{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:33:40Z","timestamp":1787337220969,"version":"build-2736575974"},"reference-count":36,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2010,1]]},"abstract":"<jats:p>Saddle-point systems arise in many applications areas, in fact in any situation where an extremum principle arises with constraints. The Stokes problem describing slow viscous flow of an incompressible fluid is a classic example coming from PDEs and in the area of optimization such problems are ubiquitous. In this paper we present a framework into which many well-known methods for solving saddle-point systems fit. Based on this description we show how new approaches for the solution of saddle-point systems arising in optimization can be derived from the Bramble\u2013Pasciak conjugate gradient approach widely used in PDEs and more recent generalizations thereof. In particular we derive a class of new solution methods based on the use of preconditioned conjugate gradients in nonstandard inner products and demonstrate how these can be understood through more standard machinery. We show connections to constraint preconditioning and give the results of numerical computations on a number of standard optimization test examples.<\/jats:p>","DOI":"10.1137\/080727129","type":"journal-article","created":{"date-parts":[[2010,2,5]],"date-time":"2010-02-05T18:13:52Z","timestamp":1265393632000},"page":"249-270","source":"Crossref","is-referenced-by-count":26,"title":["Preconditioning Saddle-Point Systems with Applications in Optimization"],"prefix":"10.1137","volume":"32","author":[{"given":"H. Sue","family":"Dollar","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Nicholas I. M.","family":"Gould","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Martin","family":"Stoll","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Andrew J.","family":"Wathen","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2010,2,5]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1137\/0727091"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1023\/A:1021961616621"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1017\/S0962492904000212"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-006-0679-9"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1988-0917816-8"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479896302898"},{"key":"R7","doi-asserted-by":"crossref","unstructured":"H. C. Elman, A. Ramage, and D. J. Silvester,\n                      Algorithm $886$: IFISS, a Matlab toolbox for modelling incompressible flow\n                      , ACM Trans. Math. Software, 33 (2007), Art. 14.","DOI":"10.1145\/1236463.1236469"},{"key":"R8","doi-asserted-by":"crossref","unstructured":"H. C. Elman, D. J. Silvester, and A. J. Wathen,\n                      Finite Elements and Fast Iterative Solvers: With Applications in Incompressible Fluid Dynamics\n                      , Numer. Math. Sci. Comput., Oxford University Press, New York, 2005.","DOI":"10.1093\/oso\/9780198528678.001.0001"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-007-0177-6"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1007\/BF02510258"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1137\/060650210"},{"key":"R12","doi-asserted-by":"crossref","unstructured":"R. W. Freund,\n                      Transpose-free quasi-minimal residual methods for non-Hermitian linear systems\n                      , in Recent Advances in Iterative Methods, IMA Vol. Math. Appl. 60, Springer, New York, 1994, pp. 69\u201394.","DOI":"10.1007\/978-1-4613-9353-5_6"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1007\/BF01585529"},{"key":"R14","unstructured":"P. E. Gill, W. Murray, and M. H. Wright,\n                      Practical Optimization\n                      , Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London-New York, 1981."},{"key":"R15","unstructured":"G. H. Golub and C. F. Van Loan,\n                      Matrix Computations\n                      , 3rd ed., Johns Hopkins Stud. Math. Sci., Johns Hopkins University Press, Baltimore, MD, 1996."},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1145\/962437.962439"},{"key":"R17","unstructured":"N. Gould, D. Orban, and P. L. 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Lazareth, eds., SIAM, Philadelphia, 1996, pp. 92\u2013100."},{"key":"R30","doi-asserted-by":"publisher","DOI":"10.1137\/0911064"},{"key":"R31","doi-asserted-by":"publisher","DOI":"10.1137\/S105262349833266X"},{"key":"R32","doi-asserted-by":"publisher","DOI":"10.1137\/060660977"},{"key":"R33","doi-asserted-by":"publisher","DOI":"10.1016\/j.apnum.2003.11.012"},{"key":"R34","unstructured":"M. Stoll,\n                      Solving Linear Systems Using the Adjoint\n                      , Ph.D. thesis, University of Oxford, Oxford, UK, 2009."},{"key":"R35","doi-asserted-by":"publisher","DOI":"10.1137\/070688961"},{"key":"R36","doi-asserted-by":"crossref","unstructured":"S. J. Wright,\n                      Primal-Dual Interior-Point Methods\n                      , SIAM, Philadelphia, 1997.","DOI":"10.1137\/1.9781611971453"}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/080727129","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:50:31Z","timestamp":1787334631000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/080727129"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,1]]},"references-count":36,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2010,1]]}},"alternative-id":["10.1137\/080727129"],"URL":"https:\/\/doi.org\/10.1137\/080727129","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,1]]}}}