# Using Optimization to Solve Positive LPs Faster in Parallel

arXiv: https://arxiv.org/abs/1407.1925 · HTML: https://mlsanity.com/paper/1407.1925/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Jul 2014.

## Built on by (most cited first, 2 of 2)

- [Online Bidding Algorithms for Return-on-Spend Constrained Advertisers](https://mlsanity.com/paper/2208.13713/) (2022)
- [LinSATNet: The Positive Linear Satisfiability Neural Networks](https://mlsanity.com/paper/2407.13917/) (2024)

## Abstract (arXiv)

Positive linear programs (LP), also known as packing and covering linear programs, are an important class of problems that bridges computer science, operations research, and optimization. Despite the consistent efforts on this problem, all known nearly-linear-time algorithms require tildeO(varepsilon-4) iterations to converge to 1± varepsilon approximate solutions. This varepsilon-4 dependence has not been improved since 1993, and limits the performance of parallel implementations for such algorithms. Moreover, previous algorithms and their analyses rely on update steps and convergence arguments that are combinatorial in nature and do not seem to arise naturally from an optimization viewpoint. In this paper, we leverage new insights from optimization theory to construct a novel algorithm that breaks the longstanding varepsilon-4 barrier. Our algorithm has a simple analysis and a clear motivation. Our work introduces a number of novel techniques, such as the combined application of gradient descent and mirror descent, and a truncated, smoothed version of the standard multiplicative weight update, which may be of independent interest.