{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:34:44Z","timestamp":1787333684122,"version":"build-2736575974"},"reference-count":25,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Optim."],"published-print":{"date-parts":[[1998,5]]},"abstract":"<jats:p>This work concerns primal--dual interior-point methods for semidefinite programming (SDP) that use a search direction originally proposed by Helmberg et al. [SIAM J. Optim., 6 (1996), pp. 342--361] and Kojima, Shindoh, and Hara [SIAM J. Optim., 7 (1997), pp. 86--125.] and recently rediscovered by Monteiro [SIAM J. Optim., 7 (1997), pp. 663--678] in a more explicit form. In analyzing these methods, a number of basic equalities and inequalities were developed in [Kojima, Shindoh, and Hara] and also in [Monteiro] through different means and in different forms.<\/jats:p>\n                  <jats:p>In this paper, we give a concise derivation of the key equalities and inequalities for complexity analysis along the exact line used in linear programming (LP), producing basic relationships that have compact forms almost identical to their counterparts in LP. We also introduce a new formulation of the central path and variable-metric measures of centrality. These results provide convenient tools for deriving polynomiality results for primal--dual algorithms extended from LP to SDP using the aforementioned and related search directions. We present examples of such extensions, including the long-step infeasible-interior-point algorithm of Zhang [SIAM J. Optim., 4 (1994), pp. 208--227].<\/jats:p>","DOI":"10.1137\/s1052623495296115","type":"journal-article","created":{"date-parts":[[2003,6,11]],"date-time":"2003-06-11T11:12:06Z","timestamp":1055329926000},"page":"365-386","source":"Crossref","is-referenced-by-count":161,"title":["On Extending Some Primal--Dual Interior-Point Algorithms From Linear Programming to Semidefinite Programming"],"prefix":"10.1137","volume":"8","author":[{"given":"Yin","family":"Zhang","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,7,31]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1137\/0805002"},{"key":"R2","first-page":"1","volume":"11","author":"Todd M.","year":"1999","journal-title":"Optim. Methods Softw."},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1007\/BF02614432"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611970777"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1137\/0806020"},{"key":"R6","doi-asserted-by":"crossref","unstructured":"R. A. Horn and C. R. Johnson,\n                      Topics in Matrix Analysis\n                      , Cambridge University Press, 1991.","DOI":"10.1017\/CBO9780511840371"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1137\/0331064"},{"key":"R8","unstructured":"M. Kojima, Private communication."},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1007\/BF01582151"},{"key":"R10","doi-asserted-by":"crossref","unstructured":"M. Kojima, S. Mizuno, and A. Yoshise,\n                      A primal\u2013dual interior point method for linear programming\n                      , In Nimrod Megiddo, editor,\n                      Progress in mathematical programming, interior\u2010point and related methods\n                      , pp. 29\u201347. Springer\u2010Verlag, New York, 1989.","DOI":"10.1007\/978-1-4613-9617-8_2"},{"key":"R11","unstructured":"M. Kojima, S. Shindoh, and S. Hara,\n                      Interior\u2010point methods for the monotone linear complementarity problem in symmetric matrices\n                      , Research Report #B\u2010282, Dept. of Mathematical and Computing Sciences, Tokyo Institute of Technology 2\u201012\u20101 Oh\u2010Okayama, Meguro\u2010ku, Tokyo 152, Japan, April 1994 (revised April 1995)."},{"key":"R12","first-page":"147","volume":"22","author":"Kojima M.","year":"1997","journal-title":"Acta Math. Vietnam."},{"key":"R13","unstructured":"M. Kojima, M. Shida, and S. Shindoh,\n                      Global and local convergence of predictor\u2010corrector infeasible\u2010interior\u2010point algorithms for semidefinite programs\n                      , Research Report #B\u2010305, Dept. of Mathematical and Computing Sciences, Tokyo Institute of Technology 2\u201012\u20101 Oh\u2010Okayama, Meguro\u2010ku, Tokyo 152, Japan, October, 1995."},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1287\/moor.18.4.964"},{"key":"R15","doi-asserted-by":"crossref","unstructured":"E.de Klerk, C.Roos, T.Terlaky, On primal\u2010dual path\u2010following algorithms for semidefinite programming, Appl. Optim., Vol. 13, Kluwer Acad. Publ., Boston, MA, 1998, 137\u20131572000b:90049","DOI":"10.1007\/978-1-4757-2878-1_11"},{"key":"R16","unstructured":"Y. E. Nesterov and A. S. Nemirovskii,\n                      Interior Point Methods in Convex Programming\u2013Theory and Applications\n                      , Society for Industrial and Applied Mathematics, Philadelphia, 1994."},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1137\/S1052623495290209"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1137\/0609021"},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1137\/0802007"},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1137\/S1052623495294955"},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1007\/BF02614433"},{"key":"R22","unstructured":"F. Rendl, R. J. Vanderbei, and H. Wolkowicz,\n                      Interior\u2010point method for max\u2010min eigenvalue problems\n                      , Manuscript, Program in Statistics and Operations Research, Princeton University, 1993."},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1137\/1038003"},{"key":"R24","doi-asserted-by":"publisher","DOI":"10.1007\/BF01585558"},{"key":"R25","doi-asserted-by":"publisher","DOI":"10.1137\/0804012"}],"container-title":["SIAM Journal on Optimization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/S1052623495296115","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:17:01Z","timestamp":1787332621000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/S1052623495296115"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1998,5]]},"references-count":25,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1998,5]]}},"alternative-id":["10.1137\/S1052623495296115"],"URL":"https:\/\/doi.org\/10.1137\/s1052623495296115","relation":{},"ISSN":["1052-6234","1095-7189"],"issn-type":[{"value":"1052-6234","type":"print"},{"value":"1095-7189","type":"electronic"}],"subject":[],"published":{"date-parts":[[1998,5]]}}}