# Faster Randomized Interior Point Methods for Tall/Wide Linear Programs

arXiv: https://arxiv.org/abs/2209.08722 · HTML: https://mlsanity.com/paper/2209.08722/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Sep 2022.

## Built on by (most cited first, 3 of 3)

- [Generalization Bound and Learning Methods for Data-Driven Projections in Linear Programming](https://mlsanity.com/paper/2309.00203/) (2023)
- [Provably data-driven projection method for quadratic programming](https://mlsanity.com/paper/2509.04524/) (2025)
- [A Provably Accurate Randomized Sampling Algorithm for Logistic Regression](https://mlsanity.com/paper/2402.16326/) (2024)

## Abstract (arXiv)

Linear programming (LP) is an extremely useful tool which 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 ℓ1-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 both feasible and 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 iterative solvers such as Conjugate Gradient or Chebyshev Iteration, provably guarantees that 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.