Distributed partition-based optimization via dual decomposition

In this paper we consider a novel partition-based framework for distributed optimization in peer-to-peer networks. In several important applications the agents of a network system have to solve an optimization problem with two important features: (i) the dimension of the decision variable is a function of the network size, and (ii) the cost function and the constraints have a sparsity structure that is related to the sparsity of the graph. For this class of problems a straightforward application of existing methods would result in all the nodes reaching consensus on the minimizer. This approach has two inefficiencies: poor scalability and redundancy of shared information. Indeed, the dimension of the vector stored by each node and the size of the local problem to be solved depend on the network size. Furthermore, all the nodes compute the entire solution. In this paper we provide a preliminary contribution in developing and analyzing novel partition based algorithms. We propose a partition-based algorithm based on dual decomposition. We show that, exploiting the problem structure, the solution can be partitioned among the nodes so that each node stores a local copy of just a portion of the decision variable (rather than a copy of the entire decision vector) and solves a small scale local problem.