Distributed Coordination for Nonsmooth Convex Optimization via Saddle-Point Dynamics

This paper considers continuous-time coordination algorithms for networks of agents that seek to collectively solve a general class of nonsmooth convex optimization problems with an inherent distributed structure. Our algorithm design builds on the characterization of the solutions of the nonsmooth...

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Bibliographic Details
Published in:Journal of nonlinear science Vol. 29; no. 4; pp. 1247 - 1272
Main Authors: Cortés, Jorge, Niederländer, Simon K.
Format: Journal Article
Language:English
Published: New York Springer US 15.08.2019
Springer Nature B.V
Subjects:
Algorithms
Analysis
Classical Mechanics
Computational geometry
Convergence
Convex analysis
Convexity
Coordination
Design parameters
Economic Theory/Quantitative Economics/Mathematical Methods
Liapunov functions
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Nonlinear programming
Optimization
Saddle points
Theoretical
Distributed multi-agent coordination
34D23
90C35
Nonsmooth convex optimization
90C25
49J52
Continuous-time optimization algorithms
68W15
Saddle-point dynamics
34A60
ISSN:0938-8974, 1432-1467
Online Access:Get full text
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Summary:This paper considers continuous-time coordination algorithms for networks of agents that seek to collectively solve a general class of nonsmooth convex optimization problems with an inherent distributed structure. Our algorithm design builds on the characterization of the solutions of the nonsmooth convex program as saddle points of an augmented Lagrangian. We show that the associated saddle-point dynamics are asymptotically correct but, in general, not distributed because of the presence of a global penalty parameter. This motivates the design of a discontinuous saddle-point-like algorithm that enjoys the same convergence properties and is fully amenable to distributed implementation. Our convergence proofs rely on the identification of a novel global Lyapunov function for saddle-point dynamics. This novelty also allows us to identify mild convexity and regularity conditions on the objective function that guarantee the exponential convergence rate of the proposed algorithms for convex optimization problems subject to equality constraints. Various examples illustrate our discussion.
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ISSN:0938-8974
1432-1467
DOI:10.1007/s00332-018-9516-4