Distributed Asynchronous Mixed-Integer Linear Programming with Feasibility Guarantees

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.

Figures (4)

• Figure 1: Agents' essential neighbors in Example \ref{['ex:comms']}.
• Figure 2: Four runs are compared based on varying communication rates for a random Generalized Assignment Problem with 100 agents and 100 tasks where there are 100 primal agents and 70 dual agents. The distance between iterates and the optimal solution obtained is compared with respect to the communication rates. As expected, a lower rate of communications results in agents requiring more iterations to converge. However, we do see that in all cases agents eventually reach a steady-state cost of approximately $10$, followed by a period of slow decrease thereafter, indicating that agents reach the same solution, regardless of their communication rate.
• Figure 3: Four runs are compared based on varying computation rates for a random Generalized Assignment Problem with 100 agents and 100 tasks where there are 100 primal agents and 70 dual agents. The distance between iterates and the optimal solution obtained is compared with respect to the computation rates. As with changing communication rates, it is expected that agents with less frequent communications do indeed require more iterations to converge, which is what is seen here.
• Figure 4: Four runs are compared based on varying levels of constraint tightening. The distance between iterates and the optimal solution obtained is compared with respect to the perturbation terms with a communication rate of $0.5$. As the constraints are tightened more, we see that agents have a higher cost at the final iterate. This is intuitive because tightening constraints leads to a smaller feasible region to optimize over, which necessarily leads to higher costs.

Theorems & Definitions (21)

• Definition 1: Section 2.2 of neumann2019feasible
• Proposition 1: neumann2019feasible
• Definition 2: Definition 2.7 in neumann2019feasible
• Remark 1
• Theorem 1: Slater's Condition Implies Granularity
• Remark 2
• Lemma 1: Constraint Tightening
• Theorem 2
• Remark 3
• Lemma 2