A Distributed Newton Method for Network Utility Maximization—Part II: Convergence

The existing distributed algorithms for network utility maximization (NUM) problems are mostly constructed using dual decomposition and first-order (gradient or subgradient) methods, which suffer from a slow rate of convergence. Part I of this paper proposed an alternative distributed Newton-type algorithm for solving NUM problems with self-concordant utility functions. For each primal iteration, this algorithm features distributed exact stepsize calculation with finite termination and decentralized computation of the dual variables using a finitely truncated iterative scheme obtained through novel matrix splitting techniques. This paper analyzes the convergence properties of a broader class of algorithms with potentially different stepsize computation schemes. In particular, we allow for errors in the stepsize computation. We show that if the error levels in the Newton direction (resulting from finite termination of dual iterations) and stepsize calculation are below a certain threshold, then the algorithm achieves local quadratic convergence rate in primal iterations to an error neighborhood of the optimal solution, where the size of the neighborhood can be explicitly characterized by the parameters of the algorithm and the error levels.