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Passivity-based Analysis of the ADMM Algorithm for Constraint-Coupled Optimization
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DOI：10.1016/j.automatica.2022.110552.png)
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Abstract
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We propose a novel, system theoretic analysis of the Alternating Direction Method of Multipliers (ADMM) applied to a convex constraint-coupled optimization problem. The resulting algorithm can be interpreted as a linear, discrete-time dynamical system (modeling the multiplier ascent update) in closed loop with a static nonlinearity (representing the minimization of the augmented Lagrangian). When expressed in suitable coordinates, we prove that the discrete-time linear dynamical system has a discrete positive-real transfer function and is interconnected in closed loop with a static, passive nonlinearity. This readily shows that the origin is a stable equilibrium for the feedback interconnection. Finally, we also show global asymptotic stability of the origin for the closed-loop system and, thus, global asymptotic convergence of ADMM to the optimal solution of the optimization problem.(c) 2022 Elsevier Ltd. All rights reserved.
Keywords:
ADMM
Constraint-coupled optimization
Passivity theory
Nonsmooth optimization
Control for optimization
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