A Note on the Convergence of ADMM for Linearly Constrained Convex Optimization Problems
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Abstract (arXiv)
This note serves two purposes. Firstly, we construct a counterexample to show that the statement on the convergence of the alternating direction method of multipliers (ADMM) for solving linearly constrained convex optimization problems in a highly influential paper by Boyd et al. [Found. Trends Mach. Learn. 3(1) 1-122 (2011)] can be false if no prior condition on the existence of solutions to all the subproblems involved is assumed to hold. Secondly, we present fairly mild conditions to guarantee the existence of solutions to all the subproblems and provide a rigorous convergence analysis on the ADMM, under a more general and useful semi-proximal ADMM (sPADMM) setting considered by Fazel et al. [SIAM J. Matrix Anal. Appl. 34(3) 946-977 (2013)], with a computationally more attractive large step-length that can even exceed the practically much preferred golden ratio of (1+√5)/2.
Key facts. Cited by 3 papers in the map (data through 14 Oct 2026). First posted on arXiv 8 Jul 2015.
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Citing papers per year: 2022: 2; 2023: 0; 2024: 1.
3 papers.
- 2022Beyond Adult and COMPAS: Fairness in Multi-Class Predictioncited by 16
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- 2024Semi-supervised Symmetric Non-negative Matrix Factorization with Low-Rank Tensor Representationcited by 2
- 2024Semi-supervised Symmetric Non-negative Matrix Factorization with Low-Rank Tensor Representation4 May 2024
- 2022A Tale of HodgeRank and Spectral Method: Target Attack Against Rank Aggregation Is the Fixed Point of Adversarial Game13 Sep 2022
- 2022Beyond Adult and COMPAS: Fairness in Multi-Class Prediction15 Jun 2022