{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,12]],"date-time":"2026-05-12T19:39:28Z","timestamp":1778614768068,"version":"3.51.4"},"reference-count":51,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2008,11,5]],"date-time":"2008-11-05T00:00:00Z","timestamp":1225843200000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Found Comput Math"],"published-print":{"date-parts":[[2009,8]]},"DOI":"10.1007\/s10208-008-9036-y","type":"journal-article","created":{"date-parts":[[2008,11,4]],"date-time":"2008-11-04T20:23:21Z","timestamp":1225830201000},"page":"485-513","source":"Crossref","is-referenced-by-count":159,"title":["Local Linear Convergence for Alternating and Averaged Nonconvex Projections"],"prefix":"10.1007","volume":"9","author":[{"given":"A. S.","family":"Lewis","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"D. R.","family":"Luke","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"J.","family":"Malick","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2008,11,5]]},"reference":[{"key":"9036_CR1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1051\/proc:071701","volume":"17","author":"F.J. Arag\u00f3n Artacho","year":"2007","unstructured":"F.J. Arag\u00f3n Artacho, A.L. Dontchev, M.H. Geoffroy, Convergence of the proximal point method for metrically regular mappings, ESAIM Proc. 17, 1\u20138 (2007).","journal-title":"ESAIM Proc."},{"key":"9036_CR2","unstructured":"H. Attouch, J. Bolte, P. Redont, A. Soubeyran, Alternating minimization and projection methods for nonconvex problems. arXiv:0801.1780v1 , 11 Jan. 2008."},{"key":"9036_CR3","doi-asserted-by":"crossref","unstructured":"A. Auslender, M\u00e9thodes Num\u00e9riques pour la R\u00e9solution des Probl\u00e8mes d\u2019Optimisation avec Contraintes. PhD thesis, Uni. Grenoble, 1969.","DOI":"10.1051\/m2an\/196903R200271"},{"key":"9036_CR4","doi-asserted-by":"crossref","first-page":"1275","DOI":"10.1090\/S0002-9947-04-03718-3","volume":"357","author":"D. Aussel","year":"2004","unstructured":"D. Aussel, A. Daniilidis, L. Thibault, Subsmooth sets: functional characterizations and related concepts, Trans. Am. Math. Soc. 357, 1275\u20131301 (2004).","journal-title":"Trans. Am. Math. Soc."},{"key":"9036_CR5","doi-asserted-by":"crossref","first-page":"185","DOI":"10.1007\/BF01027691","volume":"1","author":"H.H. Bauschke","year":"1993","unstructured":"H.H. Bauschke, J.M. Borwein, On the convergence of von Neumann\u2019s alternating projection algorithm for two sets, Set-Valued Anal. 1, 185\u2013212 (1993).","journal-title":"Set-Valued Anal."},{"key":"9036_CR6","doi-asserted-by":"crossref","first-page":"367","DOI":"10.1137\/S0036144593251710","volume":"38","author":"H.H. Bauschke","year":"1996","unstructured":"H.H. Bauschke, J.M. Borwein, On projection algorithms for solving convex feasibility problems, SIAM Rev. 38, 367\u2013426 (1996).","journal-title":"SIAM Rev."},{"issue":"7","key":"9036_CR7","doi-asserted-by":"crossref","first-page":"1334","DOI":"10.1364\/JOSAA.19.001334","volume":"19","author":"H.H. Bauschke","year":"2002","unstructured":"H.H. Bauschke, P.L. Combettes, D.R. Luke, Phase retrieval, error reduction algorithm, and Fienup variants: A view from convex optimization, J. Opt. Soc. Am. 19(7), 1334\u20131345 (2002).","journal-title":"J. Opt. Soc. Am."},{"key":"9036_CR8","first-page":"688","volume":"6","author":"L.M. Bregman","year":"1965","unstructured":"L.M. Bregman, The method of successive projection for finding a common point of convex sets, Sov. Math. Dokl. 6, 688\u2013692 (1965).","journal-title":"Sov. Math. Dokl."},{"issue":"3","key":"9036_CR9","doi-asserted-by":"crossref","first-page":"969","DOI":"10.1088\/0266-5611\/23\/3\/008","volume":"23","author":"E.J. Cand\u00e8s","year":"2007","unstructured":"E.J. Cand\u00e8s, J. Romberg, Sparsity and incoherence in compressive sampling, Inverse Probl. 23(3), 969\u2013986 (2007).","journal-title":"Inverse Probl."},{"key":"9036_CR10","doi-asserted-by":"crossref","first-page":"2417","DOI":"10.1137\/S0036142994264742","volume":"33","author":"X. Chen","year":"1996","unstructured":"X. Chen, M.T. Chu, On the least squares solution of inverse eigenvalue problems, SIAM J. Numer. Anal. 33, 2417\u20132430 (1996).","journal-title":"SIAM J. Numer. Anal."},{"key":"9036_CR11","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1137\/S0895479893243177","volume":"16","author":"M.T. Chu","year":"1995","unstructured":"M.T. Chu, Constructing a Hermitian matrix from its diagonal entries and eigenvalues, SIAM J. Matrix Anal. 16, 207\u2013217 (1995).","journal-title":"SIAM J. Matrix Anal."},{"key":"9036_CR12","volume-title":"Nonsmooth Analysis and Control Theory","author":"F.H. Clarke","year":"1998","unstructured":"F.H. Clarke, Yu.S. Ledyaev, R.J. Stern, P.R. Wolenski, Nonsmooth Analysis and Control Theory (Springer, New York, 1998)."},{"key":"9036_CR13","doi-asserted-by":"crossref","first-page":"731","DOI":"10.1137\/S0363012903427336","volume":"43","author":"P.L. Combettes","year":"2004","unstructured":"P.L. Combettes, T. Pennanen, Proximal methods for cohypomonotone operators, SIAM J. Control Opt. 43, 731\u2013742 (2004).","journal-title":"SIAM J. Control Opt."},{"issue":"3","key":"9036_CR14","doi-asserted-by":"crossref","first-page":"487","DOI":"10.1007\/BF00939646","volume":"67","author":"P.L. Combettes","year":"1990","unstructured":"P.L. Combettes, H.J. Trussell, Method of successive projections for finding a common point of sets in metric spaces, J. Optim. Theory Appl. 67(3), 487\u2013507 (1990).","journal-title":"J. Optim. Theory Appl."},{"key":"9036_CR15","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4684-9298-9","volume-title":"Best Approximation in Inner Product Spaces","author":"F. Deutsch","year":"2001","unstructured":"F. Deutsch, Best Approximation in Inner Product Spaces (Springer, New York, 2001)."},{"key":"9036_CR16","doi-asserted-by":"crossref","first-page":"36","DOI":"10.1016\/j.jat.2006.02.005","volume":"142","author":"F. Deutsch","year":"2006","unstructured":"F. Deutsch, H. Hundal, The rate of convergence for the cyclic projections algorithm I: angles between convex sets, J. Approx. Theory 142, 36\u201355 (2006).","journal-title":"J. Approx. Theory"},{"key":"9036_CR17","doi-asserted-by":"crossref","first-page":"56","DOI":"10.1016\/j.jat.2006.02.006","volume":"142","author":"F. Deutsch","year":"2006","unstructured":"F. Deutsch, H. Hundal, The rate of convergence for the cyclic projections algorithm II: norms of nonlinear operators, J. Approx. Theory 142, 56\u201382 (2006).","journal-title":"J. Approx. Theory"},{"key":"9036_CR18","doi-asserted-by":"crossref","first-page":"1289","DOI":"10.1109\/TIT.2006.871582","volume":"52","author":"D. Donoho","year":"2006","unstructured":"D. Donoho, Compressed sensing, IEEE Trans. Inf. Theory 52, 1289\u20131306 (2006).","journal-title":"IEEE Trans. Inf. Theory"},{"key":"9036_CR19","doi-asserted-by":"crossref","first-page":"493","DOI":"10.1090\/S0002-9947-02-03088-X","volume":"355","author":"A.L. Dontchev","year":"2003","unstructured":"A.L. Dontchev, A.S. Lewis, R.T. Rockafellar, The radius of metric regularity, AMS Trans. 355, 493\u2013517 (2003).","journal-title":"AMS Trans."},{"key":"9036_CR20","first-page":"489","volume":"52","author":"J. Romberg","year":"2005","unstructured":"J. Romberg, E.J. Cand\u00e8s, T. Tao, Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information, IEEE Trans. Inf. Theory 52, 489\u2013509 (2005).","journal-title":"IEEE Trans. Inf. Theory"},{"key":"9036_CR21","doi-asserted-by":"crossref","first-page":"5695","DOI":"10.1109\/TSP.2007.900760","volume":"55","author":"M. Elad","year":"2007","unstructured":"M. Elad, Optimized projections for compressed-sensing, IEEE Trans. Signal Process. 55, 5695\u20135702 (2007).","journal-title":"IEEE Trans. Signal Process."},{"key":"9036_CR22","unstructured":"J. Fadili, G. Peyr\u00e9, Personal communication, 2007."},{"key":"9036_CR23","volume-title":"Advances in Linear Matrix Inequality Methods in Control","author":"K.M. Grigoriadis","year":"2000","unstructured":"K.M. Grigoriadis, E. Beran, Alternating projection algorithm for linear matrix inequalities problems with rank constraints, in Advances in Linear Matrix Inequality Methods in Control (SIAM, Philadelphia, 2000)."},{"key":"9036_CR24","doi-asserted-by":"crossref","first-page":"1117","DOI":"10.1016\/0005-1098(96)00057-X","volume":"32","author":"K.M. Grigoriadis","year":"1996","unstructured":"K.M. Grigoriadis, R.E. Skelton, Low-order control design for LMI problems using alternating projection methods, Automatica 32, 1117\u20131125 (1996).","journal-title":"Automatica"},{"key":"9036_CR25","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0041-5553(67)90113-9","volume":"7","author":"L.G. Gubin","year":"1967","unstructured":"L.G. Gubin, B.T. Polyak, E.V. Raik, The method of projections for finding the common point of convex sets, USSR Comput. Math. Math. Phys. 7, 1\u201324 (1967).","journal-title":"USSR Comput. Math. Math. Phys."},{"key":"9036_CR26","doi-asserted-by":"crossref","first-page":"1080","DOI":"10.1137\/S1052623401399587","volume":"13","author":"A.N. Iusem","year":"2003","unstructured":"A.N. Iusem, T. Pennanen, B.F. Svaiter, Inexact versions of the proximal point algorithm without monotonicity, SIAM J. Optim. 13, 1080\u20131097 (2003).","journal-title":"SIAM J. Optim."},{"key":"9036_CR27","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1007\/s10107-007-0174-9","volume":"17","author":"D. Klatte","year":"2009","unstructured":"D. Klatte, B. Kummer, Optimization methods and stability of inclusions in Banach spaces, Math. Program. 17, 305\u2013330 (2009).","journal-title":"Math. Program."},{"key":"9036_CR28","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1007\/s11228-006-0014-8","volume":"14","author":"A.Y. Kruger","year":"2006","unstructured":"A.Y. Kruger, About regularity of collections of sets, Set-Valued Anal. 14, 187\u2013206 (2006).","journal-title":"Set-Valued Anal."},{"key":"9036_CR29","unstructured":"A.S. Lewis, D.R. Luke, J. Malick, Local convergence for alternating and averaged nonconvex projections. arXiv:0709.0109v1 , 2 Sep. 2007."},{"key":"9036_CR30","doi-asserted-by":"crossref","first-page":"216","DOI":"10.1287\/moor.1070.0291","volume":"33","author":"A.S. Lewis","year":"2008","unstructured":"A.S. Lewis, J. Malick, Alternating projections on manifolds, Math. Oper. Res. 33, 216\u2013234 (2008).","journal-title":"Math. Oper. Res."},{"key":"9036_CR31","doi-asserted-by":"crossref","first-page":"960","DOI":"10.1016\/0021-8928(76)90136-2","volume":"40","author":"B.S. Mordukhovich","year":"1976","unstructured":"B.S. Mordukhovich, Maximum principle in the problem of time optimal response with nonsmooth constraints, J. Appl. Math. Mech. 40, 960\u2013969 (1976).","journal-title":"J. Appl. Math. Mech."},{"key":"9036_CR32","first-page":"976","volume":"28","author":"B.S. Mordukhovich","year":"1984","unstructured":"B.S. Mordukhovich, Nonsmooth analysis with nonconvex generalized differentials and adjoint mappings, Dokl. Akad. Nauk BSSR 28, 976\u2013979 (1984).","journal-title":"Dokl. Akad. Nauk BSSR"},{"key":"9036_CR33","doi-asserted-by":"crossref","DOI":"10.1007\/3-540-31247-1","volume-title":"Variational Analysis and Generalized Differentiation, I: Basic Theory; II: Applications","author":"B.S. Mordukhovich","year":"2006","unstructured":"B.S. Mordukhovich, Variational Analysis and Generalized Differentiation, I: Basic Theory; II: Applications (Springer, New York, 2006)."},{"key":"9036_CR34","doi-asserted-by":"crossref","first-page":"918","DOI":"10.1016\/j.na.2008.02.073","volume":"69","author":"B.S. Mordukhovich","year":"2008","unstructured":"B.S. Mordukhovich, Failure of metric regularity for major classes of variational systems, Nonlinear Anal. 69, 918\u2013924 (2008).","journal-title":"Nonlinear Anal."},{"key":"9036_CR35","doi-asserted-by":"crossref","first-page":"190","DOI":"10.1137\/050634529","volume":"28","author":"R. Orsi","year":"2006","unstructured":"R. Orsi, Numerical methods for solving inverse eigenvalue problems for nonnegative matrices, SIAM J. Matrix Anal. 28, 190\u2013212 (2006).","journal-title":"SIAM J. Matrix Anal."},{"key":"9036_CR36","doi-asserted-by":"crossref","first-page":"1875","DOI":"10.1016\/j.automatica.2006.05.026","volume":"42","author":"R. Orsi","year":"2006","unstructured":"R. Orsi, U. Helmke, J. Moore, A Newton-like method for solving rank constrained linear matrix inequalities, Automatica 42, 1875\u20131882 (2006).","journal-title":"Automatica"},{"key":"9036_CR37","doi-asserted-by":"crossref","first-page":"170","DOI":"10.1287\/moor.27.1.170.331","volume":"27","author":"T. Pennanen","year":"2002","unstructured":"T. Pennanen, Local convergence of the proximal point algorithm and multiplier methods without monotonicity, Math. Oper. Res. 27, 170\u2013191 (2002).","journal-title":"Math. Oper. Res."},{"key":"9036_CR38","doi-asserted-by":"crossref","first-page":"200","DOI":"10.1007\/3-540-07623-9_288","volume":"41","author":"G. Pierra","year":"1976","unstructured":"G. Pierra, Eclatement de contraintes en parall\u00e8le pour la minimisation d\u2019une forme quadratique, Lect. Notes Comput. Sci. 41, 200\u2013218 (1976).","journal-title":"Lect. Notes Comput. Sci."},{"key":"9036_CR39","doi-asserted-by":"crossref","first-page":"96","DOI":"10.1007\/BF02612715","volume":"28","author":"G. Pierra","year":"1984","unstructured":"G. Pierra, Decomposition through formalization in a product space, Math. Program. 28, 96\u2013115 (1984).","journal-title":"Math. Program."},{"key":"9036_CR40","doi-asserted-by":"crossref","first-page":"5231","DOI":"10.1090\/S0002-9947-00-02550-2","volume":"352","author":"R.A. Poliquin","year":"2000","unstructured":"R.A. Poliquin, R.T. Rockafellar, L. Thibault, Local differentiability of distance functions, AMS Trans. 352, 5231\u20135249 (2000).","journal-title":"AMS Trans."},{"key":"9036_CR41","doi-asserted-by":"crossref","first-page":"506","DOI":"10.1137\/0805026","volume":"5","author":"J. Renegar","year":"1995","unstructured":"J. Renegar, Incorporating condition measures into the complexity theory of linear programming, SIAM J. Optim. 5, 506\u2013524 (1995).","journal-title":"SIAM J. Optim."},{"key":"9036_CR42","first-page":"279","volume":"70","author":"J. Renegar","year":"1995","unstructured":"J. Renegar, Linear programming, complexity theory and elementary functional analysis, Math. Program. 70, 279\u2013351 (1995).","journal-title":"Math. Program."},{"key":"9036_CR43","doi-asserted-by":"crossref","first-page":"879","DOI":"10.1137\/S105262349427532X","volume":"6","author":"J. Renegar","year":"1996","unstructured":"J. Renegar, Condition numbers, the barrier method, and the conjugate gradient method, SIAM J. Optim. 6, 879\u2013912 (1996).","journal-title":"SIAM J. Optim."},{"key":"9036_CR44","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-02431-3","volume-title":"Variational Analysis","author":"R.T. Rockafellar","year":"1998","unstructured":"R.T. Rockafellar, R.J.-B. Wets, Variational Analysis (Springer, Berlin, 1998)."},{"key":"9036_CR45","doi-asserted-by":"crossref","first-page":"130","DOI":"10.1137\/0804006","volume":"4","author":"A. Shapiro","year":"1994","unstructured":"A. Shapiro, Existence and differentiability of metric projections in Hilbert space, SIAM J. Optim. 4, 130\u2013141 (1994).","journal-title":"SIAM J. Optim."},{"key":"9036_CR46","doi-asserted-by":"crossref","first-page":"948","DOI":"10.1214\/aos\/1015952006","volume":"28","author":"A. Shapiro","year":"2000","unstructured":"A. Shapiro, On the asymptotics of constrained local M-estimation, Ann. Stat. 28, 948\u2013960 (2000).","journal-title":"Ann. Stat."},{"key":"9036_CR47","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1007\/BF00940785","volume":"77","author":"A. Shapiro","year":"1993","unstructured":"A. Shapiro, F. Al-Khayyal, First-order conditions for isolated locally optimal solutions, J. Optim. Theory Appl. 77, 189\u2013196 (1993).","journal-title":"J. Optim. Theory Appl."},{"key":"9036_CR48","doi-asserted-by":"crossref","first-page":"188","DOI":"10.1109\/TIT.2004.839492","volume":"51","author":"J.A. Tropp","year":"2005","unstructured":"J.A. Tropp, I.S. Dhillon, R.W. Heath, T. Strohmer, Designing structured tight frames via an alternating projection method, IEEE Trans. Inf. Theory 51, 188\u2013209 (2005).","journal-title":"IEEE Trans. Inf. Theory"},{"key":"9036_CR49","volume-title":"Functional Operators","author":"J. Neumann von","year":"1950","unstructured":"J. von Neumann, Functional Operators, vol.\u00a0II (Princeton University Press, Princeton, 1950). Reprint of notes distributed in 1933."},{"key":"9036_CR50","unstructured":"C.A. Weber, J.P. Allebach, Reconstruction of frequency-offset Fourier data by alternating projection on constraint sets, in 24th Allerton Conference Proc., pp. 194\u2013201. Urbana-Champaign, IL, 1986."},{"key":"9036_CR51","doi-asserted-by":"crossref","first-page":"2143","DOI":"10.1016\/j.automatica.2006.06.021","volume":"42","author":"K. Yang","year":"2006","unstructured":"K. Yang, R. Orsi, Generalized pole placement via static output feedback: a methodology based on projections, Automatica 42, 2143\u20132150 (2006).","journal-title":"Automatica"}],"container-title":["Foundations of Computational Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10208-008-9036-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10208-008-9036-y\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10208-008-9036-y","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,9,19]],"date-time":"2021-09-19T22:45:14Z","timestamp":1632091514000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10208-008-9036-y"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,11,5]]},"references-count":51,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2009,8]]}},"alternative-id":["9036"],"URL":"https:\/\/doi.org\/10.1007\/s10208-008-9036-y","relation":{},"ISSN":["1615-3375","1615-3383"],"issn-type":[{"value":"1615-3375","type":"print"},{"value":"1615-3383","type":"electronic"}],"subject":[],"published":{"date-parts":[[2008,11,5]]}}}