行极 LP-Newton 线性规划算法与 Corral 修复
Row-Polar LP-Newton for Linear Programming with Corral Repair
- Shandong University（山东大学）
- Seikei University（成蹊大学）
- Tokyo University of Science（东京理科大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出带 Corral 修复的行极 LP-Newton 算法，通过重用终止 corral 加速线性规划求解，在多数测试中显著快于原算法和 HiGHS。
AI中文摘要：
LP-Newton 通过一系列最近点问题求解线性规划。给定不等式形式线性规划的一个内点可行解，我们构造由原点及其归一化约束行构成的紧致包络。极 LP-Newton（P-LPN）沿目标射线行进至该行极包络的边界，在此处恢复原始-对偶最优解或证明无界性的衰退方向。对于有理行极数据，我们证明了外部迭代次数以维度二次和二进制输入长度线性为界（不包括内部 Wolfe 计算）。在每次外部迭代时重启 Wolfe 会丢弃其终止的 corral，即使包络未改变且下一个目标位于同一条射线上。Corral 修复 P-LPN（CR-P-LPN）转而修复该 corral 并用它启动下一次投影。对整个包络的验证在精确算术下保留了 P-LPN 的投影、外部目标和线性规划结论。Julia 和 MATLAB 中的实验展示了修复在何处有帮助。在共同初始化后，180 个单线性规划测试中，Julia 中 P-LPN 的中位时间为相应 CR-P-LPN 时间的 1.62 倍，MATLAB 中为 1.55 倍。在三个具有 5,000 至 100,000 行的问题上，相应的几何平均因子分别为 2.03 和 1.46。一项独立的端到端研究包括对 19 个应用衍生线性规划的初始化。在每种语言的 19 个工作负载中，CR-P-LPN 在 18 个上的中位总时间低于更快的冷启动 HiGHS 模式（对偶单纯形法或带交叉的内点法）。HiGHS 时间与 CR-P-LPN 时间之比的几何平均值为 4.93 和 3.06。组件比较表明终止 corral 重用是有用的组件，而仅候选优先排序未显示一致收益。修复并非总是有益：在 Klee-Minty 测试上它较慢，而在保留的 Netlib 模型上，作者实现的单纯形法更快。
英文摘要：
LP-Newton solves a linear program through a sequence of nearest-point problems. Given an interior feasible point for an inequality-form LP, we construct the compact hull formed by the origin and its normalized constraint rows. Polar LP-Newton (P-LPN) follows the objective ray to the boundary of this row-polar hull, where it recovers a primal-dual optimum or a recession direction proving unboundedness. For rational row-polar data, we prove an outer-iteration bound quadratic in dimension and linear in binary input length, excluding inner Wolfe work. Restarting Wolfe at every outer iteration discards its terminal corral even though the hull is unchanged and the next target lies on the same ray. Corral-repair P-LPN (CR-P-LPN) instead repairs that corral and uses it to start the next projection. Verification over the full hull preserves P-LPN's projections, outer targets, and LP conclusion in exact arithmetic. Experiments in Julia and MATLAB show where repair helps. After common initialization, median P-LPN times across 180 single-LP tests are 1.62 times the corresponding CR-P-LPN times in Julia and 1.55 times in MATLAB. On three problems with 5,000 to 100,000 rows, the corresponding geometric-mean factors are 2.03 and 1.46. A separate end-to-end study includes initialization on 19 application-derived LPs. CR-P-LPN has lower median total time than the faster cold HiGHS mode, either dual simplex or the interior-point method with crossover, on 18 of the 19 workloads in each language. The geometric means of the ratios of HiGHS time to CR-P-LPN time are 4.93 and 3.06. Component comparisons identify terminal-corral reuse as the useful component, whereas candidate-first ordering alone shows no consistent gain. Repair is not always beneficial: it is slower on Klee-Minty tests, and an author-implemented simplex is faster on the retained Netlib models.
