Variable Partitioning for Distributed Optimization

This paper is about how to partition decision variables while decomposing a large-scale optimization problem for the best performance of distributed solution methods. Solving a large-scale optimization problem sequen- tially can be computationally challenging. One classic approach is to decompose the problem into smaller sub-problems and solve them in a distributed fashion. However, there is little discussion in the literature on which variables should be grouped together to form the sub-problems, especially when the optimization formulation involves complex constraints. We focus on one of the most popular distributed approaches, dual decomposition and distributed sub-gradient methods. Based on a theoretical guarantee on its convergence rate, we explain that a partition of variables can critically affect the speed of convergence and highlight the importance of the number of dualized constraints. Then, we introduce a novel approach to find a partition that reduces the number of dualized constraints by utilizing a community detection algorithm from physics literature. Roughly speaking, the proposed method groups decision variables that appear together in con- straints and solves the resulting sub-problems with blocks of variables in parallel. Empirical experiments on a real application show that the proposed method significantly accelerates the convergence of the distributed sub-gradient method. The advantage of our approach becomes more significant as the size of the problem increases and each constraint involves more variables.