This work presents a Multiple Depot Traveling Salesman Problem with revisit period constraints. The revisit period constraints are relevant to persistent routing applications, where these constraints represent maximum time between successive visits to a target. This problem is first posed as a Mixed Integer Linear Program. The coupling constraints in the primal problem are then relaxed via Lagrangian relaxation. Minimizing the resulting Lagrangian over the primal variables can be separated into an individual subproblem for each salesman. An algorithm to solve the subproblems to near-optimality that scales well for larger instances is presented. The dual function is then maximized, where three different dual update methods are studied: the subgradient method, the ellipsoid algorithm, and a bundle algorithm. Further, a primal reconstruction algorithm is presented to reconstruct feasible solutions from the solution of the dual algorithm. The quality of these solutions are compared to the optimal solutions obtained from the CPLEX MILP optimizer. The results of extensive numerical testing show that the dual algorithm presented was computationally efficient, capable of finding high quality solutions, and scale well compared to CPLEX. Further, it was seen that the bundle method used showed better convergence than the other update methods within the dual algorithm. Note to Practitioners— This work was motivated by the problem of routing UAVs in the presence of timing constraints, specifically motivated by military applications. This problem has constraints on the frequency at which targets are visited, such that high priority targets must be visited more frequently. Similar prior work exists in the literature for a variety of routing problems for different timing or resource constraints. Techniques which are successful on one set of constraints may be less useful when applied to different constraints. We present a Lagrangian based approach to the multiple depot traveling salesman variant. The results show the utility of a Lagrangian based technique to this problem, as both the solution quality and computation time scale well with problem size. The Lagrangian based technique presented here is seen to provide, on average, high quality solutions with improved computational time over a branch-and-bound algorithm, which solves the problem exactly. While the solutions returned from the algorithm are on average of good quality, they do exhibit some variance. Alternative algorithms may give more consistent results. The technique presented in this paper is general and may be applied to other routing problems where the revisit period constraints are applied. Future research includes application of this method to dynamic environments, where the targets’ states are unknown to the UAVs except when being monitored. Further, alternative techniques can be developed for this problem and compared to both the branch-and-bound and the Lagrange algorithm presented here.

中文翻译：

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求解带重访期约束的多站点旅行商问题的拉格朗日算法


这项工作提出了一个具有重访期限制的多仓库旅行商问题。重访周期约束与持久路由应用相关，其中这些约束表示对目标的连续访问之间的最大时间。该问题首先被提出为混合整数线性规划。然后通过拉格朗日松弛来松弛原始问题中的耦合约束。最小化原始变量上产生的拉格朗日量可以分为每个推销员的单独子问题。提出了一种将子问题解决到接近最优的算法，该算法可以很好地适应更大的实例。然后最大化对偶函数，其中研究了三种不同的对偶更新方法：次梯度方法、椭球算法和束算法。此外，提出了一种原始重构算法，用于从对偶算法的解中重构可行解。将这些解决方案的质量与从 CPLEX MILP 优化器获得的最佳解决方案进行比较。大量数值测试的结果表明，所提出的对偶算法计算效率高，能够找到高质量的解决方案，并且与 CPLEX 相比具有良好的扩展性。此外，可以看出，所使用的捆绑方法比对偶算法中的其他更新方法表现出更好的收敛性。从业者注意事项——这项工作的动机是在存在时间限制的情况下对无人机进行路由问题，特别是军事应用。该问题对访问目标的频率有限制，因此必须更频繁地访问高优先级目标。 对于不同时序或资源限制的各种路由问题，文献中存在类似的先前工作。在一组约束上成功的技术在应用于不同的约束时可能不太有用。我们提出了一种基于拉格朗日的方法来处理多站点旅行推销员变体。结果显示了基于拉格朗日的技术对于该问题的实用性，因为解的质量和计算时间都与问题的大小成正比。平均而言，这里提出的基于拉格朗日的技术可以提供高质量的解决方案，并且比分支定界算法缩短了计算时间，从而准确地解决了问题。虽然算法返回的解决方案平均质量良好，但它们确实表现出一些差异。替代算法可能会给出更一致的结果。本文提出的技术是通用的，可以应用于应用重访周期约束的其他路由问题。未来的研究包括将该方法应用于动态环境，在动态环境中，除了被监控之外，无人机无法得知目标的状态。此外，可以针对该问题开发替代技术，并将其与此处介绍的分支定界算法和拉格朗日算法进行比较。