Convergence Rates of Min-Cost Subgraph Algorithms for Multicast in Coded Networks

The problem of establishing minimum-cost multicast connections in coded networks can be viewed as an optimization problem, and decentralized algorithms were proposed by Lun et al. to compute the optimal subgraph using the subgradient method on the dual problem. However, the convergence rate problem for these algorithms remains open. There are limited results in the literature on the convergence rate of the subgradient method in the dual space, but the convergence rate of the primal solutions was not known. In this paper, we analyze the convergence rates of the min-cost subgraph algorithms in both the dual and the primal spaces. We show that using the incremental subgradient method on the dual problem with appropriately chosen step sizes yields linear convergence rate to a neighborhood of the optimal solution. Also, if we use constant step sizes in the subgradient method and simple averaging for primal recovery, the primal solutions recovered from the dual iterations converge to a neighborhood of the optimal solution with rate O(1/n).