Linear programming (LP) is an extremely useful tool and has been successfully applied to solve various problems in a wide range of areas, including operations research, engineering, economics, or even more abstract mathematical areas such as combinatorics. It is also used in many machine learning applications, such as $\ell_1$-regularized SVMs, basis pursuit, nonnegative matrix factorization, etc. Interior Point Methods (IPMs) are one of the most popular methods to solve LPs both in theory and in practice. Their underlying complexity is dominated by the cost of solving a system of linear equations at each iteration. In this paper, we consider \emph{infeasible} IPMs for the special case where the number of variables is much larger than the number of constraints. Using tools from Randomized Linear Algebra, we present a preconditioning technique that, when combined with the Conjugate Gradient iterative solver, provably guarantees that infeasible IPM algorithms (suitably modified to account for the error incurred by the approximate solver), converge to a feasible, approximately optimal solution, without increasing their iteration complexity. Our empirical evaluations verify our theoretical results on both real-world and synthetic data.

中文翻译：

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使用随机线性代数加速线性规划

线性规划 (LP) 是一种非常有用的工具，已成功应用于解决广泛领域的各种问题，包括运筹学、工程、经济学，甚至更抽象的数学领域，如组合学。它也用于许多机器学习应用程序，例如 $\ell_1$-regularized SVM、基追踪、非负矩阵分解等。 内点方法 (IPM) 是解决 LP 的最流行的方法之一，无论是在理论上还是在实践中实践。它们的潜在复杂性取决于在每次迭代中求解线性方程组的成本。在本文中，我们针对变量数量远大于约束数量的特殊情况考虑 \emph{infeasible} IPM。使用来自随机线性代数的工具，我们提出了一种预处理技术，当与共轭梯度迭代求解器结合使用时，可证明保证不可行的 IPM 算法（适当修改以解决近似求解器引起的错误）收敛到可行的、近似最优的解决方案，而不会增加它们的迭代复杂。我们的实证评估验证了我们在现实世界和合成数据上的理论结果。