mlsanityData to 7 Oct 2026

Convergence Analysis of Alternating Direction Method of Multipliers for a Family of Nonconvex Problems

arXiv 1410.1390 ↗First posted 6 Oct 2014math.OCcited by 64

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Abstract (arXiv)

The alternating direction method of multipliers (ADMM) is widely used to solve large-scale linearly constrained optimization problems, convex or nonconvex, in many engineering fields. However there is a general lack of theoretical understanding of the algorithm when the objective function is nonconvex. In this paper we analyze the convergence of the ADMM for solving certain nonconvex consensus and sharing problems, and show that the classical ADMM converges to the set of stationary solutions, provided that the penalty parameter in the augmented Lagrangian is chosen to be sufficiently large. For the sharing problems, we show that the ADMM is convergent regardless of the number of variable blocks. Our analysis does not impose any assumptions on the iterates generated by the algorithm, and is broadly applicable to many ADMM variants involving proximal update rules and various flexible block selection rules.

Key facts. Cited by 64 papers in the map (data through 7 Oct 2026). First posted on arXiv 6 Oct 2014.

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Citing papers per year: 2021: 6; 2022: 13; 2023: 14; 2024: 7; 2025: 13; 2026: 11.

64 papers.

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  1. 2023Learning to Warm-Start Fixed-Point Optimization Algorithmscited by 32
  2. 2021Towards Federated Bayesian Network Structure Learning with Continuous Optimizationcited by 31
  3. 2022FedADMM: A Robust Federated Deep Learning Framework with Adaptivity to System Heterogeneitycited by 25
  4. 2022Robust and Explainable Autoencoders for Unsupervised Time Series Outlier Detection---Extended Versioncited by 19
  5. 2024ALPS: Improved Optimization for Highly Sparse One-Shot Pruning for Large Language Modelscited by 18
  6. 2022Rethinking Bayesian Learning for Data Analysis: The Art of Prior and Inference in Sparsity-Aware Modelingcited by 12
  7. 2022A Single-Loop Gradient Descent and Perturbed Ascent Algorithm for Nonconvex Functional Constrained Optimizationcited by 12
  8. 2022Federated Learning and Meta Learning: Approaches, Applications, and Directionscited by 12
  9. 2023Oracle Complexity of Single-Loop Switching Subgradient Methods for Non-Smooth Weakly Convex Functional Constrained Optimizationcited by 11
  10. 2021FORMS: Fine-grained Polarized ReRAM-based In-situ Computation for Mixed-signal DNN Acceleratorcited by 10
  11. 2021GRIM: A General, Real-Time Deep Learning Inference Framework for Mobile Devices based on Fine-Grained Structured Weight Sparsitycited by 10
  12. 2022Improving Privacy-Preserving Vertical Federated Learning by Efficient Communication with ADMMcited by 10
38 more
  1. 2021Training Recurrent Neural Networks by Sequential Least Squares and the Alternating Direction Method of Multiplierscited by 8
  2. 2023vSHARP: variable Splitting Half-quadratic Admm algorithm for Reconstruction of inverse-Problemscited by 8
  3. 2025AugFL: Augmenting Federated Learning with Pretrained Modelscited by 6
  4. 2022Federated PCA on Grassmann Manifold for Anomaly Detection in IoT Networkscited by 5
  5. 2022Fully Decentralized, Scalable Gaussian Processes for Multi-Agent Federated Learningcited by 4
  6. 2023Smoothing ADMM for Sparse-Penalized Quantile Regression with Non-Convex Penaltiescited by 4
  7. 2025PnP-CM: Consistency Models as Plug-and-Play Priors for Inverse Problemscited by 4
  8. 2022Distributed saddle point problems for strongly concave-convex functionscited by 3
  9. 2024Federated PCA on Grassmann Manifold for IoT Anomaly Detectioncited by 3
  10. 2025Communication-Efficient Stochastic Distributed Learningcited by 3
  11. 2025Alternating Direction Method of Multipliers for Nonlinear Matrix Decompositionscited by 3
  12. 2023Sparsity-Aware Distributed Learning for Gaussian Processes with Linear Multiple Kernelcited by 2
  13. 2023Privacy-preserving Federated Primal-dual Learning for Non-convex and Non-smooth Problems with Model Sparsificationcited by 2
  14. 2024Decentralized Smoothing ADMM for Quantile Regression with Non-Convex Sparse Penaltiescited by 2
  15. 2024HR-Bandit: Human-AI Collaborated Linear Recourse Banditcited by 2
  16. 2026Dual-space posterior sampling for Bayesian inference in constrained inverse problemscited by 2
  17. 2026Learning Multi-Agent Coordination via Sheaf-ADMMcited by 2
  18. 2021Efficient Algorithms for Estimating the Parameters of Mixed Linear Regression Modelscited by 1
  19. 2021RIFLE: Imputation and Robust Inference from Low Order Marginalscited by 1
  20. 2022Low-rank Tensor Learning with Nonconvex Overlapped Nuclear Norm Regularizationcited by 1
  21. 2024Learn What You Need in Personalized Federated Learningcited by 1
  22. 2025Toward Temporal Causal Representation Learning with Tensor Decompositioncited by 1
  23. 2025Large Language Model-Assisted Planning of Electric Vehicle Charging Infrastructure with Real-World Case Studycited by 1
  24. 2026SALAAD: Sparse And Low-Rank Adaptation via ADMM for Large Language Model Inferencecited by 1
  25. 2026NanoQuant: Efficient Sub-1-Bit Quantization of Large Language Modelscited by 1
  26. 2026An Exterior Method for Nonnegative Matrix Factorizationcited by 1
  27. 2022Nonlinear Kernel Support Vector Machine with 0-1 Soft Margin Losscited by 0
  28. 2022Study of Robust Sparsity-Aware RLS algorithms with Jointly-Optimized Parameters for Impulsive Noise Environmentscited by 0
  29. 2022Sparsity-Aware Robust Normalized Subband Adaptive Filtering algorithms based on Alternating Optimizationcited by 0
  30. 2023First-order Methods for Affinely Constrained Composite Non-convex Non-smooth Problems: Lower Complexity Bound and Near-optimal Methodscited by 0
  31. 2023A Novel Truncated Norm Regularization Method for Multi-channel Color Image Denoisingcited by 0
  32. 2023Untrained neural network embedded Fourier phase retrieval from few measurementscited by 0
  33. 2023MKL-L0/1-SVMcited by 0
  34. 2023Moreau Envelope ADMM for Decentralized Weakly Convex Optimizationcited by 0
  35. 2023Sparse Index Tracking: Simultaneous Asset Selection and Capital Allocation via ℓ0-Constrained Portfoliocited by 0
  36. 2023Federated Deep Equilibrium Learning: Harnessing Compact Global Representations to Enhance Personalizationcited by 0
  37. 2023A Distributed ADMM-based Deep Learning Approach for Thermal Control in Multi-Zone Buildings under Demand Response Eventscited by 0
  38. 2024Aggregation Models with Optimal Weights for Distributed Gaussian Processescited by 0

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