Fast Convergence of Dynamical ADMM via Time Scaling of Damped Inertial Dynamics

In this paper, we propose in a Hilbertian setting a second-order time-continuous dynamic system with fast convergence guarantees to solve structured convex minimization problems with an affine constraint. The system is associated with the augmented Lagrangian formulation of the minimization problem....

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Bibliographic Details
Published inJournal of optimization theory and applications Vol. 193; no. 1-3; pp. 704 - 736
Main Authors Attouch, Hedy, Chbani, Zaki, Fadili, Jalal, Riahi, Hassan
Format Journal Article
LanguageEnglish
Published New York Springer US 01.06.2022
Springer Nature B.V
Springer Verlag
Subjects
Algorithms
Applications of Mathematics
Approximation
Calculus of Variations and Optimal Control; Optimization
Computer Science
Convergence
Convex analysis
Dynamical Systems
Engineering
Lagrange multiplier
Machine Learning
Mathematics
Mathematics and Statistics
Methods
Operations Research/Decision Theory
Optimization
Optimization and Control
Parameters
Signal and Image Processing
Statistics
Theory of Computation
Viscosity
Viscous damping
Lyapunov analysis
65K05
Temporal scaling
Convergence rates
37N40
65B99
49M30
Nesterov accelerated gradient method
90C25
Augmented Lagrangian
Damped inertial dynamics
Convex constrained minimization
65K10
90B50
ADMM
46N10
convex constrained minimization
damped inertial dynamics
augmented Lagrangian
convergence rates
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ISSN0022-3239
1573-2878
DOI10.1007/s10957-021-01859-2

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Summary:In this paper, we propose in a Hilbertian setting a second-order time-continuous dynamic system with fast convergence guarantees to solve structured convex minimization problems with an affine constraint. The system is associated with the augmented Lagrangian formulation of the minimization problem. The corresponding dynamics brings into play three general time-varying parameters, each with specific properties, and which are, respectively, associated with viscous damping, extrapolation and temporal scaling. By appropriately adjusting these parameters, we develop a Lyapunov analysis which provides fast convergence properties of the values and of the feasibility gap. These results will naturally pave the way for developing corresponding accelerated ADMM algorithms, obtained by temporal discretization.
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ISSN:0022-3239
1573-2878
DOI:10.1007/s10957-021-01859-2