Journal of Nonlinear Science

, Volume 29, Issue 4, pp 1247–1272 | Cite as

Distributed Coordination for Nonsmooth Convex Optimization via Saddle-Point Dynamics

  • Jorge CortésEmail author
  • Simon K. Niederländer


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.


Distributed multi-agent coordination Nonsmooth convex optimization Saddle-point dynamics Continuous-time optimization algorithms 

Mathematics Subject Classification

90C25 49J52 34A60 34D23 68W15 90C35 


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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mechanical and Aerospace Engineering, Jacobs School of EngineeringUniversity of California, San DiegoLa JollaUSA
  2. 2.Institute for Systems Theory and Automatic ControlUniversity of StuttgartStuttgartGermany

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