Cluster-based Distributed Augmented Lagrangian Algorithm for a Class of Constrained Convex Optimization Problems
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We propose a distributed solution for a constrained convex optimization problem over a network of clustered agents each consisted of a set of subagents. The communication range of the clustered agents is such that they can form a connected undirected graph topology. The total cost in this optimization problem is the sum of the local convex cost of the subagents of each cluster. We seek a minimizer of this cost subject to a set of affine equality constraints, and a set of affine inequality constrains specifying the bounds on the decision variables if such bounds exist. Our proposed distributed algorithm is a novel continuous-time algorithm that is linked to the augmented Lagrangian approach. It converges asymptotically when the local cost functions are convex and exponentially when they are strongly convex and have Lipschitz gradients. For efficient communication and computation resource management, we only require the agents that are coupled through an equality constraint to form a communication topology to address that coupling in a distributed manner. We use an ϵ-exact penalty function to address the inequality constraints, and drive an explicit lower bound on the penalty function weight to guarantee convergence to ϵ-neighborhood of the global minimum value of the cost. We demonstrate our results via an optimal resource allocation problem for power generators, and an optimal multi-sensor deployment problem.
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