从内点到角点：智能交叉
From an Interior Point to a Corner Point: Smart Crossover
AI总结：
本文提出了一种基于树结构的交叉方法，利用网络结构和生成树特性，提高线性规划求解效率。
摘要：
识别线性规划问题的最优基本可行解是混合整数规划和其他应用中的关键任务。交叉方法旨在从次优解（如内点法或一阶方法的输出）中推导出最优极点，是这一过程中的关键步骤。与起始方法相比，交叉方法在实际应用中经常成为主要的计算瓶颈。我们提出了一些方法来克服这一瓶颈，通过利用问题特征并实施定制策略。对于来自网络应用且具有网络结构的问题，我们利用问题的图结构和最优解的树结构。基于这些结构，我们提出了一种基于树的交叉方法，旨在通过识别附近的生成树结构来恢复基本解。对于一般的线性规划问题，我们提出通过识别最优面并利用基于内点方法提供的次优解的受控扰动来恢复最优基本解。我们证明了扰动问题的最优解是极点，其目标值至少与初始内点解的目标值一样好。计算实验显示，我们的方法在经典线性规划问题基准、网络流问题基准和最优运输问题上相比最先进的商业求解器实现了显著的速度提升。
英文摘要：
Identifying optimal basic feasible solutions to linear programming problems is a critical task for mixed integer programming and other applications. The crossover method, which aims at deriving an optimal extreme point from a suboptimal solution (the output of a starting method such as interior-point methods or first-order methods), is crucial in this process. This method, compared to the starting method, frequently represents the primary computational bottleneck in practical applications. We propose approaches to overcome this bottleneck by exploiting problem characteristics and implementing customized strategies. For problems arising from network applications and exhibiting network structures, we take advantage of the graph structure of the problem and the tree structure of the optimal solutions. Based on these structures, we propose a tree-based crossover method, aiming to recovering basic solutions by identifying nearby spanning tree structures. For general linear programs, we propose recovering an optimal basic solution by identifying the optimal face and employing controlled perturbations based on the suboptimal solution provided by interior-point methods. We prove that an optimal solution for the perturbed problem is an extreme point, and its objective value is at least as good as that of the initial interior point solution. Computational experiments show significant speed-ups achieved by our methods compared to state-of-the-art commercial solvers on classical linear programming problem benchmarks, network flow problem benchmarks, and optimal transport problems.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
