Parallel Subgradient Algorithm with Block Dual Decomposition for Large-scale Optimization

This paper studies computational approaches for solving large-scale optimization problems using a Lagrangian dual reformulation, solved by parallel sub-gradient methods. Since there are many possible reformulations for a given problem, an important question is: Which reformulation leads to the fastest solution time? One approach is to detect a block diagonal structure in the constraint matrix, and reformulate the problem by dualizing the constraints outside of the blocks; the approach is defined herein as block dual decomposition. Main advantage of such a reformulation is that the Lagrangian relaxation has a block diagonal constraint matrix, thus decomposable into smaller sub-problems that can solved in parallel. We show that the block decomposition can critically affect convergence rate of the sub-gradient method. We propose various decomposition methods that use domain knowledge or apply algorithms using knowledge about the structure in the constraint matrix or the dependence in the decision variables, towards reducing the computational effort to solve large-scale optimization problems. In particular, we introduce a block decomposition approach that reduces the number of dualized constraints by utilizing a community detection algorithm. We present empirical experiments on an extensive set of problem instances including a real application. We illustrate that if the number of the dualized constraints in the decomposition increases, the computational effort within each iteration of the sub-gradient method decreases while the number of iterations required for convergence increases. The key message is that it is crucial to employ prior knowledge about the structure of the problem when solving large scale optimization problems using dual decomposition.