{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T15:00:49Z","timestamp":1776783649521,"version":"3.51.2"},"reference-count":20,"publisher":"American Mathematical Society (AMS)","issue":"248","license":[{"start":{"date-parts":[[2004,9,26]],"date-time":"2004-09-26T00:00:00Z","timestamp":1096156800000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n A nonlinear least squares problem with nonlinear constraints may be ill posed or even rank-deficient in two ways. Considering the problem formulated as\n \n \n \n \n \n min<\/mml:mo>\n \n x<\/mml:mi>\n <\/mml:mrow>\n <\/mml:munder>\n 1<\/mml:mn>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n 2<\/mml:mn>\n \n \u2016\n \n <\/mml:mo>\n \n f<\/mml:mi>\n \n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n (<\/mml:mo>\n x<\/mml:mi>\n )<\/mml:mo>\n \n \n \u2016\n \n <\/mml:mo>\n \n 2<\/mml:mn>\n <\/mml:mrow>\n \n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msubsup>\n <\/mml:mrow>\n \\min _{x} 1\/2\\| f_{2}(x) \\|_{2}^{2}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n subject to the constraints\n \n \n \n \n \n f<\/mml:mi>\n \n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n (<\/mml:mo>\n x<\/mml:mi>\n )<\/mml:mo>\n =<\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n f_{1}(x) = 0<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , the Jacobian\n \n \n \n \n \n J<\/mml:mi>\n \n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n =<\/mml:mo>\n \n \u2202\n \n <\/mml:mi>\n \n f<\/mml:mi>\n \n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n \n \u2202\n \n <\/mml:mi>\n x<\/mml:mi>\n <\/mml:mrow>\n J_{1} = \\partial f_{1}\/ \\partial x<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and\/or the Jacobian\n \n \n \n \n J<\/mml:mi>\n =<\/mml:mo>\n \n \u2202\n \n <\/mml:mi>\n f<\/mml:mi>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n \n \u2202\n \n <\/mml:mi>\n x<\/mml:mi>\n <\/mml:mrow>\n J = \\partial f\/ \\partial x<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n ,\n \n \n \n \n f<\/mml:mi>\n =<\/mml:mo>\n [<\/mml:mo>\n \n f<\/mml:mi>\n \n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n ;<\/mml:mo>\n \n f<\/mml:mi>\n \n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n ]<\/mml:mo>\n <\/mml:mrow>\n f = [f_{1};f_{2}]<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , may be ill conditioned at the solution. We analyze the important special case when\n \n \n \n \n J<\/mml:mi>\n \n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n J_{1}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and\/or\n \n \n \n J<\/mml:mi>\n J<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n do not have full rank at the solution. In order to solve such a problem, we formulate a nonlinear least norm problem. Next we describe a truncated Gauss-Newton method. We show that the local convergence rate is determined by the maximum of three independent Rayleigh quotients related to three different spaces in\n \n \n \n \n \n R<\/mml:mi>\n <\/mml:mrow>\n \n n<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n \\mathbb {R}^{n}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . Another way of solving an ill-posed nonlinear least squares problem is to regularize the problem with some parameter that is reduced as the iterates converge to the minimum. Our approach is a Tikhonov based local linear regularization that converges to a minimum norm problem. This approach may be used both for almost and rank-deficient Jacobians. Finally we present computational tests on constructed problems verifying the local analysis.\n <\/p>","DOI":"10.1090\/s0025-5718-03-01611-9","type":"journal-article","created":{"date-parts":[[2004,6,11]],"date-time":"2004-06-11T15:05:00Z","timestamp":1086966300000},"page":"1865-1883","source":"Crossref","is-referenced-by-count":8,"title":["Local results for the Gauss-Newton method on constrained rank-deficient nonlinear least squares"],"prefix":"10.1090","volume":"73","author":[{"given":"Jerry","family":"Eriksson","sequence":"first","affiliation":[]},{"given":"M\u00e5rten","family":"Gulliksson","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2003,9,26]]},"reference":[{"key":"1","series-title":"Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1002\/9780470316757","volume-title":"Nonlinear regression analysis and its applications","author":"Bates, Douglas M.","year":"1988","ISBN":"https:\/\/id.crossref.org\/isbn\/0471816434"},{"key":"2","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611971484","volume-title":"Numerical methods for least squares problems","author":"Bj\u00f6rck, \u00c5ke","year":"1996","ISBN":"https:\/\/id.crossref.org\/isbn\/0898713609"},{"issue":"2","key":"3","doi-asserted-by":"publisher","first-page":"134","DOI":"10.1007\/bf01932285","article-title":"Algorithms for the regularization of ill-conditioned least squares problems","volume":"17","author":"Eld\u00e9n, Lars","year":"1977","journal-title":"Nordisk Tidskr. Informationsbehandling (BIT)","ISSN":"https:\/\/id.crossref.org\/issn\/0901-246X","issn-type":"print"},{"issue":"4","key":"4","doi-asserted-by":"publisher","first-page":"487","DOI":"10.1007\/BF01934412","article-title":"A weighted pseudoinverse, generalized singular values, and constrained least squares problems","volume":"22","author":"Eld\u00e9n, Lars","year":"1982","journal-title":"BIT","ISSN":"https:\/\/id.crossref.org\/issn\/0006-3835","issn-type":"print"},{"key":"5","unstructured":"J. Eriksson, Optimization and regularization of nonlinear least squares problems, Tech. Report UMINF 96.09 (Ph.D. Thesis), Dept. of Comp. Science, Ume\u00e5 University, Ume\u00e5, Sweden, 1996."},{"key":"6","unstructured":"J. Eriksson, M. E. Gulliksson, P. Lindstr\u00f6m, and P.-\u00c5. Wedin, Regularization tools for training feed-forward neural networks part I: Theory and basic algorithms, Tech. Report UMINF 96.05, Dept. of Comp. 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Optim.","ISSN":"https:\/\/id.crossref.org\/issn\/1052-6234","issn-type":"print"},{"key":"10","series-title":"SIAM Monographs on Mathematical Modeling and Computation","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898719697","volume-title":"Rank-deficient and discrete ill-posed problems","author":"Hansen, Per Christian","year":"1998","ISBN":"https:\/\/id.crossref.org\/isbn\/0898714036"},{"key":"11","series-title":"Prentice-Hall Series in Automatic Computation","volume-title":"Solving least squares problems","author":"Lawson, Charles L.","year":"1974"},{"key":"12","series-title":"Frontiers in Applied Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611971002","volume-title":"The total least squares problem","volume":"9","author":"Van Huffel, Sabine","year":"1991","ISBN":"https:\/\/id.crossref.org\/isbn\/0898712750"},{"key":"13","unstructured":"P. Lindstr\u00f6m and P.-\u00c5. Wedin, Methods and software for nonlinear least squares problems, Tech. Report UMINF-133.87, Inst. of Info. Proc., Univ. of Ume\u00e5, Ume\u00e5, Sweden, 1988."},{"key":"14","isbn-type":"print","first-page":"105","article-title":"The Levenberg-Marquardt algorithm: implementation and theory","author":"Mor\u00e9, Jorge J.","year":"1978","ISBN":"https:\/\/id.crossref.org\/isbn\/3540085386"},{"key":"15","volume-title":"Iterative solution of nonlinear equations in several variables","author":"Ortega, J. M.","year":"1970"},{"key":"16","isbn-type":"print","first-page":"679","article-title":"Newton-like methods for underdetermined systems","author":"Walker, Homer F.","year":"1990","ISBN":"https:\/\/id.crossref.org\/isbn\/0821811312"},{"key":"17","doi-asserted-by":"publisher","first-page":"217","DOI":"10.1007\/bf01933494","article-title":"Perturbation theory for pseudo-inverses","volume":"13","author":"Wedin, Per-\u0226ke","year":"1973","journal-title":"Nordisk Tidskr. 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Proc., Univ. of Ume\u00e5, Ume\u00e5, Sweden, 1987."}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2004-73-248\/S0025-5718-03-01611-9\/S0025-5718-03-01611-9.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2004-73-248\/S0025-5718-03-01611-9\/S0025-5718-03-01611-9.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T13:54:33Z","timestamp":1776779673000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2004-73-248\/S0025-5718-03-01611-9\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2003,9,26]]},"references-count":20,"journal-issue":{"issue":"248","published-print":{"date-parts":[[2004,10]]}},"alternative-id":["S0025-5718-03-01611-9"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-03-01611-9","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2003,9,26]]}}}