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Distributed coordination for nonsmooth convex optimization via saddle-point dynamics
J. Cortés, S. K. Niederländer
Journal of Nonlinear Science 29 (4) (2019), 1247-1272

Abstract

This paper designs 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-points 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 a set of convexity and regularity conditions on the objective functions that guarantee the exponential convergence rate of the proposed algorithms for optimization problems that involve either equality or inequality constraints. Various examples illustrate our discussion.

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