{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,22]],"date-time":"2025-12-22T04:32:13Z","timestamp":1766377933139},"reference-count":48,"publisher":"Springer Science and Business Media LLC","issue":"1-2","license":[{"start":{"date-parts":[[2017,5,15]],"date-time":"2017-05-15T00:00:00Z","timestamp":1494806400000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Math. Program."],"published-print":{"date-parts":[[2018,3]]},"DOI":"10.1007\/s10107-017-1162-3","type":"journal-article","created":{"date-parts":[[2017,5,15]],"date-time":"2017-05-15T17:39:44Z","timestamp":1494869984000},"page":"509-531","update-policy":"http:\/\/dx.doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":16,"title":["Spectral operators of matrices"],"prefix":"10.1007","volume":"168","author":[{"given":"Chao","family":"Ding","sequence":"first","affiliation":[]},{"given":"Defeng","family":"Sun","sequence":"additional","affiliation":[]},{"given":"Jie","family":"Sun","sequence":"additional","affiliation":[]},{"given":"Kim-Chuan","family":"Toh","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2017,5,15]]},"reference":[{"key":"1162_CR1","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4612-0653-8","volume-title":"Matrix Analysis","author":"R Bhatia","year":"1997","unstructured":"Bhatia, R.: Matrix Analysis. Springer, New York (1997)"},{"key":"1162_CR2","doi-asserted-by":"crossref","first-page":"717","DOI":"10.1007\/s10208-009-9045-5","volume":"9","author":"EJ Cand\u00e8s","year":"2008","unstructured":"Cand\u00e8s, E.J., Recht, B.: Exact matrix completion via convex optimization. Found. Comput. Math. 9, 717\u2013772 (2008)","journal-title":"Found. Comput. Math."},{"key":"1162_CR3","doi-asserted-by":"crossref","first-page":"2053","DOI":"10.1109\/TIT.2010.2044061","volume":"56","author":"EJ Cand\u00e8s","year":"2009","unstructured":"Cand\u00e8s, E.J., Tao, T.: The power of convex relaxation: near-optimal matrix completion. IEEE Trans. Inf. Theory 56, 2053\u20132080 (2009)","journal-title":"IEEE Trans. Inf. Theory"},{"key":"1162_CR4","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1145\/1970392.1970395","volume":"58","author":"EJ Cand\u00e8s","year":"2011","unstructured":"Cand\u00e8s, E.J., Li, X., Ma, Y., Wright, J.: Robust principal component analysis? J. ACM 58, 11 (2011)","journal-title":"J. ACM"},{"key":"1162_CR5","doi-asserted-by":"crossref","first-page":"370","DOI":"10.1137\/070681235","volume":"19","author":"ZX Chan","year":"2008","unstructured":"Chan, Z.X., Sun, D.F.: Constraint nondegeneracy, strong regularity, and nonsingularity in semidefinite programming. SIAM J. Optim. 19, 370\u2013396 (2008)","journal-title":"SIAM J. Optim."},{"key":"1162_CR6","doi-asserted-by":"crossref","first-page":"572","DOI":"10.1137\/090761793","volume":"21","author":"V Chandrasekaran","year":"2011","unstructured":"Chandrasekaran, V., Sanghavi, S., Parrilo, P.A., Willsky, A.: Rank-sparsity incoherence for matrix decomposition. SIAM J. Optim. 21, 572\u2013596 (2011)","journal-title":"SIAM J. Optim."},{"key":"1162_CR7","doi-asserted-by":"crossref","first-page":"435","DOI":"10.1007\/s10107-014-0853-2","volume":"155","author":"CH Chen","year":"2016","unstructured":"Chen, C.H., Liu, Y.J., Sun, D.F., Toh, K.C.: A semismooth Newton-CG dual proximal point algorithm for matrix spectral norm approximation problems. Math. Program. 155, 435\u2013470 (2016)","journal-title":"Math. Program."},{"key":"1162_CR8","doi-asserted-by":"crossref","first-page":"960","DOI":"10.1137\/S1052623400380584","volume":"13","author":"X Chen","year":"2003","unstructured":"Chen, X., Qi, H.D., Tseng, P.: Analysis of nonsmooth symmetric-matrix-valued functions with applications to semidefinite complement problems. SIAM J. Optim. 13, 960\u2013985 (2003)","journal-title":"SIAM J. Optim."},{"key":"1162_CR9","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1016\/S0024-3795(02)00505-0","volume":"366","author":"M Chu","year":"2003","unstructured":"Chu, M., Funderlic, R., Plemmons, R.: Structured low rank approximation. Linear Algebra Appl. 366, 157\u2013172 (2003)","journal-title":"Linear Algebra Appl."},{"key":"1162_CR10","first-page":"3","volume":"14","author":"VF Demyanov","year":"1983","unstructured":"Demyanov, V.F., Rubinov, A.M.: On quasidifferentiable mappings. Optimization 14, 3\u201321 (1983)","journal-title":"Optimization"},{"key":"1162_CR11","unstructured":"Ding, C.: An introduction to a class of matrix optimization problems. PhD thesis, National University of Singapore. http:\/\/www.math.nus.edu.sg\/~matsundf\/DingChao_Thesis_final.pdf (2012)"},{"key":"1162_CR12","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1007\/s10107-012-0619-7","volume":"144","author":"C Ding","year":"2014","unstructured":"Ding, C., Sun, D.F., Toh, K.C.: An introduction to a class of matrix cone programming. Math. Program. 144, 141\u2013179 (2014)","journal-title":"Math. Program."},{"key":"1162_CR13","doi-asserted-by":"crossref","first-page":"539","DOI":"10.1007\/s10107-013-0735-z","volume":"147","author":"C Ding","year":"2014","unstructured":"Ding, C., Sun, D.F., Ye, J.J.: First order optimality conditions for mathematical programs with semidefinite cone complementarity constraints. Math. Program. 147, 539\u2013579 (2014)","journal-title":"Math. Program."},{"key":"1162_CR14","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1016\/S0012-365X(03)00308-X","volume":"276","author":"V Dobrynin","year":"2004","unstructured":"Dobrynin, V.: On the rank of a matrix associated with a graph. Discrete Math. 276, 169\u2013175 (2004)","journal-title":"Discrete Math."},{"key":"1162_CR15","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511897191","volume-title":"Differential Analysis","author":"TM Flett","year":"1980","unstructured":"Flett, T.M.: Differential Analysis. Cambridge University Press, Cambridge (1980)"},{"key":"1162_CR16","doi-asserted-by":"crossref","first-page":"359","DOI":"10.1137\/0915025","volume":"15","author":"A Greenbaum","year":"1994","unstructured":"Greenbaum, A., Trefethen, L.N.: GMRES\/CR and Arnoldi\/Lanczos as matrix approximation problems. SIAM J. Sci. Comput. 15, 359\u2013368 (1994)","journal-title":"SIAM J. Sci. Comput."},{"key":"1162_CR17","doi-asserted-by":"crossref","first-page":"483","DOI":"10.1007\/BF01195002","volume":"17","author":"A Kotlov","year":"1997","unstructured":"Kotlov, A., Lov\u00e1sz, L., Vempala, S.: The Colin de Verdi\u00e8re number and sphere representations of a graph. Combinatorica 17, 483\u2013521 (1997)","journal-title":"Combinatorica"},{"key":"1162_CR18","first-page":"173","volume":"2","author":"AS Lewis","year":"1995","unstructured":"Lewis, A.S.: The convex analysis of unitarily invariant matrix functions. J. Convex Anal. 2, 173\u2013183 (1995)","journal-title":"J. Convex Anal."},{"key":"1162_CR19","doi-asserted-by":"crossref","first-page":"576","DOI":"10.1287\/moor.21.3.576","volume":"21","author":"AS Lewis","year":"1996","unstructured":"Lewis, A.S.: Derivatives of spectral functions. Math. Oper. Res. 21, 576\u2013588 (1996)","journal-title":"Math. Oper. Res."},{"key":"1162_CR20","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1017\/S0962492900002646","volume":"5","author":"AS Lewis","year":"1996","unstructured":"Lewis, A.S., Overton, M.L.: Eigenvalue optimization. Acta Numer. 5, 149\u2013190 (1996)","journal-title":"Acta Numer."},{"key":"1162_CR21","doi-asserted-by":"crossref","first-page":"368","DOI":"10.1137\/S089547980036838X","volume":"23","author":"AS Lewis","year":"2001","unstructured":"Lewis, A.S., Sendov, H.S.: Twice differentiable spectral functions. SIAM J. Matrix Anal. Appl. 23, 368\u2013386 (2001)","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"1162_CR22","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1007\/s11228-004-7197-7","volume":"13","author":"AS Lewis","year":"2005","unstructured":"Lewis, A.S., Sendov, H.S.: Nonsmooth analysis of singular values. Part I: theory. Set-Valued Anal. 13, 213\u2013241 (2005)","journal-title":"Set-Valued Anal."},{"key":"1162_CR23","doi-asserted-by":"crossref","first-page":"243","DOI":"10.1007\/s11228-004-7198-6","volume":"13","author":"AS Lewis","year":"2005","unstructured":"Lewis, A.S., Sendov, H.S.: Nonsmooth analysis of singular values. Part II: application. Set-Valued Anal. 13, 243\u2013264 (2005)","journal-title":"Set-Valued Anal."},{"key":"1162_CR24","doi-asserted-by":"crossref","first-page":"399","DOI":"10.1007\/s10107-010-0437-8","volume":"133","author":"YJ Liu","year":"2012","unstructured":"Liu, Y.J., Sun, D.F., Toh, K.C.: An implementable proximal point algorithmic framework for nuclear norm minimization. Math. Program. 133, 399\u2013436 (2012)","journal-title":"Math. Program."},{"key":"1162_CR25","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1007\/BF01170633","volume":"38","author":"K L\u00f6wner","year":"1934","unstructured":"L\u00f6wner, K.: \u00dcber monotone matrixfunktionen. Math. Z. 38, 177\u2013216 (1934)","journal-title":"Math. Z."},{"key":"1162_CR26","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1109\/TIT.1979.1055985","volume":"25","author":"L Lov\u00e1sz","year":"1979","unstructured":"Lov\u00e1sz, L.: On the Shannon capacity of a graph. IEEE Trans. Inf. Theory 25, 1\u20137 (1979)","journal-title":"IEEE Trans. Inf. Theory"},{"key":"1162_CR27","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1007\/s10107-015-0961-7","volume":"159","author":"WM Miao","year":"2016","unstructured":"Miao, W.M., Sun, D.F., Pan, S.H.: A rank-corrected procedure for matrix completion with fixed basis coefficients. Math. Program. 159, 289\u2013338 (2016)","journal-title":"Math. Program."},{"key":"1162_CR28","doi-asserted-by":"crossref","first-page":"959","DOI":"10.1137\/0315061","volume":"15","author":"R Mifflin","year":"1977","unstructured":"Mifflin, R.: Semismooth and semiconvex functions in constrained optimization. SIAM J. Control Optim. 15, 959\u2013972 (1977)","journal-title":"SIAM J. Control Optim."},{"key":"1162_CR29","doi-asserted-by":"crossref","first-page":"691","DOI":"10.1080\/01630560701380999","volume":"28","author":"H Mohebi","year":"2007","unstructured":"Mohebi, H., Salemi, A.: Analysis of symmetric matrix valued functions. Numer. Funct. Anal. Optim. 28, 691\u2013715 (2007)","journal-title":"Numer. Funct. Anal. Optim."},{"key":"1162_CR30","doi-asserted-by":"crossref","first-page":"226","DOI":"10.1287\/moor.2014.0669","volume":"40","author":"BS Mordukhovich","year":"2015","unstructured":"Mordukhovich, B.S., Nghia, T.T.A., Rockafellar, R.T.: Full stability in finite-dimensional optimization. Math. Oper. Res. 40, 226\u2013252 (2015)","journal-title":"Math. Oper. Res."},{"key":"1162_CR31","first-page":"1067","volume":"93","author":"J-J Moreau","year":"1965","unstructured":"Moreau, J.-J.: Proximit\u00e9 et dualit\u00e9 dans un espace hilbertien. Bull. Soc. Math. Fr. 93, 1067\u20131070 (1965)","journal-title":"Bull. Soc. Math. Fr."},{"key":"1162_CR32","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1016\/B978-0-12-576350-9.50006-0","volume-title":"Nonlinear Functional Analysis and Applications","author":"MZ Nashed","year":"1971","unstructured":"Nashed, M.Z.: Differentiability and related properties of nonlinear operators: some aspects of the role of differentials in nonlinear functional analysis. In: Rall, L.B. (ed.) Nonlinear Functional Analysis and Applications, pp. 103\u2013309. Academic Press, New York (1971)"},{"key":"1162_CR33","volume-title":"Iterative Solution of Nonlinear Equations in Several Variables","author":"JM Ortega","year":"1970","unstructured":"Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. SIAM, Philadelphia (1970)"},{"key":"1162_CR34","doi-asserted-by":"crossref","first-page":"766","DOI":"10.1137\/S0895479802417921","volume":"25","author":"HD Qi","year":"2003","unstructured":"Qi, H.D., Yang, X.Q.: Semismoothness of spectral functions. SIAM J. Matrix Anal. Appl. 25, 766\u2013783 (2003)","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"1162_CR35","doi-asserted-by":"crossref","first-page":"353","DOI":"10.1007\/BF01581275","volume":"58","author":"L Qi","year":"1993","unstructured":"Qi, L., Sun, J.: A nonsmooth version of Newton\u2019s method. Math. Program. 58, 353\u2013367 (1993)","journal-title":"Math. Program."},{"key":"1162_CR36","doi-asserted-by":"crossref","first-page":"471","DOI":"10.1137\/070697835","volume":"52","author":"B Recht","year":"2010","unstructured":"Recht, B., Fazel, M., Parrilo, P.A.: Guaranteed minimum rank solutions to linear matrix equations via nuclear norm minimization. SIAM Rev. 52, 471\u2013501 (2010)","journal-title":"SIAM Rev."},{"key":"1162_CR37","doi-asserted-by":"crossref","DOI":"10.1515\/9781400873173","volume-title":"Convex Analysis","author":"RT Rockafellar","year":"1970","unstructured":"Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)"},{"key":"1162_CR38","doi-asserted-by":"crossref","first-page":"761","DOI":"10.1287\/moor.1060.0195","volume":"31","author":"DF Sun","year":"2006","unstructured":"Sun, D.F.: The strong second order sufficient condition and constraint nondegeneracy in nonlinear semidefinite programming and their implications. Math. Oper. Res. 31, 761\u2013776 (2006)","journal-title":"Math. Oper. Res."},{"key":"1162_CR39","doi-asserted-by":"crossref","first-page":"150","DOI":"10.1287\/moor.27.1.150.342","volume":"27","author":"DF Sun","year":"2002","unstructured":"Sun, D.F., Sun, J.: Semismooth matrix-valued functions. Math. Oper. Res. 27, 150\u2013169 (2002)","journal-title":"Math. Oper. Res."},{"key":"1162_CR40","doi-asserted-by":"crossref","first-page":"421","DOI":"10.1287\/moor.1070.0300","volume":"33","author":"DF Sun","year":"2008","unstructured":"Sun, D.F., Sun, J.: L\u00f6wner\u2019s operator and spectral functions in Euclidean Jordan algebras. Math. Oper. Res. 33, 421\u2013445 (2008)","journal-title":"Math. Oper. Res."},{"key":"1162_CR41","doi-asserted-by":"crossref","first-page":"515","DOI":"10.1017\/S0962492901000071","volume":"10","author":"MJ Todd","year":"2001","unstructured":"Todd, M.J.: Semidefinite optimization. Acta Numer. 10, 515\u2013560 (2001)","journal-title":"Acta Numer."},{"key":"1162_CR42","doi-asserted-by":"crossref","first-page":"30","DOI":"10.1137\/S089547989427909X","volume":"18","author":"KC Toh","year":"1997","unstructured":"Toh, K.C.: GMRES vs. ideal GMRES. SIAM J. Matrix Anal. Appl. 18, 30\u201336 (1997)","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"1162_CR43","doi-asserted-by":"crossref","first-page":"400","DOI":"10.1137\/S0895479896303739","volume":"20","author":"KC Toh","year":"1998","unstructured":"Toh, K.C., Trefethen, L.N.: The Chebyshev polynomials of a matrix. SIAM J. Matrix Anal. Appl. 20, 400\u2013419 (1998)","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"1162_CR44","unstructured":"Wright, J., Ma, Y., Ganesh, A., Rao, S.: Robust principal component analysis: exact recovery of corrupted low-rank matrices via convex optimization. In: Bengio, Y., Schuurmans, D., Lafferty, J., Williams, C. (eds.), Advances in Neural Information Processing Systems 22 (2009)"},{"key":"1162_CR45","doi-asserted-by":"crossref","first-page":"766","DOI":"10.1137\/110827144","volume":"24","author":"B Wu","year":"2014","unstructured":"Wu, B., Ding, C., Sun, D.F., Toh, K.C.: On the Moreau\u2013Yosida regularization of the vector k-norm related functions. SIAM J. Optim. 24, 766\u2013794 (2014)","journal-title":"SIAM J. Optim."},{"key":"1162_CR46","doi-asserted-by":"crossref","first-page":"331","DOI":"10.1007\/s12532-015-0082-6","volume":"7","author":"LQ Yang","year":"2015","unstructured":"Yang, L.Q., Sun, D.F., Toh, K.C.: SDPNAL $$+$$ + : a majorized semismooth Newton-CG augmented Lagrangian method for semidefinite programming with nonnegative constraints. Math. Program. Comput. 7, 331\u2013366 (2015)","journal-title":"Math. Program. Comput."},{"key":"1162_CR47","unstructured":"Yang, Z.: A study on nonsymmetric matrix-valued functions. Master\u2019s Thesis, National University of Singapore. http:\/\/www.math.nus.edu.sg\/~matsundf\/Main_YZ.pdf (2009)"},{"key":"1162_CR48","doi-asserted-by":"crossref","first-page":"1737","DOI":"10.1137\/080718206","volume":"20","author":"XY Zhao","year":"2010","unstructured":"Zhao, X.Y., Sun, D.F., Toh, K.C.: A Newton-CG augmented Lagrangian method for semidefinite programming. SIAM J. Optim. 20, 1737\u20131765 (2010)","journal-title":"SIAM J. Optim."}],"container-title":["Mathematical Programming"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10107-017-1162-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10107-017-1162-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10107-017-1162-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,9,24]],"date-time":"2019-09-24T17:31:04Z","timestamp":1569346264000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10107-017-1162-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,5,15]]},"references-count":48,"journal-issue":{"issue":"1-2","published-print":{"date-parts":[[2018,3]]}},"alternative-id":["1162"],"URL":"https:\/\/doi.org\/10.1007\/s10107-017-1162-3","relation":{},"ISSN":["0025-5610","1436-4646"],"issn-type":[{"value":"0025-5610","type":"print"},{"value":"1436-4646","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,5,15]]}}}