  This paper presents a robust, distributed algorithm to solve general linear
programs. The algorithm design builds on the characterization of the solutions
of the linear program as saddle points of a modified Lagrangian function. We
show that the resulting continuous-time saddle-point algorithm is provably
correct but, in general, not distributed because of a global parameter
associated with the nonsmooth exact penalty function employed to encode the
inequality constraints of the linear program. This motivates the design of a
discontinuous saddle-point dynamics that, while enjoying the same convergence
guarantees, is fully distributed and scalable with the dimension of the
solution vector. We also characterize the robustness against disturbances and
link failures of the proposed dynamics. Specifically, we show that it is
integral-input-to-state stable but not input-to-state stable. The latter fact
is a consequence of a more general result, that we also establish, which states
that no algorithmic solution for linear programming is input-to-state stable
when uncertainty in the problem data affects the dynamics as a disturbance. Our
results allow us to establish the resilience of the proposed distributed
dynamics to disturbances of finite variation and recurrently disconnected
communication among the agents. Simulations in an optimal control application
illustrate the results.
