In this paper, we study fast first-order algorithms that approximately solve linear programs (LPs). More specifically, we apply algorithms from online linear programming to offline LPs and derive algorithms that are free of any matrix multiplication. To further improve the applicability of the proposed methods, we propose a variable-duplication technique that achieves $\mathcal{O}(\sqrt{mn/K})$ optimality gap by copying each variable $K$ times. Moreover, we identify that online algorithms can be efficiently incorporated into a column generation framework for large-scale LPs. Finally, numerical experiments show that our proposed methods can be applied either as an approximate direct solver or as an initialization subroutine in frameworks of exact LP solving.

翻译：本文研究近似求解线性规划问题的快速一阶算法。具体而言，我们将在线线性规划算法应用于离线线性规划问题，推导出无需任何矩阵乘法的算法。为进一步提升所提方法的适用性，我们提出一种变量复制技术，通过将每个变量复制 $K$ 次实现 $\mathcal{O}(\sqrt{mn/K})$ 的最优性间隙。此外，我们发现在线算法可有效融入大规模线性规划的列生成框架中。最后，数值实验表明，所提方法既可作为近似直接求解器使用，也可作为精确线性规划求解框架中的初始化子程序。