[{"article_processing_charge":"Yes (in subscription journal)","OA_type":"hybrid","quality_controlled":"1","_id":"22607","title":"Duality theory in linear optimization and its extensions -- formally verified","abstract":[{"text":"Farkas established that a system of linear inequalities has a solution if and only if we cannot obtain a contradiction by taking a linear combination of the inequalities. We state and formally prove several Farkas-like theorems over linearly ordered fields in Lean 4. Furthermore, we extend duality theory to the case when some coefficients are allowed to take \"infinite values\".\r\nCode: https://github.com/madvorak/duality/tree/v3.2.0","lang":"eng"}],"type":"journal_article","keyword":["Farkas lemma","linear programming","extended reals","calculus of inductive constructions"],"related_material":{"record":[{"relation":"earlier_version","id":"20071","status":"public"}]},"department":[{"_id":"VlKo"},{"_id":"GradSch"}],"supplementarymaterial":"no","year":"2026","intvolume":"         2","status":"public","has_accepted_license":"1","volume":2,"mathsc":["68V20","15A39","90C05"],"researchdata_availability":"no","day":"13","language":[{"iso":"eng"}],"publication_identifier":{"eissn":["3117-4604"]},"arxiv":1,"publication_status":"published","publisher":"EPI Sciences","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","acknowledgement":"We would like to thank David Bartl and Jasmin Blanchette for frequent consultations.\r\nWe would also like to express gratitude to Henrik Böving for a help with generalization\r\nfrom extended rationals to extended linearly ordered fields and to Andrew Yang for the\r\nproof of Finset.univ_sum_of_zero_when_not. We would also like to acknowledge Antoine\r\nChambert-Loir, Apurva Nakade, Yaël Dillies, Richard Copley, Edward van de Meent, Markus\r\nHimmel, Mario Carneiro, and Kevin Buzzard.","das_tickbox":"0","date_created":"2026-07-29T09:06:55Z","publication":"Annals of Formalized Mathematics","citation":{"mla":"Dvorak, Martin, and Vladimir Kolmogorov. “Duality Theory in Linear Optimization and Its Extensions -- Formally Verified.” <i>Annals of Formalized Mathematics</i>, vol. 2, 14253, EPI Sciences, 2026, doi:<a href=\"https://doi.org/10.46298/afm.14253\">10.46298/afm.14253</a>.","apa":"Dvorak, M., &#38; Kolmogorov, V. (2026). Duality theory in linear optimization and its extensions -- formally verified. <i>Annals of Formalized Mathematics</i>. EPI Sciences. <a href=\"https://doi.org/10.46298/afm.14253\">https://doi.org/10.46298/afm.14253</a>","ama":"Dvorak M, Kolmogorov V. Duality theory in linear optimization and its extensions -- formally verified. <i>Annals of Formalized Mathematics</i>. 2026;2. doi:<a href=\"https://doi.org/10.46298/afm.14253\">10.46298/afm.14253</a>","chicago":"Dvorak, Martin, and Vladimir Kolmogorov. “Duality Theory in Linear Optimization and Its Extensions -- Formally Verified.” <i>Annals of Formalized Mathematics</i>. EPI Sciences, 2026. <a href=\"https://doi.org/10.46298/afm.14253\">https://doi.org/10.46298/afm.14253</a>.","ista":"Dvorak M, Kolmogorov V. 2026. Duality theory in linear optimization and its extensions -- formally verified. Annals of Formalized Mathematics. 2, 14253.","short":"M. Dvorak, V. Kolmogorov, Annals of Formalized Mathematics 2 (2026).","ieee":"M. Dvorak and V. Kolmogorov, “Duality theory in linear optimization and its extensions -- formally verified,” <i>Annals of Formalized Mathematics</i>, vol. 2. EPI Sciences, 2026."},"ddc":["500"],"corr_author":"1","PlanS_conform":"1","month":"03","doi":"10.46298/afm.14253","OA_place":"publisher","article_type":"original","author":[{"last_name":"Dvorak","full_name":"Dvorak, Martin","first_name":"Martin","id":"40ED02A8-C8B4-11E9-A9C0-453BE6697425","orcid":"0000-0001-5293-214X"},{"first_name":"Vladimir","full_name":"Kolmogorov, Vladimir","last_name":"Kolmogorov","id":"3D50B0BA-F248-11E8-B48F-1D18A9856A87"}],"date_published":"2026-03-13T00:00:00Z","date_updated":"2026-07-29T10:50:17Z","oa_version":"Published Version","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"external_id":{"arxiv":["2409.08119"]},"article_number":"14253"}]