{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/on-the-optimal-objective-value-of-random","title":"On the optimal objective value of random linear programs","arxiv_id":"2401.17530","date":"2024-01-31","proceeding":null,"authors":["Marzieh Bakhshi","James Ostrowski","Konstantin Tikhomirov"],"abstract":"We consider the problem of maximizing $\\langle c,x \\rangle$ subject to the constraints $Ax \\leq \\mathbf{1}$, where $x\\in R^n$, $A$ is an $m\\times n$ matrix with mutually independent centered subgaussian entries of unit variance, and $c$ is a cost vector of unit Euclidean length. In the asymptotic regime $n\\to\\infty$, $\\frac{m}{n}\\to\\infty$, and under some mild assumptions on $c$, we prove that the optimal objective value $z^*$ of the linear program satisfies $$ \\lim\\limits_{n\\to\\infty}\\sqrt{2\\log(m/n)}\\,z^*= 1\\quad \\mbox{almost surely}. $$ We provide numerical experiments as supporting data for the theoretical predictions. Further, we carry out numerical studies of the limiting distribution and the standard deviation of $z^*$.","url_abs":"https://arxiv.org/abs/2401.17530v3","url_pdf":"https://arxiv.org/pdf/2401.17530v3.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"on-the-optimal-objective-value-of-random","repo_url":"https://github.com/marzb93/randomlinearprogram","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}