{"title":"A generalization of the linear complementarity problem","authors":"Richard W. Cottle , George B. Dantzig","doi":"10.1016/S0021-9800(70)80010-2","DOIUrl":null,"url":null,"abstract":"<div><p>The linear complementarity problem: find <em>z</em>∈<em>R<sup>p</sup></em> satisfying <span><math><mtable><mtr><mtd><mi>w</mi><mo>=</mo><mi>q</mi><mo>+</mo><mi>M</mi><mi>z</mi></mtd></mtr><mtr><mtd><mi>w</mi><mo>⩾</mo><mn>0</mn><mo>,</mo><mi>z</mi><mo>⩾</mo><mn>0</mn><mo>(</mo><mo>LCP</mo><mo>)</mo></mtd></mtr><mtr><mtd><msup><mo>z</mo><mi>T</mi></msup><mi>w</mi><mo>=</mo><mn>0</mn></mtd></mtr></mtable></math></span> is generalized to a problem in which the matrix <em>M</em> is not square. A solution technique similar to <span>C. E. Lemke's (1965)</span> method for solving (LCP) is given. The method is discussed from a graph-theoretic viewpoint and closely parallels a proof of Sperner's lemma by <span>D. I. A. Cohen (1967)</span> and some work of <span>H. Scarf (1967)</span> on approximating fixed points of a continuous mapping of a simplex into itself.</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"8 1","pages":"Pages 79-90"},"PeriodicalIF":0.0000,"publicationDate":"1970-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(70)80010-2","citationCount":"171","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Combinatorial Theory","FirstCategoryId":"1085","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0021980070800102","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 171
Abstract
The linear complementarity problem: find z∈Rp satisfying is generalized to a problem in which the matrix M is not square. A solution technique similar to C. E. Lemke's (1965) method for solving (LCP) is given. The method is discussed from a graph-theoretic viewpoint and closely parallels a proof of Sperner's lemma by D. I. A. Cohen (1967) and some work of H. Scarf (1967) on approximating fixed points of a continuous mapping of a simplex into itself.
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