  Motivated by broad applications in various fields of engineering, we study a
network resource allocation problem where the goal is to optimally allocate a
fixed quantity of resources over a network of nodes. We consider large scale
networks with complex interconnection structures, thus any solution must be
implemented in parallel and based only on local data resulting in a need for
distributed algorithms. In this paper, we study a distributed Lagrangian method
for such problems. By utilizing the so-called distributed subgradient methods
to solve the dual problem, our approach eliminates the need for central
coordination in updating the dual variables, which is often required in classic
Lagrangian methods. Our focus is to understand the performance of this
distributed algorithm when the number of resources is unknown and may be
time-varying. In particular, we obtain an upper bound on the convergence rate
of the algorithm to the optimal value, in expectation, as a function of the
size and the topology of the underlying network. The effectiveness of the
proposed method is demonstrated by its application to the economic dispatch
problem in power systems, with simulations completed on the benchmark IEEE-14
and IEEE-118 bus test systems.
