We present an efficient cutting-plane based approach to exactly solve a directed fixed charge network design (DFCND) problem, wherein the valid inequalities to the problem are generated using the polar duality approach. The biggest challenge in using this approach arises in constructing the polar dual of the problem. This would require enumerating all the extreme points of the convex hull of DFCND, which is computationally impractical for any instance of a reasonable size. Moreover, the resulting polar dual would be too large to solve efficiently, which is required at every iteration of the cutting-plane algorithm. The novelty of our solution approach lies in suggesting a way to circumvent this challenge by instead generating the violated facets, using polar duality, of the smaller substructures involving only a small subset of constraints and variables, obtained from 2-, 3-and 4-partitions of the underlying graph. For problem instances based on sparse graphs with zero flow costs, addition of these inequalities closes more than 20% of the optimality gap remaining after the addition of the knapsack cover inequalities used in the literature. This allows us to solve the problem instances in less than 400 seconds, on average, which otherwise take around 1,000 seconds with the addition of only the knapsack cover inequalities, and around 4 hours for the Cplex MIP solver at its default setting.

我们提出了一种有效的基于切割平面的方法来精确解决定向固定电荷网络设计 (DFCND) 问题，其中使用极性对偶方法生成问题的有效不等式。使用这种方法的最大挑战在于构建问题的极对偶。这将需要枚举 DFCND 凸包的所有极值点，这对于任何合理大小的实例在计算上都是不切实际的。此外，生成的极对偶太大而无法有效求解，这在切割平面算法的每次迭代中都是必需的。我们的解决方案方法的新颖之处在于提出了一种规避这一挑战的方法，即通过使用极性二元性生成违反面，较小的子结构仅涉及约束和变量的一小部分子集，从底层图的 2、3 和 4 分区获得。对于基于具有零流量成本的稀疏图的问题实例，在添加文献中使用的背包覆盖不等式之后，这些不等式的添加可以缩小剩余的最优性差距的 20% 以上。这使我们能够在平均不到 400 秒的时间内解决问题实例，否则在仅添加背包覆盖不等式的情况下需要大约 1,000 秒，而在默认设置下，Cplex MIP 求解器大约需要 4 小时。加上这些不等式，在加上文献中使用的背包盖不等式后，剩余的最优性差距缩小了 20% 以上。这使我们能够在平均不到 400 秒的时间内解决问题实例，否则在仅添加背包覆盖不等式的情况下需要大约 1,000 秒，而在默认设置下，Cplex MIP 求解器大约需要 4 小时。加上这些不等式，在加上文献中使用的背包盖不等式后，剩余的最优性差距缩小了 20% 以上。这使我们能够在平均不到 400 秒的时间内解决问题实例，否则在仅添加背包覆盖不等式的情况下需要大约 1,000 秒，而在默认设置下，Cplex MIP 求解器大约需要 4 小时。