Distributed and time-varying primal-dual dynamics via contraction analysis
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In this paper, we provide a holistic analysis of the primal-dual dynamics associated to linear equality-constrained optimization problems and aimed at computing saddle-points of the associated Lagrangian using contraction analysis. We analyze the well-known standard version of the problem: we establish convergence results for convex objective functions and further characterize its convergence rate under strong convexity. Then, we consider a popular implementation of a distributed optimization problem and, using weaker notations of contraction theory, we establish the global exponential convergence of its associated distributed primal-dual dynamics. Moreover, based on this analysis, we propose a new distributed solver for the least-squares problem with global exponential convergence guarantees. Finally, we consider time-varying versions of the centralized and distributed implementations of primal-dual dynamics and exploit their contractive nature to establish asymptotic bounds on their tracking error. To support our convergence analyses, we introduce novel results on contraction theory and specifically use them in the cases where the analyzed systems are weakly contractive (i.e., has zero contraction rate) and/or converge to specific points belonging to a subspace of equilibria.
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