{"title":"线性优化及其扩展中的对偶理论 -- 正式验证","authors":"Martin Dvorak, Vladimir Kolmogorov","doi":"arxiv-2409.08119","DOIUrl":null,"url":null,"abstract":"Farkas established that a system of linear inequalities has a solution if and\nonly if we cannot obtain a contradiction by taking a linear combination of the\ninequalities. We state and formally prove several Farkas-like theorems over\nlinearly ordered fields in Lean 4. Furthermore, we extend duality theory to the\ncase when some coefficients are allowed to take ``infinite values''.","PeriodicalId":501286,"journal":{"name":"arXiv - MATH - Optimization and Control","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Duality theory in linear optimization and its extensions -- formally verified\",\"authors\":\"Martin Dvorak, Vladimir Kolmogorov\",\"doi\":\"arxiv-2409.08119\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Farkas established that a system of linear inequalities has a solution if and\\nonly if we cannot obtain a contradiction by taking a linear combination of the\\ninequalities. We state and formally prove several Farkas-like theorems over\\nlinearly ordered fields in Lean 4. Furthermore, we extend duality theory to the\\ncase when some coefficients are allowed to take ``infinite values''.\",\"PeriodicalId\":501286,\"journal\":{\"name\":\"arXiv - MATH - Optimization and Control\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-12\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Optimization and Control\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2409.08119\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Optimization and Control","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.08119","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Duality theory in linear optimization and its extensions -- formally verified
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''.