{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T15:29:06Z","timestamp":1787239746481,"version":"build-2736575974"},"reference-count":7,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM Rev."],"published-print":{"date-parts":[[1991,6]]},"abstract":"<jats:p>The projective scaling algorithm in Karmarkar [Combinatorica, 4 (1984), pp. 373\u2013395] and Dantzig\u2019s simplex method (see [Linear Programming and Extensions, Princeton University Press, 1963]) are usually thought of as fundamentally different approaches to linear programming. When viewed in Dantzig\u2019s column space geometry, however, both algorithms turn out to be iteratively reweighted least squares methods. The projective scaling algorithm (and the affine scaling method in Dikin [Soviet Math. Dokl., 8 (1967), pp. 674\u2013675]) can then be derived as a natural generalization of the simplex method. This derivation shows how the dual variables arise in the interior methods. The insight is essentially geometric; suitable figures are provided.<\/jats:p>","DOI":"10.1137\/1033049","type":"journal-article","created":{"date-parts":[[2005,3,7]],"date-time":"2005-03-07T02:21:47Z","timestamp":1110162107000},"page":"220-237","source":"Crossref","is-referenced-by-count":27,"title":["The Simplex and Projective Scaling Algorithms as Iteratively Reweighted Least Squares Methods"],"prefix":"10.1137","volume":"33","author":[{"given":"Richard E.","family":"Stone","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Craig A.","family":"Tovey","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,7,18]]},"reference":[{"key":"R1","unstructured":"V. Chandru, B. Kochar,  A class of algorithms for linear programming, Research Memorandum, 85-14, School of Industrial Engineering, Purdue University, West Lafayette, IN,  1986"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1515\/9781400884179"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1016\/0167-6377(82)90043-8"},{"key":"R4","first-page":"674","volume":"8","author":"Dikin I. I.","year":"1967","journal-title":"Soviet Math. Dokl."},{"key":"R5","volume-title":"Breakthrough in problem solving","author":"Gleick J.","year":"1984"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1007\/BF02579150"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1007\/BF01840454"}],"container-title":["SIAM Review"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/1033049","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T14:48:40Z","timestamp":1787237320000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/1033049"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1991,6]]},"references-count":7,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1991,6]]}},"alternative-id":["10.1137\/1033049"],"URL":"https:\/\/doi.org\/10.1137\/1033049","relation":{},"ISSN":["0036-1445","1095-7200"],"issn-type":[{"value":"0036-1445","type":"print"},{"value":"1095-7200","type":"electronic"}],"subject":[],"published":{"date-parts":[[1991,6]]}}}