This paper presents fast first-order methods for solving linear programs (LP) approximately. We adapt online linear programming algorithms to offline LPs and obtain algorithms that avoid any matrix multiplication. We also introduce a variable-duplication technique that copies each variable $K$ times and reduces optimality gap and constraint violation by a factor of $\sqrt{K}$. Furthermore, we show how online algorithms can be effectively integrated into sifting, a column generation scheme for large-scale LPs. Numerical experiments demonstrate that our methods can serve as either an approximate direct solver, or an initialization subroutine for exact LP solving.

翻译：本文提出了求解线性规划问题的快速一阶方法，用于近似求解线性规划。我们将在线线性规划算法适配到离线线性规划问题中，并获得了无需任何矩阵乘法的算法。我们还引入了一种变量复制技术，该技术将每个变量复制 $K$ 次，并使得最优性间隙和约束违反度降低 $\sqrt{K}$ 倍。此外，我们展示了在线算法如何有效集成到筛选法中——一种用于大规模线性规划的列生成方案。数值实验表明，我们的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的初始化子程序。