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In this paper we propose a distributed dual gradient algorithm for minimizinglinearly constrained separable convex problems and analyze its rate ofconvergence. In particular, we prove that under the assumption of strongconvexity and Lipshitz continuity of the gradient of the primal objectivefunction we have a global error bound type property for the dual problem. Usingthis error bound property we devise a fully distributed dual gradient scheme,i.e. a gradient scheme based on a weighted step size, for which we deriveglobal linear rate of convergence for both dual and primal suboptimality andfor primal feasibility violation. Many real applications, e.g. distributedmodel predictive control, network utility maximization or optimal power flow,can be posed as linearly constrained separable convex problems for which dualgradient type methods from literature have sublinear convergence rate. In thepresent paper we prove for the first time that in fact we can achieve linearconvergence rate for such algorithms when they are used for solving theseapplications. Numerical simulations are also provided to confirm our theory.

On linear convergence of a distributed dual gradient algorithm for linearly constrained separable convex problems