用于大规模最优传输问题的多尺度原始-对偶内点松弛方法
A Multiscale Primal-Dual Interior-Point Relaxation Method for Large-Scale Optimal Transport Problems
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中文总结 AI 辅助
针对大规模最优传输问题，提出多尺度原始-对偶内点松弛方法，通过多尺度框架与稀疏子问题求解降低内存需求，实验表明其效率与扩展性优于现有求解器。
中文摘要 AI 辅助
大规模最优传输（OT）问题涉及数量庞大的传输变量，会导致内存和计算成本过高。为应对这些挑战，我们提出了多尺度原始-对偶内点松弛方法（MSIPRM）。多尺度外层框架构建了一系列逐步精细化的标准OT问题层级。在每一层级，OT问题会在一系列自适应精细化的活动集上求解，这些活动集基于前一层级的解支撑集初始化。这会生成一系列紧密相关的稀疏子问题，从而大幅降低内存需求。原始-对偶内点松弛方法（IPRM）作为每个稀疏子问题的内层求解器，由于IPRM不需要严格的内点迭代，因此可以直接将前一个子问题的解作为热启动。为了高效获取牛顿方向，我们求解了由法方程推导得到的简化舒尔补系统。此外，我们还基于IPRM得到的近似解开发了一种有效的支撑识别策略。我们对舒尔补矩阵建立了条件数估计，并分析了算法的全局和局部收敛特性。针对大规模测试问题的数值实验证明了MSIPRM的计算效率和可扩展性，且表明其性能优于现有求解器。特别地，MSIPRM可处理完整表述包含数万亿个传输变量的实例。
英文摘要
Large-scale optimal transport (OT) problems involve a vast number of transport variables, leading to prohibitive memory and computational costs. To address these challenges, we propose a multiscale primal-dual interior-point relaxation method (MSIPRM). The multiscale outer framework constructs a hierarchy of standard OT problems at progressively finer levels. At each level, the OT problem is solved over a sequence of adaptively refined active sets initialized based on the solution support at the previous level. This yields a sequence of closely related sparse subproblems, thereby substantially reducing memory requirements. The primal-dual interior-point relaxation method (IPRM) serves as the inner solver for each sparse subproblem. Since IPRM does not require strictly interior iterates, it can readily use the solution of the previous subproblem as a warm start. To efficiently obtain the Newton direction, we solve a reduced Schur complement system derived from the normal equations. Furthermore, we develop an effective support-identification strategy based on the approximate solutions obtained by IPRM. We establish condition number estimates for the Schur complement matrices and analyze the global and local convergence properties of the algorithm. Numerical experiments on large-scale test problems demonstrate the computational efficiency and scalability of MSIPRM and show that it compares favorably with existing solvers. In particular, MSIPRM can handle instances whose full formulations contain trillions of transport variables.
