{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T10:42:15Z","timestamp":1740134535782,"version":"3.37.3"},"reference-count":31,"publisher":"Wiley","license":[{"start":{"date-parts":[[2014,1,1]],"date-time":"2014-01-01T00:00:00Z","timestamp":1388534400000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"funder":[{"DOI":"10.13039\/501100001868","name":"National Science Council","doi-asserted-by":"publisher","award":["NSC-102-2221-E-002-125-MY3"],"award-info":[{"award-number":["NSC-102-2221-E-002-125-MY3"]}],"id":[{"id":"10.13039\/501100001868","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Applied Mathematics"],"published-print":{"date-parts":[[2014]]},"abstract":"<jats:p>To solve an unconstrained nonlinear minimization problem, we propose an optimal algorithm (OA) as well as a globally optimal algorithm (GOA), by deflecting the gradient direction to the best descent direction at each iteration step, and with an optimal parameter being derived explicitly. An invariant manifold defined for the model problem in terms of a locally quadratic function is used to derive a purely iterative algorithm and the convergence is proven. Then, the rank-two updating techniques of BFGS are employed, which result in several novel algorithms as being faster than the steepest descent method (SDM) and the variable metric method (DFP). Six numerical examples are examined and compared with exact solutions, revealing that the new algorithms of OA, GOA, and the updated ones have superior computational efficiency and accuracy.<\/jats:p>","DOI":"10.1155\/2014\/324181","type":"journal-article","created":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T17:02:18Z","timestamp":1394643738000},"page":"1-14","source":"Crossref","is-referenced-by-count":2,"title":["Optimal Algorithms and the BFGS Updating Techniques for Solving Unconstrained Nonlinear Minimization Problems"],"prefix":"10.1155","volume":"2014","author":[{"given":"Chein-Shan","family":"Liu","sequence":"first","affiliation":[{"name":"Department of Civil Engineering, National Taiwan University, Taipei 106-17, Taiwan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","reference":[{"key":"1","doi-asserted-by":"publisher","DOI":"10.1093\/imanum\/8.1.141"},{"key":"2","doi-asserted-by":"publisher","DOI":"10.1093\/imanum\/13.3.321"},{"issue":"1","key":"3","doi-asserted-by":"crossref","first-page":"26","DOI":"10.1137\/S1052623494266365","volume":"7","year":"1997","journal-title":"SIAM Journal on Optimization"},{"issue":"1","key":"4","first-page":"275","volume":"36","year":"1999","journal-title":"SIAM Journal on Numerical Analysis"},{"key":"5","doi-asserted-by":"publisher","DOI":"10.1023\/A:1013708715892"},{"key":"6","doi-asserted-by":"publisher","DOI":"10.1023\/A:1014838419611"},{"key":"7","doi-asserted-by":"publisher","DOI":"10.1093\/imanum\/22.1.1"},{"key":"8","doi-asserted-by":"publisher","DOI":"10.1093\/imanum\/23.3.377"},{"key":"9","doi-asserted-by":"publisher","DOI":"10.1007\/0-387-24255-4_10"},{"issue":"2","key":"10","first-page":"149","volume":"24","year":"2006","journal-title":"Journal of Computational Mathematics"},{"key":"11","doi-asserted-by":"publisher","DOI":"10.1007\/s00245-001-0003-0"},{"key":"12","doi-asserted-by":"publisher","DOI":"10.1007\/s10589-007-9055-7"},{"key":"13","doi-asserted-by":"publisher","DOI":"10.1016\/j.aml.2007.05.002"},{"key":"14","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2010.05.002"},{"key":"15","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2008.07.016"},{"key":"16","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2010.01.052"},{"key":"17","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2008.05.012"},{"key":"19","doi-asserted-by":"publisher","DOI":"10.1093\/comjnl\/6.2.163"},{"key":"20","doi-asserted-by":"publisher","DOI":"10.1016\/S0020-7462(99)00027-X"},{"key":"21","doi-asserted-by":"publisher","DOI":"10.1093\/imamat\/6.3.222"},{"issue":"3","key":"22","doi-asserted-by":"crossref","first-page":"317","DOI":"10.1093\/comjnl\/13.3.317","volume":"13","year":"1970","journal-title":"The Computer Journal"},{"key":"23","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1970-0258249-6"},{"key":"24","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1970-0274029-X"},{"key":"25","doi-asserted-by":"publisher","DOI":"10.1137\/S1052623401383455"},{"key":"26","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-012-0522-2"},{"key":"28","doi-asserted-by":"publisher","DOI":"10.1093\/comjnl\/3.3.175"},{"issue":"3","key":"29","first-page":"170","volume":"14","year":"2006","journal-title":"Journal of Marine Science and Technology"},{"issue":"2","key":"30","first-page":"155","volume":"34","year":"2008","journal-title":"Computer Modeling in Engineering & Sciences"},{"key":"31","doi-asserted-by":"publisher","DOI":"10.1016\/S0020-0255(99)00116-4"},{"key":"32","doi-asserted-by":"publisher","DOI":"10.1162\/evco.2009.17.3.437"},{"key":"33","doi-asserted-by":"publisher","DOI":"10.1093\/comjnl\/5.2.147"}],"container-title":["Journal of Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2014\/324181.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2014\/324181.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2014\/324181.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2017,6,22]],"date-time":"2017-06-22T05:30:44Z","timestamp":1498109444000},"score":1,"resource":{"primary":{"URL":"http:\/\/www.hindawi.com\/journals\/jam\/2014\/324181\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014]]},"references-count":31,"alternative-id":["324181","324181"],"URL":"https:\/\/doi.org\/10.1155\/2014\/324181","relation":{},"ISSN":["1110-757X","1687-0042"],"issn-type":[{"type":"print","value":"1110-757X"},{"type":"electronic","value":"1687-0042"}],"subject":[],"published":{"date-parts":[[2014]]}}}