Random Projections for Linear Programming

• 成果类型: Article
• 署名作者: Vu, Ky; Poirion, Pierre-Louis; Liberti, Leo
• 署名单位: Chinese University of Hong Kong; Huawei Technologies; Institut Polytechnique de Paris; Ecole Polytechnique
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X
• DOI: 10.1287/moor.2017.0894
• 发表日期: 2018
• 页码: 1051-1071
• 关键词: johnson-lindenstrauss

摘要

Random projections are random linear maps, sampled from appropriate distributions, which approximately preserve certain geometrical invariants so that the approximation improves as the dimension of the space grows. The well known Johnson-Lindenstrauss lemma states that there are random matrices with surprisingly few rows which approximately preserve pairwise Euclidean distances among a set of points. This is commonly used to speed up algorithms based on Euclidean distances. We prove that these matrices also preserve other quantities, such as the distance to a cone. We exploit this result to devise a probabilistic algorithm to approximately solve linear programs. We show that this algorithm can approximately solve very large randomly generated LP instances. We also showcase its application to an error correction coding problem.