direction method of multipliers: adaptive step-sizes and convergence,

    D. A. Lorenz and Q. Tran-Dinh, “Non-stationary Douglas-Rachford and alternating direction method of multipliers: adaptive step-sizes and convergence,” Computational Optimization and Applications , vol. 74, no. 1, pp. 67–92, May 2019. doi:10.1007/s10589-019-00106-9

    work page doi:10.1007/s10589-019-00106-9 2019

  30. [30]

    Robust and simple ADMM penalty parameter selection,

    M. T. McCann and B. Wohlberg, “Robust and simple ADMM penalty parameter selection,” IEEE Open Journal of Signal Processing , vol. 5, pp. 402–420, Jan. 2024. doi:10.1109/OJSP.2023.3349115

    work page doi:10.1109/ojsp.2023.3349115 2024

  31. [31]

    Alternating optimization: Constrained problems, adversarial net- works, and robust models,

    Z. Xu, “Alternating optimization: Constrained problems, adversarial net- works, and robust models,” Ph.D. dissertation, University of Maryland, College Park, 2019

    work page 2019

  32. [32]

    On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone oper- ators,

    J. Eckstein and D. P. Bertsekas, “On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone oper- ators,” Mathematical Programming, vol. 55, no. 1-3, pp. 293–318, Apr

    work page

  33. [33]

    doi:10.1007/bf01581204

    work page doi:10.1007/bf01581204

  34. [34]

    Self equivalence of the alternating direction method of multipliers,

    M. Yan and W. Yin, “Self equivalence of the alternating direction method of multipliers,” Splitting Methods in Communication, Imaging, Science, and Engineering, pp. 165–194, 2016. doi:10.1007/978-3-319-41589-5 5

    work page doi:10.1007/978-3-319-41589-5 2016

  35. [35]

    Nesterov, Introductory Lectures on Convex Optimization

    Y . Nesterov, Introductory Lectures on Convex Optimization . Springer US, 2004

    work page 2004

  36. [36]

    Sequential quadratic programming,

    P. T. Boggs and J. W. Tolle, “Sequential quadratic programming,” Acta numerica, vol. 4, pp. 1–51, 1995

    work page 1995

  37. [37]

    Convex quadratic approximation,

    J. B. Rosen and R. F. Marcia, “Convex quadratic approximation,” Computational Optimization and Applications , vol. 28, pp. 173–184, 2004

    work page 2004

  38. [38]

    A modified local quadratic approxi- mation algorithm for penalized optimization problems,

    S. Lee, S. Kwon, and Y . Kim, “A modified local quadratic approxi- mation algorithm for penalized optimization problems,” Computational Statistics & Data Analysis , vol. 94, pp. 275–286, 2016

    work page 2016

  39. [39]

    Global and fine approximation of convex functions,

    D. Azagra, “Global and fine approximation of convex functions,” Pro- ceedings of the London Mathematical Society , vol. 107, no. 4, pp. 799– 824, 2013

    work page 2013

  40. [40]

    Quasi-Newton methods, motivation and theory,

    J. E. Dennis, Jr and J. J. Mor ´e, “Quasi-Newton methods, motivation and theory,” SIAM review, vol. 19, no. 1, pp. 46–89, 1977

    work page 1977

  41. [41]

    Nonsmooth optimization via quasi- Newton methods,

    A. S. Lewis and M. L. Overton, “Nonsmooth optimization via quasi- Newton methods,” Mathematical Programming, vol. 141, pp. 135–163, 2013

    work page 2013

  42. [42]

    Subspace sys- tem identification via weighted nuclear norm optimization,

    A. Hansson, Z. Liu, and L. Vandenberghe, “Subspace sys- tem identification via weighted nuclear norm optimization,” in IEEE Conference on Decision and Control (CDC) , Dec. 2012. doi:10.1109/CDC.2012.6426980

    work page doi:10.1109/cdc.2012.6426980 2012

  43. [43]

    Nuclear norm system iden- tification with missing inputs and outputs,

    Z. Liu, A. Hansson, and L. Vandenberghe, “Nuclear norm system iden- tification with missing inputs and outputs,” Systems & Control Letters , vol. 62, no. 8, pp. 605 – 612, 2013. doi:10.1016/j.sysconle.2013.04.005

    work page doi:10.1016/j.sysconle.2013.04.005 2013

  44. [44]

    Fantope projection and selection: A near-optimal convex relaxation of sparse PCA,

    V . Q. Vu, J. Cho, J. Lei, and K. Rohe, “Fantope projection and selection: A near-optimal convex relaxation of sparse PCA,” inAdvances in Neural Information Processing Systems 26 , 2013, pp. 2670–2678

    work page 2013

  45. [45]

    Collaborative sparse regression for hyperspectral unmixing,

    M.-D. Iordache, J. M. Bioucas-Dias, and A. Plaza, “Collaborative sparse regression for hyperspectral unmixing,” IEEE Transactions on Geoscience and Remote Sensing, vol. 52, no. 1, pp. 341–354, Jan. 2014. doi:10.1109/TGRS.2013.2240001

    work page doi:10.1109/tgrs.2013.2240001 2014

  46. [46]

    Phase retrieval of sparse signals using optimization transfer and ADMM,

    D. S. Weller, A. Pnueli, O. Radzyner, G. Divon, Y . C. Eldar, and J. A. Fessler, “Phase retrieval of sparse signals using optimization transfer and ADMM,” in IEEE International Conference on Image Processing (ICIP), Oct. 2014, pp. 1342–1346

    work page 2014

  47. [47]

    Efficient convolutional sparse coding,

    B. Wohlberg, “Efficient convolutional sparse coding,” in IEEE International Conference on Acoustics, Speech, and Signal Pro- cessing (ICASSP) , Florence, Italy, May 2014, pp. 7173–7177. doi:10.1109/ICASSP.2014.6854992

    work page doi:10.1109/icassp.2014.6854992 2014

  48. [48]

    Adap- tive relaxed ADMM: Convergence theory and practical implementation,

    Z. Xu, M. A. T. Figueiredo, X. Yuan, C. Studer, and T. Goldstein, “Adap- tive relaxed ADMM: Convergence theory and practical implementation,” in IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Jul. 2017. doi:10.1109/cvpr.2017.765

    work page doi:10.1109/cvpr.2017.765 2017

  49. [49]

    Two-point step size gradient methods,

    J. Barzilai and J. M. Borwein, “Two-point step size gradient methods,” IMA Journal of Numerical Analysis , vol. 8, no. 1, pp. 141–148, 1988

    work page 1988

  50. [50]

    Nonlinear total variation based noise removal algorithms,

    L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D: nonlinear phenomena , vol. 60, no. 1-4, pp. 259–268, 1992

    work page 1992

  51. [51]

    Scientific computational imaging code (SCICO),

    T. Balke, F. Davis, C. Garcia-Cardona, S. Majee, M. McCann, L. Pfister, and B. Wohlberg, “Scientific computational imaging code (SCICO),” Journal of Open Source Software , vol. 7, no. 78, p. 4722, 2022. doi:10.21105/joss.04722

    work page doi:10.21105/joss.04722 2022