DC-DistADMM: ADMM Algorithm for Contrained Distributed Optimization over Directed Graphs

We present a distributed algorithm to solve a multi-agent optimization problem, where the global objective function is the sum $n$ convex objective functions. Our focus is on constrained problems where the agents' estimates are restricted to be in different convex sets. The interconnection topology among the $n$ agents has directed links and each agent $i$ can only communicate with agents in its neighborhood determined by a directed graph. In this article, we propose an algorithm called \underline{D}irected \underline{C}onstrained-\underline{Dist}ributed \underline{A}lternating \underline{D}irection \underline{M}ethod of \underline{M}ultipliers (DC-DistADMM) to solve the above multi-agent convex optimization problem. During every iteration of the DC-DistADMM algorithm, each agent solves a local convex optimization problem and utilizes a finite-time "approximate" consensus protocol to update its local estimate of the optimal solution. To the best of our knowledge the proposed algorithm is the first ADMM based algorithm to solve distributed multi-agent optimization problems in directed interconnection topologies with convergence guarantees. We show that in case of individual functions being convex and not-necessarily differentiable the proposed DC-DistADMM algorithm converges at a rate of $O(1/k)$, where $k$ is the iteration counter. We further establish a linear rate of convergence for the DC-DistADMM algorithm when the global objective function is strongly convex and smooth. We numerically evaluate our proposed algorithm by solving a constrained distributed $\ell_1$-regularized logistic regression problem. Additionally, we provide a numerical comparison of the proposed DC-DistADMM algorithm with the other state-of-the-art algorithms in solving a distributed least squares problem to show the efficacy of the DC-DistADMM algorithm over the existing methods in the literature.Comment: 17 pages, 8 Figures, includes an appendi

Topology Identification under Spatially Correlated Noise

This article addresses the problem of reconstructing the topology of a network of agents interacting via linear dynamics, while being excited by exogenous stochastic sources that are possibly correlated across the agents, from time-series measurements alone. It is shown, under the assumption that the correlations are affine in nature, such network of nodal interactions is equivalent to a network with added agents, represented by nodes that are latent, where no corresponding time-series measurements are available; however, here all exogenous excitements are spatially (that is, across agents) uncorrelated. Generalizing affine correlations, it is shown that, under polynomial correlations, the latent nodes in the expanded networks can be excited by clusters of noise sources, where the clusters are uncorrelated with each other. The clusters can be replaced with a single noise source if the latent nodes are allowed to have non-linear interactions. Finally, using the sparse plus low-rank matrix decomposition of the imaginary part of the inverse power spectral density matrix (IPSDM) of the time-series data, the topology of the network is reconstructed. Under non conservative assumptions, the correlation graph is retrieved.Comment: 14 pages, 5 figure

Distributed Apportioning in a Power Network for providing Demand Response Services

Greater penetration of Distributed Energy Resources (DERs) in power networks requires coordination strategies that allow for self-adjustment of contributions in a network of DERs, owing to variability in generation and demand. In this article, a distributed scheme is proposed that enables a DER in a network to arrive at viable power reference commands that satisfies the DERs local constraints on its generation and loads it has to service, while, the aggregated behavior of multiple DERs in the network and their respective loads meet the ancillary services demanded by the grid. The Net-load Management system for a single unit is referred to as the Local Inverter System (LIS) in this article . A distinguishing feature of the proposed consensus based solution is the distributed finite time termination of the algorithm that allows each LIS unit in the network to determine power reference commands in the presence of communication delays in a distributed manner. The proposed scheme allows prioritization of Renewable Energy Sources (RES) in the network and also enables auto-adjustment of contributions from LIS units with lower priority resources (non-RES). The methods are validated using hardware-in-the-loop simulations with Raspberry PI devices as distributed control units, implementing the proposed distributed algorithm and responsible for determining and dispatching realtime power reference commands to simulated power electronics interface emulating LIS units for demand response.Comment: 7 pages, 11 Figures, IEEE International Conference on Smart Grid Communication