A note on the convergence of ADMM for linearly constrained convex optimization problems

Liang Chen (), Defeng Sun () and Kim-Chuan Toh ()
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Liang Chen: Hunan University
Defeng Sun: National University of Singapore
Kim-Chuan Toh: National University of Singapore

Computational Optimization and Applications, 2017, vol. 66, issue 2, No 6, 327-343

Abstract: Abstract 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 of the ADMM and provide a rigorous convergence analysis on the ADMM with a computationally more attractive large step-length that can even exceed the practically much preferred golden ratio of $$(1+\sqrt{5})/2$$ ( 1 + 5 ) / 2 .

Keywords: Alternating direction method of multipliers (ADMM); Convergence; Counterexample; Large step-length; 65K05; 90C25; 90C46 (search for similar items in EconPapers)
Date: 2017
References: View complete reference list from CitEc
Citations: View citations in EconPapers (4)

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DOI: 10.1007/s10589-016-9864-7

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