{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,12]],"date-time":"2026-02-12T11:59:50Z","timestamp":1770897590013,"version":"3.50.1"},"reference-count":48,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2022,11,14]],"date-time":"2022-11-14T00:00:00Z","timestamp":1668384000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2022,11,14]],"date-time":"2022-11-14T00:00:00Z","timestamp":1668384000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comput Optim Appl"],"published-print":{"date-parts":[[2022,12]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In this paper we present general-purpose preconditioners for regularized augmented systems, and their corresponding normal equations, arising from optimization problems. We discuss positive definite preconditioners, suitable for CG and MINRES. We consider \u201csparsifications\" which avoid situations in which eigenvalues of the preconditioned matrix may become complex. Special attention is given to systems arising from the application of regularized interior point methods to linear or nonlinear convex programming problems.<\/jats:p>","DOI":"10.1007\/s10589-022-00424-5","type":"journal-article","created":{"date-parts":[[2022,11,14]],"date-time":"2022-11-14T20:16:48Z","timestamp":1668457008000},"page":"727-757","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["General-purpose preconditioning for regularized interior point methods"],"prefix":"10.1007","volume":"83","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6270-4666","authenticated-orcid":false,"given":"Jacek","family":"Gondzio","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7903-9335","authenticated-orcid":false,"given":"Spyridon","family":"Pougkakiotis","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6063-1766","authenticated-orcid":false,"given":"John W.","family":"Pearson","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,11,14]]},"reference":[{"issue":"1\u20134","key":"424_CR1","doi-asserted-by":"publisher","first-page":"275","DOI":"10.1080\/10556789908805754","volume":"11","author":"A Altman","year":"1999","unstructured":"Altman, A., Gondzio, J.: Regularized symmetric indefinite systems in interior point methods for linear and quadratic optimization. Optim. Methods Softw. 11(1\u20134), 275\u2013302 (1999). https:\/\/doi.org\/10.1080\/10556789908805754","journal-title":"Optim. Methods Softw."},{"key":"424_CR2","volume-title":"Interior Point Methods in Mathematical Programming","author":"ED Andersen","year":"1996","unstructured":"Andersen, E.D., Gondzio, J., M\u00e9sz\u00e1ros, C., Xu, X.: Implementation of interior point methods for large scale linear programming. In: Terlaky, T. (ed.) Interior Point Methods in Mathematical Programming. Kluwer Academic Publishers, Dordrecht, 199\u2013252 (1996)"},{"key":"424_CR3","doi-asserted-by":"publisher","first-page":"523","DOI":"10.1007\/s10957-017-1071-x","volume":"173","author":"P Armand","year":"2017","unstructured":"Armand, P., Omheni, R.: A mixed logarithmic barrier-augmented Lagrangian method for nonlinear optimization. J. Optim. Theor. Appl. 173, 523\u2013547 (2017). https:\/\/doi.org\/10.1007\/s10957-017-1071-x","journal-title":"J. Optim. Theor. Appl."},{"key":"424_CR4","doi-asserted-by":"publisher","first-page":"339","DOI":"10.1007\/s10589-016-9830-4","volume":"65","author":"S Bellavia","year":"2016","unstructured":"Bellavia, S., De Simone, V., di Serafino, D., Morini, B.: On the update of constraint preconditioners for regularized KKT systems. Comput. Optim. Appl. 65, 339\u2013360 (2016). https:\/\/doi.org\/10.1007\/s10589-016-9830-4","journal-title":"Comput. Optim. Appl."},{"issue":"2","key":"424_CR5","doi-asserted-by":"publisher","first-page":"418","DOI":"10.1006\/jcph.2002.7176","volume":"182","author":"M Benzi","year":"2002","unstructured":"Benzi, M.: Preconditioning techniques for large linear systems: a survey. J. Comput. Phys. 182(2), 418\u2013477 (2002). https:\/\/doi.org\/10.1006\/jcph.2002.7176","journal-title":"J. Comput. Phys."},{"key":"424_CR6","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1017\/S0962492904000212","volume":"14","author":"M Benzi","year":"2005","unstructured":"Benzi, M., Golub, G.H., Liesen, J.: Numerical solution of saddle point systems. Acta Numer. 14, 1\u2013137 (2005). https:\/\/doi.org\/10.1017\/S0962492904000212","journal-title":"Acta Numer."},{"issue":"4","key":"424_CR7","doi-asserted-by":"publisher","first-page":"e2361","DOI":"10.1002\/nla.2361","volume":"28","author":"L Bergamaschi","year":"2020","unstructured":"Bergamaschi, L., Gondzio, J., Mart\u00ednez, A., Pearson, J.W., Pougkakiotis, S.: A new preconditioning approach for an interior point-proximal method of multipliers for linear and convex quadratic programming. Numer. Linear Algebra Appl. 28(4), e2361 (2020). https:\/\/doi.org\/10.1002\/nla.2361","journal-title":"Numer. Linear Algebra Appl."},{"key":"424_CR8","doi-asserted-by":"publisher","unstructured":"Bergamaschi, L., Gondzio, J., Venturin, M., Zilli, G.: Inexact constraint preconditioners for linear systems arising in interior point methods. Comput. Optim. Appl. 36, 137\u2013147 (2007). https:\/\/doi.org\/10.1007\/s10589-006-9001-0. See also Erratum, Comput. Optim. Appl. 49, 401\u2013406 (2011)","DOI":"10.1007\/s10589-006-9001-0"},{"key":"424_CR9","doi-asserted-by":"publisher","first-page":"149","DOI":"10.1023\/B:COAP.0000026882.34332.1b","volume":"28","author":"L Bergamaschi","year":"2004","unstructured":"Bergamaschi, L., Gondzio, J., Zilli, G.: Preconditioning indefinite systems in interior point methods for optimization. Comput. Optim. Appl. 28, 149\u2013171 (2004). https:\/\/doi.org\/10.1023\/B:COAP.0000026882.34332.1b","journal-title":"Comput. Optim. Appl."},{"key":"424_CR10","doi-asserted-by":"publisher","first-page":"149","DOI":"10.1007\/s10589-006-9009-5","volume":"36","author":"S Bocanegra","year":"2007","unstructured":"Bocanegra, S., Campos, F., Oliveira, A.: Using a hybrid preconditioner for solving large-scale linear systems arising from interior point methods. Comput. Optim. Appl. 36, 149\u2013164 (2007). https:\/\/doi.org\/10.1007\/s10589-006-9009-5","journal-title":"Comput. Optim. Appl."},{"key":"424_CR11","doi-asserted-by":"publisher","first-page":"165","DOI":"10.1007\/s10589-006-9007-7","volume":"36","author":"S Cafieri","year":"2007","unstructured":"Cafieri, S., D\u2019Appuzo, M., De Simone, V., di Serafino, D.: Stopping criteria for inner iterations in inexact potential reduction methods: a computational study. Comput. Optim. Appl. 36, 165\u2013193 (2007). https:\/\/doi.org\/10.1007\/s10589-006-9007-7","journal-title":"Comput. Optim. Appl."},{"key":"424_CR12","doi-asserted-by":"publisher","first-page":"283","DOI":"10.1007\/s10589-008-9226-1","volume":"45","author":"M D\u2019Apuzzo","year":"2010","unstructured":"D\u2019Apuzzo, M., De Simone, V., di Serafino, D.: On mutual impact of numerical linear algebra and large-scale optimization with focus on interior point methods. Comput. Optim. Appl. 45, 283\u2013310 (2010). https:\/\/doi.org\/10.1007\/s10589-008-9226-1","journal-title":"Comput. Optim. Appl."},{"key":"424_CR13","doi-asserted-by":"publisher","unstructured":"De Simone, V., di Serafino, D., Gondzio, J., Pougkakiotis, S., Viola, M.: Sparse approximations with interior point methods. SIAM Rev. 64(4), 954\u2013988 (2022). https:\/\/doi.org\/10.1137\/21M1401103","DOI":"10.1137\/21M1401103"},{"issue":"2","key":"424_CR14","doi-asserted-by":"publisher","first-page":"A1001","DOI":"10.1137\/19M1291753","volume":"43","author":"D di Serafino","year":"2021","unstructured":"di Serafino, D., Orban, D.: Constraint-preconditioned Krylov solvers for regularized saddle-point systems. SIAM J. Sci. Comput. 43(2), A1001\u2013A1026 (2021). https:\/\/doi.org\/10.1137\/19M1291753","journal-title":"SIAM J. Sci. Comput."},{"issue":"2","key":"424_CR15","doi-asserted-by":"publisher","first-page":"672","DOI":"10.1137\/050626168","volume":"29","author":"HS Dollar","year":"2007","unstructured":"Dollar, H.S.: Constraint-style preconditioners for regularized saddle point problems. SIAM J. Matrix Anal. Appl. 29(2), 672\u2013684 (2007). https:\/\/doi.org\/10.1137\/050626168","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"424_CR16","doi-asserted-by":"publisher","first-page":"249","DOI":"10.1007\/s10589-006-9004-x","volume":"36","author":"HS Dollar","year":"2007","unstructured":"Dollar, H.S., Gould, N.I.M., Schilders, W.H.A., Wathen, A.J.: Using constraint preconditioners with regularized saddle-point problems. Comput. Optim. Appl. 36, 249\u2013270 (2007). https:\/\/doi.org\/10.1007\/s10589-006-9004-x","journal-title":"Comput. Optim. Appl."},{"issue":"2","key":"424_CR17","doi-asserted-by":"publisher","first-page":"14","DOI":"10.1145\/1236463.1236469","volume":"33","author":"HC Elman","year":"2007","unstructured":"Elman, H.C., Ramage, A., Silvester, D.J.: Algorithm 866: IFISS, a Matlab toolbox for modelling incompressible flow. ACM Trans. Math. Softw. 33(2), 14 (2007). https:\/\/doi.org\/10.1145\/1236463.1236469","journal-title":"ACM Trans. Math. Softw."},{"issue":"2","key":"424_CR18","doi-asserted-by":"publisher","first-page":"261","DOI":"10.1137\/120891393","volume":"52","author":"HC Elman","year":"2014","unstructured":"Elman, H.C., Ramage, A., Silvester, D.J.: IFISS: A computational laboratory for investigating incompressible flow problems. SIAM Rev. 52(2), 261\u2013273 (2014). https:\/\/doi.org\/10.1137\/120891393","journal-title":"SIAM Rev."},{"key":"424_CR19","doi-asserted-by":"publisher","first-page":"71","DOI":"10.1007\/s12532-012-0035-2","volume":"4","author":"MP Friedlander","year":"2012","unstructured":"Friedlander, M.P., Orban, D.: A primal-dual regularized interior-point method for convex quadratic programs. Math. Program. Comput. 4, 71\u2013107 (2012). https:\/\/doi.org\/10.1007\/s12532-012-0035-2","journal-title":"Math. Program. Comput."},{"issue":"1","key":"424_CR20","doi-asserted-by":"publisher","first-page":"292","DOI":"10.1137\/0613022","volume":"13","author":"PE Gill","year":"1992","unstructured":"Gill, P.E., Murray, W., Poncele\u00f3n, D.B., Saunders, M.A.: Preconditioners for indefinite systems arising in optimization. SIAM J. Matrix Anal. Appl. 13(1), 292\u2013311 (1992). https:\/\/doi.org\/10.1137\/0613022","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"424_CR21","doi-asserted-by":"publisher","unstructured":"Greenbaum, A.: Iterative Methods for Solving Linear Systems. Frontiers in Applied Mathematics. SIAM, Philadelphia, USA (1997). https:\/\/doi.org\/10.1137\/1.9781611970937","DOI":"10.1137\/1.9781611970937"},{"issue":"3","key":"424_CR22","doi-asserted-by":"publisher","first-page":"465","DOI":"10.1137\/S0895479894275030","volume":"17","author":"A Greenbaum","year":"1996","unstructured":"Greenbaum, A., Pt\u00e1k, V., Strako\u0161, Z.: Any nonincreasing convergence curve is possible for GMRES. SIAM J. Matrix Anal. Appl. 17(3), 465\u2013469 (1996). https:\/\/doi.org\/10.1137\/S0895479894275030","journal-title":"SIAM J. Matrix Anal. Appl."},{"issue":"6","key":"424_CR23","doi-asserted-by":"publisher","first-page":"409","DOI":"10.6028\/jres.049.044","volume":"49","author":"MR Hestenes","year":"1952","unstructured":"Hestenes, M.R., Stiefel, E.: Method of conjugate gradients for solving linear systems. J. Res. Natl. Bur. Stand. 49(6), 409\u2013436 (1952). https:\/\/doi.org\/10.6028\/jres.049.044","journal-title":"J. Res. Natl. Bur. Stand."},{"key":"424_CR24","unstructured":"Hughes, T.J.R., Brooks, A.N.: A multidimensional upwind scheme with no crosswind diffusion. In: Hughes, T.J.R. (ed.) Finite Element Methods for Convection Dominated Flows, vol. 34, pp. 19\u201335. American Society of Mechanical Engineers (1979)"},{"issue":"3","key":"424_CR25","doi-asserted-by":"publisher","first-page":"1050","DOI":"10.1137\/S1064827500377435","volume":"23","author":"ICF Ipsen","year":"2001","unstructured":"Ipsen, I.C.F.: A note on preconditioning non-symmetric matrices. SIAM J. Sci. Comput. 23(3), 1050\u20131051 (2001). https:\/\/doi.org\/10.1137\/S1064827500377435","journal-title":"SIAM J. Sci. Comput."},{"issue":"4","key":"424_CR26","doi-asserted-by":"publisher","first-page":"1300","DOI":"10.1137\/S0895479899351805","volume":"21","author":"C Keller","year":"2000","unstructured":"Keller, C., Gould, N.I.M., Wathen, A.J.: Constraint preconditioning for symmetric indefinite linear systems. SIAM J. Matrix Anal. Appl. 21(4), 1300\u20131317 (2000). https:\/\/doi.org\/10.1137\/S0895479899351805","journal-title":"SIAM J. Matrix Anal. Appl."},{"issue":"3","key":"424_CR27","doi-asserted-by":"publisher","first-page":"2410","DOI":"10.1137\/19M1251795","volume":"30","author":"X Li","year":"2020","unstructured":"Li, X., Sun, D., Toh, K.C.: An asymptotically superilinearly convergent semismooth Newton augmented Lagrangian method for linear programming. SIAM J. Optim. 30(3), 2410\u20132440 (2020). https:\/\/doi.org\/10.1137\/19M1251795","journal-title":"SIAM J. Optim."},{"issue":"3","key":"424_CR28","doi-asserted-by":"publisher","first-page":"720","DOI":"10.1016\/S0377-2217(97)00074-X","volume":"107","author":"I Maros","year":"1998","unstructured":"Maros, I., M\u00e9sz\u00e1ros, C.: The role of the augmented system in interior point methods. Eur. J. Oper. Res. 107(3), 720\u2013736 (1998). https:\/\/doi.org\/10.1016\/S0377-2217(97)00074-X","journal-title":"Eur. J. Oper. Res."},{"issue":"1\u20134","key":"424_CR29","doi-asserted-by":"publisher","first-page":"671","DOI":"10.1080\/10556789908805768","volume":"11","author":"I Maros","year":"1999","unstructured":"Maros, I., M\u00e9sz\u00e1ros, C.: A repository of convex quadratic programming problems. Optim. Methods Softw. 11(1\u20134), 671\u2013681 (1999). https:\/\/doi.org\/10.1080\/10556789908805768","journal-title":"Optim. Methods Softw."},{"issue":"6","key":"424_CR30","doi-asserted-by":"publisher","first-page":"1969","DOI":"10.1137\/S1064827599355153","volume":"21","author":"MF Murphy","year":"2000","unstructured":"Murphy, M.F., Golub, G.H., Wathen, A.J.: A note on preconditioning for indefinite linear systems. SIAM J. Sci. Comput. 21(6), 1969\u20131972 (2000). https:\/\/doi.org\/10.1137\/S1064827599355153","journal-title":"SIAM J. Sci. Comput."},{"key":"424_CR31","unstructured":"Netlib: http:\/\/netlib.org\/lp (2011)"},{"issue":"1","key":"424_CR32","doi-asserted-by":"publisher","first-page":"143","DOI":"10.1137\/130911962","volume":"35","author":"Y Notay","year":"2014","unstructured":"Notay, Y.: A new analysis of block preconditioners for saddle point systems. SIAM J. Matrix Anal. Appl. 35(1), 143\u2013173 (2014). https:\/\/doi.org\/10.1137\/130911962","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"424_CR33","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.laa.2004.08.019","volume":"394","author":"ARL Oliveira","year":"2005","unstructured":"Oliveira, A.R.L., Sorensen, D.C.: A new class of preconditioners for large-scale linear systems from interior point methods for linear programming. Linear Algebra Appl. 394, 1\u201324 (2005). https:\/\/doi.org\/10.1016\/j.laa.2004.08.019","journal-title":"Linear Algebra Appl."},{"key":"424_CR34","doi-asserted-by":"publisher","first-page":"9","DOI":"10.1007\/s11075-014-9933-x","volume":"70","author":"D Orban","year":"2015","unstructured":"Orban, D.: Limited-memory LD$$L^T$$ factorization of symmetric quasi-definite matrices with application to constrained optimization. Numer. Algorithms 70, 9\u201341 (2015). https:\/\/doi.org\/10.1007\/s11075-014-9933-x","journal-title":"Numer. Algorithms"},{"issue":"4","key":"424_CR35","doi-asserted-by":"publisher","first-page":"617","DOI":"10.1137\/0712047","volume":"12","author":"CC Paige","year":"1975","unstructured":"Paige, C.C., Saunders, M.A.: Solution of sparse indefinite systems of linear equations. SIAM J. Numer. Anal. 12(4), 617\u2013629 (1975). https:\/\/doi.org\/10.1137\/0712047","journal-title":"SIAM J. Numer. Anal."},{"key":"424_CR36","doi-asserted-by":"publisher","first-page":"959","DOI":"10.1007\/s00211-017-0892-8","volume":"137","author":"JW Pearson","year":"2017","unstructured":"Pearson, J.W., Gondzio, J.: Fast interior point solution of quadratic programming problems arising from PDE-constrained optimization. Numer. Math. 137, 959\u2013999 (2017). https:\/\/doi.org\/10.1007\/s00211-017-0892-8","journal-title":"Numer. Math."},{"issue":"4","key":"424_CR37","doi-asserted-by":"publisher","first-page":"e202000015","DOI":"10.1002\/gamm.202000015","volume":"43","author":"JW Pearson","year":"2020","unstructured":"Pearson, J.W., Pestana, J.: Preconditioners for Krylov subspace methods: an overview. GAMM-Mitt. 43(4), e202000015 (2020). https:\/\/doi.org\/10.1002\/gamm.202000015","journal-title":"GAMM-Mitt."},{"issue":"2","key":"424_CR38","doi-asserted-by":"publisher","first-page":"e2276","DOI":"10.1002\/nla.2276","volume":"27","author":"JW Pearson","year":"2019","unstructured":"Pearson, J.W., Porcelli, M., Stoll, M.: Interior-point methods and preconditioning for PDE-constrained optimization problems involving sparsity terms. Numer. Linear Algebra Appl. 27(2), e2276 (2019). https:\/\/doi.org\/10.1002\/nla.2276","journal-title":"Numer. Linear Algebra Appl."},{"issue":"3","key":"424_CR39","doi-asserted-by":"publisher","first-page":"905","DOI":"10.1007\/s10957-019-01491-1","volume":"181","author":"S Pougkakiotis","year":"2019","unstructured":"Pougkakiotis, S., Gondzio, J.: Dynamic non-diagonal regularization in interior point methods for linear and convex quadratic programming. J. Optim. Theory Appl. 181(3), 905\u2013945 (2019). https:\/\/doi.org\/10.1007\/s10957-019-01491-1","journal-title":"J. Optim. Theory Appl."},{"key":"424_CR40","doi-asserted-by":"publisher","first-page":"307","DOI":"10.1007\/s10589-020-00240-9","volume":"78","author":"S Pougkakiotis","year":"2021","unstructured":"Pougkakiotis, S., Gondzio, J.: An interior point-proximal method of multipliers for convex quadratic programming. Comput. Optim. Appl. 78, 307\u2013351 (2021). https:\/\/doi.org\/10.1007\/s10589-020-00240-9","journal-title":"Comput. Optim. Appl."},{"key":"424_CR41","doi-asserted-by":"publisher","first-page":"97","DOI":"10.1007\/s10957-021-01954-4","volume":"192","author":"S Pougkakiotis","year":"2022","unstructured":"Pougkakiotis, S., Gondzio, J.: An interior point-proximal method of multipliers for linear positive semi-definite programming. J. Optim. Theory Appl. 192, 97\u2013129 (2022). https:\/\/doi.org\/10.1007\/s10957-021-01954-4","journal-title":"J. Optim. Theory Appl."},{"issue":"3","key":"424_CR42","doi-asserted-by":"publisher","first-page":"856","DOI":"10.1137\/0907058","volume":"7","author":"Y Saad","year":"1986","unstructured":"Saad, Y., Schultz, M.H.: GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Comput. 7(3), 856\u2013869 (1986).\u00a0https:\/\/doi.org\/10.1137\/0907058","journal-title":"SIAM J. Sci. Comput."},{"key":"424_CR43","unstructured":"Saunders, M., Tomlin, J.A.: Solving regularized linear programs using barrier methods and KKT systems. Tech Rep SOL 96-4. Systems Optimization Laboratory, Department of Operations Research, Stanford University, Stanford (1996)"},{"key":"424_CR44","doi-asserted-by":"publisher","first-page":"1147","DOI":"10.1007\/s11075-018-0478-2","volume":"79","author":"J Scott","year":"2018","unstructured":"Scott, J., T\u016fma, M.: A Schur complement approach to preconditioning sparse linear least-squares problems with some dense rows. Numer. Algorithms 79, 1147\u20131168 (2018). https:\/\/doi.org\/10.1007\/s11075-018-0478-2","journal-title":"Numer. Algorithms"},{"issue":"5","key":"424_CR45","doi-asserted-by":"publisher","first-page":"1352","DOI":"10.1137\/0731070","volume":"31","author":"DJ Silvester","year":"1994","unstructured":"Silvester, D.J., Wathen, A.J.: Fast iterative solution of stabilized Stokes systems. Part II: using general block preconditioners. SIAM J. Numer. Anal. 31(5), 1352\u20131367 (1994). https:\/\/doi.org\/10.1137\/0731070","journal-title":"SIAM J. Numer. Anal."},{"issue":"1","key":"424_CR46","doi-asserted-by":"publisher","first-page":"100","DOI":"10.1137\/0805005","volume":"5","author":"RJ Vanderbei","year":"1995","unstructured":"Vanderbei, R.J.: Symmetric quasidefinite matrices. SIAM J. Optim. 5(1), 100\u2013113 (1995). https:\/\/doi.org\/10.1137\/0805005","journal-title":"SIAM J. Optim."},{"issue":"4","key":"424_CR47","doi-asserted-by":"publisher","first-page":"449","DOI":"10.1093\/imanum\/7.4.449","volume":"7","author":"AJ Wathen","year":"1987","unstructured":"Wathen, A.J.: Realistic eigenvalue bounds for the Galerkin mass matrix. IMA J. Numer. Anal. 7(4), 449\u2013457 (1987). https:\/\/doi.org\/10.1093\/imanum\/7.4.449","journal-title":"IMA J. Numer. Anal."},{"key":"424_CR48","doi-asserted-by":"publisher","first-page":"329","DOI":"10.1017\/S0962492915000021","volume":"24","author":"AJ Wathen","year":"2015","unstructured":"Wathen, A.J.: Preconditioning. Acta Numer. 24, 329\u2013376 (2015). https:\/\/doi.org\/10.1017\/S0962492915000021","journal-title":"Acta Numer."}],"container-title":["Computational Optimization and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10589-022-00424-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10589-022-00424-5\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10589-022-00424-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,11,23]],"date-time":"2022-11-23T11:21:07Z","timestamp":1669202467000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10589-022-00424-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,11,14]]},"references-count":48,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2022,12]]}},"alternative-id":["424"],"URL":"https:\/\/doi.org\/10.1007\/s10589-022-00424-5","relation":{},"ISSN":["0926-6003","1573-2894"],"issn-type":[{"value":"0926-6003","type":"print"},{"value":"1573-2894","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,11,14]]},"assertion":[{"value":"14 July 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 October 2022","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"14 November 2022","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare no competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}