In this paper, we consider the problem of optimizing the flows in a lossless flow network with additional nonconvex cycle constraints on nodal variables; such constraints appear in several applications, including electric power, water distribution, and natural gas networks. This problem is a nonconvex version of the minimum-cost network flow problem (NFP), and to solve it, we propose three different approaches. One approach is based on solving a convex approximation of the problem, obtained by augmenting the cost function with an entropy-like term to relax the nonconvex constraints. We show that the approximation error, for which we give an upper bound, can be made small enough for practical use. An alternative approach is to solve the classical NFP, that is, without the cycle constraints, and solve a separate optimization problem afterwards in order to recover the actual flows satisfying the cycle constraints; the solution of this separate problem maximizes the individual entropies of the cycles. The third approach is based on replacing the nonconvex constraint set with a convex inner approximation, which yields a suboptimal solution for the cyclic networks with each edge belonging to at most two cycles. We validate the practical usefulness of the theoretical results through numerical examples, in which we study the standard test systems for water and electric power distribution networks.

中文翻译：

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循环约束下网络流量问题的凸松弛

在本文中，我们考虑了在无变量流网络中优化流的问题，该流网络对节点变量具有附加的非凸循环约束。这种限制出现在几种应用中，包括电力，水分配和天然气网络。此问题是最低成本网络流问题（NFP）的非凸版本，为解决此问题，我们提出了三种不同的方法。一种方法是基于求解问题的凸近似值，该凸近似值是通过用类似于熵的项扩展成本函数来放松非凸约束而获得的。我们表明，可以给出逼近误差的上限，可以将其减小到足以实际使用的程度。一种替代方法是解决经典NFP，即没有周期限制，然后解决一个单独的优化问题，以恢复满足周期约束的实际流量；这个独立问题的解决方案最大化了循环的各个熵。第三种方法是基于用凸内部近似替换非凸约束集，这为循环网络产生了次优解，每个边缘最多包含两个循环。我们通过数值示例验证了理论结果的实用性，其中我们研究了水和电力分配网络的标准测试系统。第三种方法是基于用凸内部近似替换非凸约束集，这为循环网络产生了次优解，每个边最多包含两个循环。我们通过数值示例验证了理论结果的实用性，其中我们研究了水和电力分配网络的标准测试系统。第三种方法是基于用凸内部近似替换非凸约束集，这为循环网络产生了次优解，每个边缘最多包含两个循环。我们通过数值示例验证了理论结果的实用性，其中我们研究了水和电力分配网络的标准测试系统。