Various first order approaches have been proposed in the literature to solve Linear Programming (LP) problems, recently leading to practically efficient solvers for large-scale LPs. From a theoretical perspective, linear convergence rates have been established for first order LP algorithms, despite the fact that the underlying formulations are not strongly convex. However, the convergence rate typically depends on the Hoffman constant of a large matrix that contains the constraint matrix, as well as the right hand side, cost, and capacity vectors. We introduce a first order approach for LP optimization with a convergence rate depending polynomially on the circuit imbalance measure, which is a geometric parameter of the constraint matrix, and depending logarithmically on the right hand side, capacity, and cost vectors. This provides much stronger convergence guarantees. For example, if the constraint matrix is totally unimodular, we obtain polynomial-time algorithms, whereas the convergence guarantees for approaches based on primal-dual formulations may have arbitrarily slow convergence rates for this class. Our approach is based on a fast gradient method due to Necoara, Nesterov, and Glineur (Math. Prog. 2019); this algorithm is called repeatedly in a framework that gradually fixes variables to the boundary. This technique is based on a new approximate version of Tardos's method, that was used to obtain a strongly polynomial algorithm for combinatorial LPs (Oper. Res. 1986).

翻译：文献中提出了多种一阶方法用于求解线性规划问题，近期这些方法为大规模线性规划提供了实际高效的求解器。从理论角度看，尽管底层问题并非强凸，但一阶线性规划算法仍建立了线性收敛速率。然而，该收敛速率通常依赖于包含约束矩阵、右端项、成本向量和容量向量的一个大矩阵的Hoffman常数。我们提出了一种针对线性规划优化的一阶方法，其收敛速率与电路不平衡性度量（约束矩阵的几何参数）呈多项式关系，且与右端项、容量和成本向量呈对数关系。这提供了更强的收敛保证。例如，当约束矩阵为全幺模矩阵时，我们得到多项式时间算法，而基于原始-对偶公式的方法对此类问题可能具有任意慢的收敛速率。我们的方法基于Necoara、Nesterov和Glineur（《数学规划》2019年）提出的快速梯度法；该算法在逐步固定变量至边界的框架中被重复调用。此技术基于Tardos方法的近似新版本，后者曾用于获得组合线性规划的强多项式算法（《运筹学》1986年）。