{"language":[{"iso":"eng"}],"type":"preprint","external_id":{"arxiv":["2409.08119"]},"day":"12","_id":"20071","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2409.08119","open_access":"1"}],"arxiv":1,"doi":"10.48550/arXiv.2409.08119","keyword":["Farkas lemma","linear programming","extended reals","calculus of inductive constructions"],"publication_status":"submitted","author":[{"first_name":"Martin","last_name":"Dvorak","full_name":"Dvorak, Martin","orcid":"0000-0001-5293-214X","id":"40ED02A8-C8B4-11E9-A9C0-453BE6697425"},{"id":"3D50B0BA-F248-11E8-B48F-1D18A9856A87","full_name":"Kolmogorov, Vladimir","last_name":"Kolmogorov","first_name":"Vladimir"}],"date_created":"2025-07-23T11:21:52Z","article_processing_charge":"No","status":"public","year":"2024","acknowledgement":"We would like to thank David Bartl and Jasmin Blanchette for frequent consultations. We would also like to express gratitude to Andrew Yang for the proof of Finset.univ sum of zero when not and to Henrik B¨oving for a help with generalization from extended rationals to extended linearly ordered fields. We would also like to acknowledge Antoine Chambert-Loir, Apurva Nakade, Ya¨el Dillies, Richard Copley, Edward van de Meent, Markus Himmel, Mario Carneiro, and Kevin Buzzard.","month":"09","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","oa":1,"citation":{"mla":"Dvorak, Martin, and Vladimir Kolmogorov. “Duality Theory in Linear Optimization and Its Extensions -- Formally  Verified.” <i>ArXiv</i>, 2409.08119, doi:<a href=\"https://doi.org/10.48550/arXiv.2409.08119\">10.48550/arXiv.2409.08119</a>.","chicago":"Dvorak, Martin, and Vladimir Kolmogorov. “Duality Theory in Linear Optimization and Its Extensions -- Formally  Verified.” <i>ArXiv</i>, n.d. <a href=\"https://doi.org/10.48550/arXiv.2409.08119\">https://doi.org/10.48550/arXiv.2409.08119</a>.","ama":"Dvorak M, Kolmogorov V. Duality theory in linear optimization and its extensions -- formally  verified. <i>arXiv</i>. doi:<a href=\"https://doi.org/10.48550/arXiv.2409.08119\">10.48550/arXiv.2409.08119</a>","ista":"Dvorak M, Kolmogorov V. Duality theory in linear optimization and its extensions -- formally  verified. arXiv, 2409.08119.","short":"M. Dvorak, V. Kolmogorov, ArXiv (n.d.).","apa":"Dvorak, M., &#38; Kolmogorov, V. (n.d.). Duality theory in linear optimization and its extensions -- formally  verified. <i>arXiv</i>. <a href=\"https://doi.org/10.48550/arXiv.2409.08119\">https://doi.org/10.48550/arXiv.2409.08119</a>","ieee":"M. Dvorak and V. Kolmogorov, “Duality theory in linear optimization and its extensions -- formally  verified,” <i>arXiv</i>. ."},"abstract":[{"lang":"eng","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\"."}],"title":"Duality theory in linear optimization and its extensions -- formally  verified","department":[{"_id":"GradSch"},{"_id":"VlKo"}],"related_material":{"link":[{"description":"full version of all definitions, statement, and proofs","relation":"software","url":"https://github.com/madvorak/duality/tree/v3.2"}]},"date_updated":"2025-07-31T09:29:30Z","publication":"arXiv","date_published":"2024-09-12T00:00:00Z","OA_type":"green","corr_author":"1","article_number":"2409.08119","oa_version":"Preprint","OA_place":"repository"}