Distributed Asynchronous Mixed-Integer Linear Programming with Feasibility Guarantees

TL;DR

This paper introduces a distributed asynchronous algorithm for solving mixed-integer linear programs (MILPs) by relaxing, regularizing, and computing saddle points, with guarantees on feasibility and convergence in multi-agent systems.

Contribution

It develops a novel parallelized, asynchronous saddle point method for MILPs with feasibility guarantees and analyzes its convergence and suboptimality bounds.

Findings

Algorithm tolerates asynchrony in distributed computations.

Relaxation and regularization have mild impact on solution suboptimality.

Simulation confirms theoretical guarantees and practical effectiveness.

Abstract

In this paper we solve mixed-integer linear programs (MILPs) via distributed asynchronous saddle point computation. This work is motivated by the MILPs being able to model problems in multi-agent autonomy, such as task assignment problems and trajectory planning with collision avoidance constraints in multi-robot systems. To solve a MILP, we relax it with a linear program approximation. We first show that if the linear program relaxation satisfies Slater's condition, then relaxing the problem, solving it, and rounding the relaxed solution produces a point that is guaranteed to satisfy the constraints of the original MILP. Next, we form a Lagrangian saddle point problem that is equivalent to the linear program relaxation, and then we regularize the Lagrangian in both the primal and dual spaces. Doing so gives a regularized Lagrangian that is strongly convex-strongly concave. We then develop a parallelized algorithm to compute saddle points of the regularized Lagrangian, and we show that it is tolerant to asynchrony in the computations and communications of primal and dual variables. Suboptimality bounds and convergence rates are presented for convergence to a saddle point. The suboptimality bound accounts for (i) the error induced by regularizing the Lagrangian and (ii) the suboptimality gap between the solution to the original MILP and the solution to its relaxed form. Simulation results illustrate these theoretical developments in practice, and show that relaxation and regularization combined typically have only a mild impact on the suboptimality of the solution obtained.