Ahmad Abdi, Gérard Cornuéjols, Bertrand Guenin, Levent Tunçel
{"title":"并矢线性规划及其扩展。","authors":"Ahmad Abdi, Gérard Cornuéjols, Bertrand Guenin, Levent Tunçel","doi":"10.1007/s10107-024-02146-4","DOIUrl":null,"url":null,"abstract":"<p><p>A rational number is <i>dyadic</i> if it has a finite binary representation <math><mrow><mi>p</mi> <mo>/</mo> <msup><mn>2</mn> <mi>k</mi></msup> </mrow> </math> , where <i>p</i> is an integer and <i>k</i> is a nonnegative integer. Dyadic rationals are important for numerical computations because they have an exact representation in floating-point arithmetic on a computer. A vector is <i>dyadic</i> if all its entries are dyadic rationals. We study the problem of finding a dyadic optimal solution to a linear program, if one exists. We show how to solve dyadic linear programs in polynomial time. We give bounds on the size of the support of a solution as well as on the size of the denominators. We identify properties that make the solution of dyadic linear programs possible: closure under addition and negation, and density, and we extend the algorithmic framework beyond the dyadic case.</p>","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":"213 1-2","pages":"473-516"},"PeriodicalIF":2.5000,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12402050/pdf/","citationCount":"0","resultStr":"{\"title\":\"Dyadic linear programming and extensions.\",\"authors\":\"Ahmad Abdi, Gérard Cornuéjols, Bertrand Guenin, Levent Tunçel\",\"doi\":\"10.1007/s10107-024-02146-4\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p><p>A rational number is <i>dyadic</i> if it has a finite binary representation <math><mrow><mi>p</mi> <mo>/</mo> <msup><mn>2</mn> <mi>k</mi></msup> </mrow> </math> , where <i>p</i> is an integer and <i>k</i> is a nonnegative integer. Dyadic rationals are important for numerical computations because they have an exact representation in floating-point arithmetic on a computer. A vector is <i>dyadic</i> if all its entries are dyadic rationals. We study the problem of finding a dyadic optimal solution to a linear program, if one exists. We show how to solve dyadic linear programs in polynomial time. We give bounds on the size of the support of a solution as well as on the size of the denominators. We identify properties that make the solution of dyadic linear programs possible: closure under addition and negation, and density, and we extend the algorithmic framework beyond the dyadic case.</p>\",\"PeriodicalId\":18297,\"journal\":{\"name\":\"Mathematical Programming\",\"volume\":\"213 1-2\",\"pages\":\"473-516\"},\"PeriodicalIF\":2.5000,\"publicationDate\":\"2025-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12402050/pdf/\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Mathematical Programming\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s10107-024-02146-4\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"2024/10/3 0:00:00\",\"PubModel\":\"Epub\",\"JCR\":\"Q2\",\"JCRName\":\"COMPUTER SCIENCE, SOFTWARE ENGINEERING\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Programming","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10107-024-02146-4","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"2024/10/3 0:00:00","PubModel":"Epub","JCR":"Q2","JCRName":"COMPUTER SCIENCE, SOFTWARE ENGINEERING","Score":null,"Total":0}
A rational number is dyadic if it has a finite binary representation , where p is an integer and k is a nonnegative integer. Dyadic rationals are important for numerical computations because they have an exact representation in floating-point arithmetic on a computer. A vector is dyadic if all its entries are dyadic rationals. We study the problem of finding a dyadic optimal solution to a linear program, if one exists. We show how to solve dyadic linear programs in polynomial time. We give bounds on the size of the support of a solution as well as on the size of the denominators. We identify properties that make the solution of dyadic linear programs possible: closure under addition and negation, and density, and we extend the algorithmic framework beyond the dyadic case.
期刊介绍:
Mathematical Programming publishes original articles dealing with every aspect of mathematical optimization; that is, everything of direct or indirect use concerning the problem of optimizing a function of many variables, often subject to a set of constraints. This involves theoretical and computational issues as well as application studies. Included, along with the standard topics of linear, nonlinear, integer, conic, stochastic and combinatorial optimization, are techniques for formulating and applying mathematical programming models, convex, nonsmooth and variational analysis, the theory of polyhedra, variational inequalities, and control and game theory viewed from the perspective of mathematical programming.