Robust distributed linear programming

TL;DR

This paper introduces a fully distributed, robust saddle-point algorithm for solving linear programs, capable of handling disturbances and communication failures, with proven convergence and scalability.

Contribution

It develops a novel discontinuous saddle-point dynamics that is fully distributed, scalable, and robust to disturbances, advancing distributed linear programming methods.

Findings

The algorithm converges to the optimal solution under disturbances.

It is integral-input-to-state stable but not input-to-state stable.

The method is resilient to communication failures and disturbances of finite variation.

Abstract

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.