Self Equivalence of the Alternating Direction Method of Multipliers

By Ming Yan, Wotao Yin. 2014

68 citations recorded by the source corpus.

The alternating direction method of multipliers (ADM or ADMM) breaks a complex optimization problem into much simpler subproblems. The ADM algorithms are typically short and easy to implement yet exhibit (nearly) state-of-the-art performance for large-scale optimization problems.

Subfields: Aerospace Engineering, Computational Mechanics, Mathematical Physics, Optimization & Training

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Citations in the corpus

Frank-Wolfe Splitting via Augmented Lagrangian Method (2018, methodology): “One of the most popular algorithms in this framework is the alternating direction method of multipliers (ADMM), proposed by Glowinski and Marroco (1975), studied by Gabay and Mercier (1976), and revisited many times; see for instance (Boyd et al., 2011; Yan and Yin, 2016).”
Conic Optimization via Operator Splitting and Homogeneous Self-Dual Embedding (2013, background): “In [7] and [38] it was shown that equivalences exist between ADMM applied to the primal problem, the dual problem, and a saddle point formulation of the problem; in other words, ADMM is (in a sense) itself self-dual.”
On the Global and Linear Convergence of the Generalized Alternating Direction Method of Multipliers (2016, methodology): “Specifically, [35] shows that applying ADM is equivalent to applying the DRSM to the same problem.”
A New Primal–Dual Algorithm for Minimizing the Sum of Three Functions with a Linear Operator (2016, methodology): “ADMMapplied on a convex optimization problem was shown to be equivalent to Douglas–Rachford Splitting (DRS) applied on the dual problem by [19], and [35] showed recently that it is also equivalent to DRS applied on the same primal problem.”
Global Convergence of ADMM in Nonconvex Nonsmooth Optimization (2015, background): “However, in general, permutation including the last block y could cause ADMM to diverge (A convex example can be found in [62]).”
ARock: an Algorithmic Framework for Asynchronous Parallel Coordinate Updates (2015, background): “Note that the order in which the proximal operators are applied to f and g affects both z [57] and whether coordinate-wise updates can be efficiently computed.”
Proximal Splitting Algorithms: A Tour of Recent Advances, with New Twists. (2020, methodology): “…using ﬁne properties of the operators, we have: Let us show that, like the Chambolle–Pock algorithm, the Douglas–Rachford algorithm is self-dual [62, Lemma 3.6] [63]: it is equivalent to apply it to the primal problem (79) or to the dual problem: minimize 𝑢 ∈ X 𝑓 ∗ (− 𝑢 ) + 𝑔 ∗ ( 𝑢…”
Convergence Rate Analysis of Several Splitting Schemes (2014, background): “7 A recent result reported in Chapter 5 [55] of this volume shows the direct (non-duality) equivalence between ADMM and DRS when they are both applied to Problem (3).”
A Primer on Monotone Operator Methods (2015, methodology): “Another obtains ADMM by applying Douglas-Rachford splitting to the primal optimization problem [131].”