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REVIEW 3 major objections 4 minor 78 references

A Semismooth Newton Augmented Lagrangian Method for Sparse Spectral Risk Optimization

T0 review · 3 major / 4 minor · reviewed 2026-07-08 · grok-4.5

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Pith's one-line read A relative inexact proximal ALM with semismooth Newton solves sparse spectral-risk optimization faster than ADMM

desk verdict Solid specialized ALM–SSN solver for sparse spectral-risk ERM; real within-subfield computational progress, not a foundational advance. read the letter →

arxiv 2607.05991 v1 pith:OMRUHIGX submitted 2026-07-07 math.OC

Rufeng Xiao , Rujun Jiang , Xudong Li , Defeng Sun This is my paper · ORCID
classification math.OC MSC 90C2590C0690C3065K05
keywords spectralriskmeasureaugmentedLagrangianmethodsemismoothNewtonMoreauenvelopepool-adjacent-violatorssparseoptimizationrelativeinexactnessempiricalminimization
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

Average-loss training can hide rare but severe failures. Spectral risk measures fix that by weighting ordered losses, yet the sorting step makes the objective nonsmooth and nonseparable, so standard solvers struggle. This paper builds a relative inexact proximal augmented Lagrangian method whose subproblems are solved by a semismooth Newton iteration. A dual reformulation and Moreau-envelope structure collapse each subproblem onto a structured dual variable whose generalized Jacobian can be evaluated by a tailored pool-adjacent-violators algorithm. Under the paper's standing assumptions the method converges, and on synthetic and real instances it matches the stationarity and sparsity quality of an ADMM baseline while finishing in less wall-clock time. The practical payoff is a reliable first-order-to-Newton pipeline for high-stakes sparse learning that respects worst-case tails without sacrificing speed.

What carries the argument

The dual residual map of the Moreau-envelope reformulation of the proximal ALM subproblem; its generalized Jacobian is assembled by a pool-adjacent-violators algorithm that enforces the ordered spectral weights, turning each Newton step into a structured linear solve.

What would settle it

On a suite of sparse spectral-risk instances with known spectral weights, run the method with the stated relative inexactness tolerances and compare final stationarity residual, support cardinality, and wall-clock time against ADMM; if residuals are systematically worse or times are not lower while solutions remain sparse, the central claim fails.

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Extended reading notes

Core claim

A relative inexact proximal augmented Lagrangian method, with subproblems solved by a semismooth Newton solver that exploits a dual reformulation and Moreau-envelope structure, solves sparse spectral-risk optimization problems to stationarity and sparsity quality comparable to ADMM while attaining lower running times. Explicit generalized Jacobians of the dual residual map are obtained via a tailored pool-adjacent-violators algorithm, making each Newton step inexpensive.

Load-bearing premise

That the relative inexactness criteria together with the dual and Moreau reductions still guarantee global convergence of the proximal ALM under the paper's assumptions on spectral weights, loss, and regularizer, so that practical Newton termination yields the claimed stationarity and sparsity.

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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The manuscript develops a relative inexact proximal augmented Lagrangian method whose subproblems are solved by a semismooth Newton (SSN) solver for sparse spectral-risk-measure (SRM) optimization. By dual reformulation and Moreau-envelope structure the ALM subproblems are reduced to structured dual-variable problems; explicit generalized Jacobians of the dual residual maps are characterized and evaluated by a tailored pool-adjacent-violators algorithm. Global convergence of the outer relative-inexact proximal ALM is claimed under standing assumptions on spectral weights, loss, and regularizer. Numerical experiments on synthetic and real instances report lower runtimes than an ADMM baseline at comparable stationarity residuals and solution sparsity.

Significance. SRM-based empirical risk minimization is a practically relevant nonsmooth, nonseparable problem class in risk-sensitive learning. A dual/Moreau reduction that yields tractable SSN subproblems, together with explicit generalized-Jacobian formulae and a specialized PAVA evaluation, constitutes a concrete algorithmic contribution. If the relative-inexactness theory carries through the reduction and the reported speed-ups hold under fair baselines, the method would be a useful addition to the toolkit for spectral-risk optimization. The paper supplies comparative numerics and structured Jacobian characterizations rather than free-parameter fitting; those are genuine strengths.

major comments (3)
  1. [Theory: relative-inexact PALM and dual/Moreau subproblem reduction] The central theory-to-practice link is the inheritance of the relative inexact proximal-ALM convergence guarantees after the dual/Moreau reduction. The relative residual criteria appear to be stated for the original primal ALM; after dual reformulation the controlled residual and dual-variable geometry change. The manuscript needs an explicit argument (or equivalence lemma) showing that the concrete SSN termination used in the dual subproblems still satisfies the relative inexactness conditions that underwrite global convergence and the claimed stationarity quality versus ADMM. Without that bridge, the tabulated residuals and sparsity are only empirical observations, not consequences of the theory.
  2. [Standing assumptions; Jacobian characterizations] Standing assumptions on the spectral weight vector (monotonicity/ordering) and on the regularizer are invoked to justify both the dual reduction and the generalized-Jacobian formulae. It should be stated precisely which of these assumptions are used for (i) well-posedness of the dual residual map, (ii) the PAVA-based Jacobian evaluation, and (iii) the relative-inexact outer convergence. If any assumption is stronger than what is needed for the numerics (e.g., strictly decreasing weights), that gap should be flagged so that the scope of the guarantees is clear.
  3. [Numerical experiments / ADMM baseline] The ADMM baseline is the sole external comparator. For the claim of systematically lower running times at comparable residuals to be load-bearing, the ADMM implementation must be described at the same level of detail as the proposed method (penalty update schedule, termination tolerances, warm-start policy, and whether the same dual/Moreau structure is exploited). Otherwise the runtime advantage may reflect implementation disparity rather than algorithmic superiority. A short fairness paragraph or an additional first-order baseline would strengthen the numerical claim.
minor comments (4)
  1. [Notation / preliminaries] Notation for the spectral weight vector and the associated sorting operator should be fixed once and used consistently; occasional switches between w and lambda (or similar) make the dual residual maps harder to track.
  2. [Jacobian evaluation / PAVA] The tailored PAVA procedure is central to Jacobian evaluation; a short complexity statement (or reference to the classical O(n) bound under the paper's ordering assumptions) would help readers assess per-iteration cost.
  3. [Numerical results tables/figures] Figures and tables reporting residuals and sparsity should state the precise stationarity measure (primal residual, dual residual, or KKT residual) and the sparsity threshold used, so that 'comparable' is reproducible.
  4. [Algorithm parameters] A brief remark on how the free ALM penalty and relative-tolerance sequences are chosen in practice (fixed schedule vs adaptive) would aid reproducibility.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for a careful and constructive report. The three major comments correctly identify the places where the theory-to-algorithm link, the scope of standing assumptions, and the fairness of the ADMM comparison need to be made fully explicit. We will revise the manuscript accordingly: (i) add an equivalence argument relating the dual SSN residual to the relative inexactness criterion of the outer proximal ALM; (ii) itemize precisely which assumptions are used for dual well-posedness, PAVA Jacobian evaluation, and outer convergence, and flag any gap relative to the numerical setting; (iii) expand the ADMM baseline description to the same level of implementation detail as the proposed method and add a short fairness paragraph. None of these revisions alter the algorithmic design or the reported numerical conclusions; they strengthen the supporting arguments.

read point-by-point responses

  1. Referee: The central theory-to-practice link is the inheritance of the relative inexact proximal-ALM convergence guarantees after the dual/Moreau reduction. The relative residual criteria appear to be stated for the original primal ALM; after dual reformulation the controlled residual and dual-variable geometry change. The manuscript needs an explicit argument (or equivalence lemma) showing that the concrete SSN termination used in the dual subproblems still satisfies the relative inexactness conditions that underwrite global convergence and the claimed stationarity quality versus ADMM. Without that bridge, the tabulated residuals and sparsity are only empirical observations, not consequences of the theory.

    Authors: We agree that an explicit bridge is required and that the current draft does not state it with sufficient clarity. The outer relative-inexact proximal ALM is formulated in the primal variables; after the dual/Moreau reduction the subproblem residual is measured on a dual residual map whose geometry differs from the primal residual. Because the Moreau envelope of the dual spectral-risk term is continuously differentiable with Lipschitz gradient, and because the primal–dual pair is linked by the proximal mapping of that envelope, the dual residual norm controls the primal residual that appears in the relative inexactness criterion. We will insert a short equivalence lemma (after the dual reduction) that makes this control quantitative: any dual residual small enough relative to the current dual step size implies the relative residual bound required by the outer convergence theorem. The concrete SSN termination used in the implementation will then be shown to satisfy that dual residual bound. With the lemma in place, global convergence of the outer iteration and the stationarity quality of the returned points become consequences of the theory rather than purely empirical observations. The tabulated residuals will be re-interpreted in light of this guarantee. revision: yes

  2. Referee: Standing assumptions on the spectral weight vector (monotonicity/ordering) and on the regularizer are invoked to justify both the dual reduction and the generalized-Jacobian formulae. It should be stated precisely which of these assumptions are used for (i) well-posedness of the dual residual map, (ii) the PAVA-based Jacobian evaluation, and (iii) the relative-inexact outer convergence. If any assumption is stronger than what is needed for the numerics (e.g., strictly decreasing weights), that gap should be flagged so that the scope of the guarantees is clear.

    Authors: The referee’s request for a precise inventory is well taken. We will add a short subsection (or a clearly labeled paragraph at the beginning of the theory section) that lists each standing assumption and tags it with the roles (i)–(iii). Concretely: monotonic nonincreasing spectral weights are used for (i) dual residual well-posedness (via the dual of the ordered weighted average) and for (ii) correctness of the pool-adjacent-violators evaluation of the generalized Jacobian; they are not required for (iii) outer relative-inexact convergence once the dual residual is well-defined. Strict decrease is used only to guarantee uniqueness of certain active-set partitions inside the PAVA routine and is therefore stronger than needed for the outer theory; the numerical instances satisfy it, but we will explicitly flag that the outer guarantees remain valid under mere monotonicity. The regularizer assumptions (convexity, properness, and the existence of a computable proximal mapping) are used for (i) and (iii) but not for the PAVA formulae themselves. This itemization will make the scope of every claim transparent and will prevent readers from over-interpreting the strength of the hypotheses. revision: yes

  3. Referee: The ADMM baseline is the sole external comparator. For the claim of systematically lower running times at comparable residuals to be load-bearing, the ADMM implementation must be described at the same level of detail as the proposed method (penalty update schedule, termination tolerances, warm-start policy, and whether the same dual/Moreau structure is exploited). Otherwise the runtime advantage may reflect implementation disparity rather than algorithmic superiority. A short fairness paragraph or an additional first-order baseline would strengthen the numerical claim.

    Authors: We accept the criticism. The current draft describes the ADMM baseline only at a high level, which leaves open the possibility that the observed speed-ups are partly implementation artifacts. In the revision we will expand the experimental section with a dedicated fairness paragraph that records, for ADMM: the precise penalty-parameter update schedule (including the initial value and the multiplicative factor), the absolute and relative termination tolerances used for both the outer ADMM iteration and any inner linear-system solves, the warm-start policy for the dual and slack variables, and whether the same dual/Moreau reduction of the spectral-risk term is exploited inside the ADMM subproblems. Both codes will be confirmed to share the same linear-algebra backend and the same sparsity-aware data structures. We will also note that the dual residual maps used for termination are comparable, so that the stationarity residuals reported in the tables are measured on an equal footing. If space permits we will add a simple first-order proximal-gradient baseline on the dual formulation as a further sanity check; otherwise the expanded ADMM description alone will make the runtime comparison load-bearing. revision: yes

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: methods paper with independent algorithmic derivation, dual/Moreau reductions, and external ADMM runtime/stationarity benchmarks.

full rationale

This is a computational optimization methods paper. The claimed contributions are (i) a relative-inexact proximal ALM with SSN subproblem solver for sparse spectral-risk minimization, (ii) dual reformulation plus Moreau-envelope reductions that simplify the subproblems, (iii) explicit generalized Jacobians evaluated via a tailored PAVA, and (iv) numerical comparisons showing lower runtimes than an ADMM baseline at comparable stationarity residuals and sparsity. None of these steps reduce by construction to their inputs: the dual/Moreau identities are standard convex-analysis equivalences applied to a new problem class; the generalized Jacobians and PAVA specialization are derived characterizations, not fitted parameters renamed as predictions; and the empirical claims are measured against an external baseline on synthetic and real instances, not against quantities used to define the method. Self-citations (if any) to prior ALM/SSN work by overlapping authors are normal scaffolding for algorithmic building blocks and are not load-bearing uniqueness theorems that force the present claims. The skeptic concern about whether practical SSN termination rigorously inherits the relative-inexact PALM global-convergence guarantees is a correctness/theory-to-practice risk, not circularity. Score 1 reflects only the ordinary residual risk of self-citation typical of optimization methods papers; the central derivation chain is self-contained against external benchmarks.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

Central claim rests on standard convex/nonsmooth optimization assumptions (spectral risk weights, loss and regularizer structure allowing dual/Moreau reduction and SSN), the relative inexact proximal ALM framework, and correctness of the generalized Jacobian characterizations with PAVA evaluation. No new physical entities. Free parameters are algorithmic hyperparameters (penalty, inexactness tolerances, Newton settings), not fitted scientific constants that define the result.

free parameters (3)
  • ALM penalty / proximal parameters
    Standard algorithmic hyperparameters of proximal ALM; affect speed and practical convergence but are not scientific constants defining the risk measure.
  • Relative inexactness tolerances for subproblem solves
    Control how accurately each SSN subproblem is solved; load-bearing for the relative inexact convergence theory in practice.
  • Spectral risk weight vector / spectrum
    User-chosen SRM weights (including CVaR-like spectra); part of the problem definition rather than fitted inside the paper's central algorithmic claim.
assumptions (4)
  • domain assumption Spectral risk measure is a weighted sum of ordered losses with admissible nondecreasing spectrum weights.
    Standard SRM definition; induces sorting nonsmoothness that the algorithm targets.
  • domain assumption Relative inexact proximal augmented Lagrangian method converges under the paper's standing convexity/qualification assumptions when subproblems meet relative error criteria.
    Load-bearing algorithmic theory for the outer method.
  • standard math Moreau envelope and dual reformulation yield equivalent structured dual subproblems amenable to semismooth Newton.
    Uses classical convex analysis tools; correctness of the reduction is central.
  • ad hoc to paper Generalized Jacobians of the dual residual maps admit the stated characterizations and can be evaluated via a tailored pool-adjacent-violators algorithm.
    Paper-specific technical contribution; if wrong, SSN efficiency and correctness claims fail.

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Pith. "Pith review of A Semismooth Newton Augmented Lagrangian Method for Sparse Spectral Risk Optimization." pith.science (2026). https://pith.science/paper/OMRUHIGX

@misc{pith2026260705991,
  author       = {Pith},
  title        = {Pith review of: A Semismooth Newton Augmented Lagrangian Method for Sparse Spectral Risk Optimization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OMRUHIGX}},
  note         = {Machine review of arXiv:2607.05991}
}
read the original abstract

Empirical risk minimization is a standard and effective paradigm for learning predictive models by minimizing average loss. In high-stakes decision-making, however, an average-loss criterion may underrepresent rare but severe losses. Spectral risk measures (SRMs) provide a principled framework by incorporating weighted order statistics of losses, but the induced nonsmoothness and nonseparability from sorting make the resulting optimization problems challenging. We propose a relative inexact proximal augmented Lagrangian method with a semismooth Newton subproblem solver for solving SRM-based optimization problems. Exploiting a dual reformulation and properties of the Moreau envelope, we reduce the subproblems to structured dual-variable formulations, significantly simplifying computation. We provide explicit generalized Jacobian characterizations and tailor the pool adjacent violators algorithm for their efficient evaluation. Numerical results on synthetic and real-data instances show that the proposed method attains lower running times than the tested ADMM baseline while producing comparable stationarity residuals and sparse solutions.

Figures

Figures reproduced from arXiv: 2607.05991 by the authors.

Figure 1
Figure 1. Average computation time and average relative objective difference across problem [PITH_FULL_IMAGE:figures/full_fig_p023_1.png] view at source ↗

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Works this paper leans on

78 extracted references · 78 canonical work pages

  1. [1]

    Some continuity properties of polyhedral multifunctions , author=. Math. Programming Stud. , volume=. 1981 , publisher=

    work page 1981

  2. [2]

    Implicit function theorems for mathematical programming and for systems of inequalities , author=. Math. Program. , volume=. 1974 , publisher=

    work page 1974

  3. [3]

    2021 , publisher=

    Modern nonconvex nondifferentiable optimization , author=. 2021 , publisher=

    work page 2021

  4. [4]

    SIAM Rev

    On deriving the inverse of a sum of matrices , author=. SIAM Rev. , volume=. 1981 , publisher=

    work page 1981

  5. [5]

    1997 , publisher=

    Han, Jiye and Sun, Defeng , journal=. 1997 , publisher=

    work page 1997

  6. [6]

    2017 , publisher=

    First-order methods in optimization , author=. 2017 , publisher=

    work page 2017

  7. [7]

    2009 , publisher=

    Variational analysis , author=. 2009 , publisher=

    work page 2009

  8. [8]

    Qi, Liqun and Sun, Jie , title =. Math. Program. , month = jan, pages =. 1993 , issue_date =

    work page 1993

Show all 78 references
  1. [9]

    , title =

    Clarke, Frank H. , title =

  2. [10]

    Inverse and implicit function theorems for

    Gowda, M Seetharama , journal=. Inverse and implicit function theorems for. 2004 , publisher=

    2004

  3. [11]

    On efficiently solving the subproblems of a level-set method for fused lasso problems , author=. SIAM J. Optim. , volume=. 2018 , publisher=

    2018

  4. [12]

    A Riemannian proximal Newton method , author=. SIAM J. Optim. , volume=. 2024 , publisher=

    2024

  5. [13]

    2003 , publisher=

    Finite-dimensional variational inequalities and complementarity problems , author=. 2003 , publisher=

    2003

  6. [14]

    Risk-sensitive and robust decision-making: a

    Chow, Yinlam and Tamar, Aviv and Mannor, Shie and Pavone, Marco , journal=. Risk-sensitive and robust decision-making: a

  7. [15]

    International conference on machine learning , pages=

    Fairness risk measures , author=. International conference on machine learning , pages=. 2019 , organization=

    2019

  8. [16]

    2011 IEEE 11th international conference on data mining workshops , pages=

    Fairness-aware learning through regularization approach , author=. 2011 IEEE 11th international conference on data mining workshops , pages=. 2011 , organization=

    2011

  9. [17]

    Proceedings of the 3rd Innovations in Theoretical Computer Science Conference , pages=

    Fairness through awareness , author=. Proceedings of the 3rd Innovations in Theoretical Computer Science Conference , pages=

  10. [18]

    Portfolio optimization with spectral measures of risk , author=

  11. [19]

    Spectral measures of risk: A coherent representation of subjective risk aversion , author=. J. Bank. Finance , volume=. 2002 , publisher=

    2002

  12. [20]

    Robust unsupervised learning via

    Maurer, Andreas and Parletta, Daniela Angela and Paudice, Andrea and Pontil, Massimiliano , booktitle=. Robust unsupervised learning via. 2021 , organization=

    2021

  13. [21]

    2013 , publisher=

    The nature of statistical learning theory , author=. 2013 , publisher=

    2013

  14. [22]

    International Conference on Machine Learning , pages=

    Minimizing the maximal loss: How and why , author=. International Conference on Machine Learning , pages=. 2016 , organization=

    2016

  15. [23]

    Learning with average top-k loss , author=. Adv. Neural Inf. Process. Syst. , volume=

  16. [24]

    Set-Valued Var

    Superquantiles at work: Machine learning applications and efficient subgradient computation , author=. Set-Valued Var. Anal. , volume=. 2021 , publisher=

    2021

  17. [25]

    Extremiles: A new perspective on asymmetric least squares , author=. J. Amer. Statist. Assoc. , volume=. 2019 , publisher=

    2019

  18. [26]

    Extreme spectral risk measures: an application to futures clearinghouse margin requirements , author=. J. Bank. Finance , volume=. 2006 , publisher=

    2006

  19. [27]

    International Conference on Medical Image Computing and Computer-Assisted Intervention , pages=

    Clinical-realistic annotation for histopathology images with probabilistic semi-supervision: a worst-case study , author=. International Conference on Medical Image Computing and Computer-Assisted Intervention , pages=. 2022 , organization=

    2022

  20. [28]

    2020 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) , pages=

    Risk-aware planning and assignment for ground vehicles using uncertain perception from aerial vehicles , author=. 2020 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) , pages=. 2020 , organization=

    2020

  21. [29]

    Distributionally robust learning , author=. Found. Trends Optim. , volume=. 2020 , publisher=

    2020

  22. [30]

    Optimization of conditional value-at-risk , author=. J. Risk , volume=

  23. [31]

    Adaptive sampling for stochastic risk-averse learning , author=. Adv. Neural Inf. Process. Syst. , volume=

  24. [32]

    INFORMS J

    Convex and nonconvex risk-based linear regression at scale , author=. INFORMS J. Comput. , volume=. 2023 , publisher=

    2023

  25. [33]

    Maximum properties and inequalities for the eigenvalues of completely continuous operators , author=. Proc. Natl. Acad. Sci. USA , volume=

  26. [34]

    INFORMS J

    Decision making under cumulative prospect theory: An alternating direction method of multipliers , author=. INFORMS J. Comput. , volume=. 2025 , publisher=

    2025

  27. [35]

    A unified framework for rank-based loss minimization , author=. Adv. Neural Inf. Process. Syst. , volume=

  28. [36]

    Minimizing separable convex functions subject to simple chain constraints , author=. SIAM J. Optim. , volume=. 2000 , publisher=

    2000

  29. [37]

    International conference on artificial intelligence and statistics , pages=

    Spectral risk-based learning using unbounded losses , author=. International conference on artificial intelligence and statistics , pages=. 2022 , organization=

    2022

  30. [38]

    International Conference on Artificial Intelligence and Statistics , pages=

    Ordered sgd: A new stochastic optimization framework for empirical risk minimization , author=. International Conference on Artificial Intelligence and Statistics , pages=. 2020 , organization=

    2020

  31. [39]

    Large-scale methods for distributionally robust optimization , author=. Adv. Neural Inf. Process. Syst. , volume=

  32. [40]

    International Conference on Artificial Intelligence and Statistics , pages=

    Stochastic optimization for spectral risk measures , author=. International Conference on Artificial Intelligence and Statistics , pages=. 2023 , organization=

    2023

  33. [41]

    Accelerating stochastic gradient descent using predictive variance reduction , author=. Adv. Neural Inf. Process. Syst. , volume=

  34. [42]

    The Twelfth International Conference on Learning Representations , year=

    Distributionally Robust Optimization with Bias and Variance Reduction , author=. The Twelfth International Conference on Learning Representations , year=

  35. [43]

    DRAGO: Primal-Dual Coupled Variance Reduction for Faster Distributionally Robust Optimization , author=. Adv. Neural Inf. Process. Syst. , volume=

  36. [44]

    Yuze Ge and Rujun Jiang , booktitle=

  37. [45]

    A stochastic variance-reduced accelerated primal-dual method for finite-sum saddle-point problems , author=. Comput. Optim. Appl. , volume=. 2023 , publisher=

    2023

  38. [46]

    Stochastic Primal-Dual Algorithms with Faster Convergence than

    Yan, Yan and Xu, Yi and Lin, Qihang and Zhang, Lijun and Yang, Tianbao , journal=. Stochastic Primal-Dual Algorithms with Faster Convergence than

  39. [47]

    A primal-dual algorithm with line search for general convex-concave saddle point problems , author=. SIAM J. Optim. , volume=. 2021 , publisher=

    2021

  40. [48]

    International conference on machine learning , pages=

    Accelerated primal-dual methods for convex-strongly-concave saddle point problems , author=. International conference on machine learning , pages=. 2023 , organization=

    2023

  41. [49]

    Efficient algorithms for smooth minimax optimization , author=. Adv. Neural Inf. Process. Syst. , volume=

  42. [50]

    Zhu, Jiayi and Liang, Ling and Yang, Lei and Toh, Kim-Chuan , howpublished=

  43. [51]

    Jiayi Zhu and Ling Liang and Lei Yang and Kim-Chuan Toh , howpublished=

  44. [52]

    Augmented Lagrangians and applications of the proximal point algorithm in convex programming , author=. Math. Oper. Res. , volume=. 1976 , publisher=

    1976

  45. [53]

    On the efficient computation of a generalized

    Li, Xudong and Sun, Defeng and Toh, Kim-Chuan , journal =. On the efficient computation of a generalized. 2020 , pages =

    2020

  46. [54]

    Augmented Lagrangian methods for convex matrix optimization problems , author=. J. Oper. Res. Soc. China , volume=. 2022 , publisher=

    2022

  47. [55]

    2003 , publisher=

    Convex analysis and optimization , author=. 2003 , publisher=

    2003

  48. [56]

    An asymptotically superlinearly convergent semismooth

    Li, Xudong and Sun, Defeng and Toh, Kim-Chuan , journal=. An asymptotically superlinearly convergent semismooth. 2020 , publisher=

    2020

  49. [57]

    A corrected inexact proximal augmented

    Yang, Lei and Liang, Ling and Chu, Hong TM and Toh, Kim-Chuan , journal=. A corrected inexact proximal augmented. 2024 , publisher=

    2024

  50. [58]

    Asia-Pac

    The linear and asymptotically superlinear convergence rates of the augmented Lagrangian method with a practical relative error criterion , author=. Asia-Pac. J. Oper. Res. , volume=. 2020 , publisher=

    2020

  51. [59]

    arXiv preprint arXiv:2510.19614 , year=

    An Alternating Direction Method of Multipliers for Utility-based Shortfall Risk Portfolio Optimization , author=. arXiv preprint arXiv:2510.19614 , year=

  52. [60]

    2004 , publisher=

    Convex optimization , author=. 2004 , publisher=

    2004

  53. [61]

    2006 , publisher=

    Numerical optimization , author=. 2006 , publisher=

    2006

  54. [62]

    Local uniqueness and convergence of iterative methods for nonsmooth variational inequalities , author=. J. Math. Anal. Appl. , volume=. 1995 , publisher=

    1995

  55. [63]

    Generalized second derivatives of convex functions and saddle functions , author=. Trans. Amer. Math. Soc. , volume=

  56. [64]

    Examples Counterexamples , volume=

    Maximal monotone operators with non-maximal graphical limit , author=. Examples Counterexamples , volume=. 2022 , publisher=

    2022

  57. [65]

    Zhao, Xin-Yuan and Sun, Defeng and Toh, Kim-Chuan , journal=. A. 2010 , publisher=

    2010

  58. [66]

    Approximations and generalized

    Klatte, Diethard and Kummer, Bernd , journal=. Approximations and generalized. 2018 , publisher=

    2018

  59. [67]

    Sensitivity analysis of optimization problems under second order regular constraints , author=. Math. Oper. Res. , volume=. 1998 , publisher=

    1998

  60. [68]

    Liu, Yong-Jin and Zhou, Weimi , howpublished=. Dual

  61. [69]

    A method for large-scale

    Koh, Kwangmoo and Kim, Seung-Jean and Boyd, Stephen , booktitle=. A method for large-scale

  62. [70]

    and Chandelier, E

    Feltes, B.C. and Chandelier, E. B. and Grisci, B. I. and Dorn, M. , year = 2019, journal =

    2019

  63. [71]

    Tyrrell , title =

    Rockafellar, R. Tyrrell , title =. 1970 , publisher =

    1970

  64. [72]

    Yuan et al

    Adaptive sieving: a dimension reduction technique for sparse optimization problems: Y. Yuan et al. , author=. Math. Program. Comput. , volume=. 2025 , publisher=

    2025

  65. [73]

    Moore, Robert and DeNero, John , booktitle=

  66. [74]

    PLoS One , volume=

    Mutual information between discrete and continuous data sets , author=. PLoS One , volume=. 2014 , publisher=

    2014

  67. [75]

    Estimating mutual information , author=. Phys. Rev. E , volume=. 2004 , publisher=

    2004

  68. [76]

    2020 , eprint=

    Adaptive Sieving with PPDNA: Generating Solution Paths of Exclusive Lasso Models , author=. 2020 , eprint=

    2020

  69. [77]

    1952 , publisher =

    Inequalities , author =. 1952 , publisher =

    1952

  70. [78]

    INFORMS J

    Portfolio optimization with nonparametric value at risk: A block coordinate descent method , author=. INFORMS J. Comput. , volume=. 2018 , publisher=

    2018

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