{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,29]],"date-time":"2026-05-29T21:42:42Z","timestamp":1780090962242,"version":"3.54.0"},"reference-count":37,"publisher":"Springer Science and Business Media LLC","issue":"1-2","license":[{"start":{"date-parts":[[2015,2,21]],"date-time":"2015-02-21T00:00:00Z","timestamp":1424476800000},"content-version":"tdm","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":[[2016,3]]},"DOI":"10.1007\/s10107-015-0874-5","type":"journal-article","created":{"date-parts":[[2015,2,20]],"date-time":"2015-02-20T06:01:09Z","timestamp":1424412069000},"page":"161-187","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":39,"title":["A Lagrangian\u2013DNN relaxation: a fast method for computing tight lower bounds for a class of quadratic optimization problems"],"prefix":"10.1007","volume":"156","author":[{"given":"Sunyoung","family":"Kim","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Masakazu","family":"Kojima","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Kim-Chuan","family":"Toh","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2015,2,21]]},"reference":[{"key":"874_CR1","doi-asserted-by":"crossref","first-page":"2320","DOI":"10.1137\/120890636","volume":"23","author":"N Arima","year":"2013","unstructured":"Arima, N., Kim, S., Kojima, M.: A quadratically constrained quadratic optimization model for completely positive cone programming. SIAM J. Optim. 23, 2320\u20132340 (2013)","journal-title":"SIAM J. Optim."},{"key":"874_CR2","first-page":"437","volume":"10","author":"N Arima","year":"2013","unstructured":"Arima, N., Kim, S., Kojima, M.: Simplified copositive and Lagrangian relaxations for linearly constrained quadratic optimization problems in continuous and binary variables. Pac. J. Optim. 10, 437\u2013451 (2013)","journal-title":"Pac. J. Optim."},{"key":"874_CR3","unstructured":"Arima, N., Kim, S., Kojima, M.: Extension of completely positive cone relaxation to polynomial optimization, Research report B-471, Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Tokyo, Feb 2013"},{"key":"874_CR4","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1137\/080716542","volume":"2","author":"A Beck","year":"2009","unstructured":"Beck, A., Teboulle, M.: A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2, 183\u2013202 (2009)","journal-title":"SIAM J. Imaging Sci."},{"key":"874_CR5","volume-title":"Nonlinear Programming","author":"DP Bertsekas","year":"1999","unstructured":"Bertsekas, D.P.: Nonlinear Programming, 2nd edn. Athena Scientific, Belmont (1999)","edition":"2"},{"key":"874_CR6","unstructured":"BIQMAC Library. http:\/\/www.biqmac.uni-klu.ac.at\/biqmaclib.html"},{"key":"874_CR7","doi-asserted-by":"crossref","first-page":"613","DOI":"10.1080\/10556789908805765","volume":"11","author":"B Borcher","year":"1999","unstructured":"Borcher, B.: CSDP, a C library for semidefinite programming. Optim. Methods Softw. 11, 613\u2013623 (1999)","journal-title":"Optim. Methods Softw."},{"key":"874_CR8","doi-asserted-by":"crossref","first-page":"30","DOI":"10.1137\/070711815","volume":"20","author":"S Bundfuss","year":"2009","unstructured":"Bundfuss, S., D\u00fcr, M.: An adaptive linear approximation algorithm for copositive programs. SIAM J. Optim. 20, 30\u201353 (2009)","journal-title":"SIAM J. Optim."},{"key":"874_CR9","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1007\/s10107-008-0223-z","volume":"120","author":"S Burer","year":"2009","unstructured":"Burer, S.: On the copositive representation of binary and continuous non-convex quadratic programs. Math. Program. 120, 479\u2013495 (2009)","journal-title":"Math. Program."},{"key":"874_CR10","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s12532-010-0010-8","volume":"2","author":"S Burer","year":"2010","unstructured":"Burer, S.: Optimizating a polyhedral-semidefinite relaxation of completely positive programs. Math. Program. Comput. 2, 1\u201319 (2010)","journal-title":"Math. Program. Comput."},{"key":"874_CR11","doi-asserted-by":"crossref","unstructured":"Chen, C., He, B., Ye, Y., Yuan X.: The direct extension of admm for multi-block convex minimization problems is not necessarily convergent. Math. Program. (2015, to appear)","DOI":"10.1007\/s10107-014-0826-5"},{"key":"874_CR12","doi-asserted-by":"crossref","first-page":"875","DOI":"10.1137\/S1052623401383248","volume":"12","author":"E Klerk de","year":"2002","unstructured":"de Klerk, E., Pasechnik, D.V.: Approximation of the stability number of a graph via copositive programming. SIAM J. Optim. 12, 875\u2013892 (2002)","journal-title":"SIAM J. Optim."},{"key":"874_CR13","unstructured":"Dickinson, P.J.C., Eichfelder, G., Povh, J.: Erratum to: \u201cOn the set-semidefinite representation of nonconvex quadratic programs over arbitrary feasible sets\u201d [Optim. Letters, 2012]. Optimization Online (2012). http:\/\/www.optimization-online.org\/DB_HTML\/2012\/09\/3598.html"},{"key":"874_CR14","unstructured":"DIMACS, Second DIMACS Challenge, Test instances available at: http:\/\/dimacs.rutgers.edu\/Challenges\/"},{"key":"874_CR15","first-page":"28","volume":"17","author":"M D\u00fcr","year":"2008","unstructured":"D\u00fcr, M., Still, G.: Interior points of the completely positive cone. Electron. J. Linear Algebra 17, 28\u201333 (2008)","journal-title":"Electron. J. Linear Algebra"},{"key":"874_CR16","author":"G Eichfelder","year":"2012","unstructured":"Eichfelder, G., Povh, J.: On the set-semidefinite representation of nonconvex quadratic programs over arbitrary feasible sets. Optim. Lett. (2012). doi: 10.1007\/s11590-012-0450-3","journal-title":"Optim. Lett."},{"key":"874_CR17","doi-asserted-by":"crossref","unstructured":"Fazel, M., Pong, Ti. K., Sun, D., Tseng, P.: Hankel matrix rank minimization with applications to system identification and realization. SIAM J. Matrix Anal. A 34, 946\u2013977 (2013)","DOI":"10.1137\/110853996"},{"key":"874_CR18","unstructured":"Fujisawa, K., Fukuda, M., Kobayashi, K., Kojima, M., Nakata, K., Nakata, M., Yamashita, M.: SDPA (semidefinite programming algorithm) user\u2019s manual\u2014version 7.05. Research report B-448, Dept. of Mathematical and Computing Sciences, Tokyo Institute of Technology, Tokyo (2008)"},{"key":"874_CR19","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1016\/0898-1221(76)90003-1","volume":"2","author":"D Gabay","year":"1976","unstructured":"Gabay, D., Mercier, B.: A dual algorithm for the solution of nonlinear variational problems via finite element approximations. Comput. Math. Appl. 2, 17\u201340 (1976)","journal-title":"Comput. Math. Appl."},{"key":"874_CR20","unstructured":"Ge, D., Ye. Y.: On doubly positive semidefinite programming relaxations. Optimization Online (2010). http:\/\/www.optimization-online.org\/DB_HTML\/2010\/08\/2709.html"},{"key":"874_CR21","first-page":"188","volume":"46","author":"C Jansson","year":"2007","unstructured":"Jansson, C., Keil, C.: Rigorous error bounds for the optimal value in semidefinite programming. SIAM J. Numer. Anal. 46, 188\u2013200 (2007)","journal-title":"SIAM J. Numer. Anal."},{"key":"874_CR22","doi-asserted-by":"crossref","unstructured":"Glowinski, R., Marroco, A.: Sur l\u2019approximation, par \u00e9l\u00e9ments finis d\u2019ordre un, et la r\u00e9solution, par p\u00e9nalisation-dualit\u00e9, d\u2019une classe de probl\u00e8mes de Dirichlet non lin\u00e9ares. Revue Francaise d\u2019Automatique, Informatique et Recherche Op\u00e9rationelle 9(R-2), 41\u201376 (1975)","DOI":"10.1051\/m2an\/197509R200411"},{"key":"874_CR23","first-page":"238","volume":"255","author":"JJ Moreau","year":"1962","unstructured":"Moreau, J.J.: D\u00e9composition orthogonale d\u2019un espace hilbertien selon deux cones mutuellement polaires. C. R. Acad. Sci. 255, 238\u2013240 (1962)","journal-title":"C. R. Acad. Sci."},{"key":"874_CR24","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1007\/BF02592948","volume":"39","author":"KG Murty","year":"1987","unstructured":"Murty, K.G., Kabadi, S.N.: Some NP-complete problems in quadratic and non-linear programming. Math. Program. 39, 117\u2013129 (1987)","journal-title":"Math. Program."},{"key":"874_CR25","doi-asserted-by":"crossref","DOI":"10.1137\/1.9781611970791","volume-title":"Interior Point Methods for Convex Programming","author":"YE Nesterov","year":"1994","unstructured":"Nesterov, Y.E., Nemirovskii, A.: Interior Point Methods for Convex Programming. SIAM, Philadelphia (1994)"},{"key":"874_CR26","doi-asserted-by":"crossref","first-page":"231","DOI":"10.1016\/j.disopt.2009.01.002","volume":"6","author":"J Povh","year":"2009","unstructured":"Povh, J., Rendl, F.: Copositive and semidefinite relaxations of the quadratic assignment problem. Discrete Optim. 6, 231\u2013224 (2009)","journal-title":"Discrete Optim."},{"key":"874_CR27","unstructured":"Quadratic assignment problems. http:\/\/www.seas.upenn.edu\/qaplib"},{"key":"874_CR28","doi-asserted-by":"crossref","first-page":"307","DOI":"10.1007\/s10107-008-0235-8","volume":"121","author":"F Rendl","year":"2010","unstructured":"Rendl, F., Rinaldi, G., Wiegele, A.: Solving max-cut to optimality by intersecting semidefinite and polyhedral relaxations. Math. Program. 121, 307\u2013335 (2010)","journal-title":"Math. Program."},{"key":"874_CR29","doi-asserted-by":"crossref","unstructured":"Sarac, T., Sipahioglu, A.: A genetic algorithm for the quadratic multiple knapsack problem. In: Advances in Brain, Vision, and Artificial Intelligence, vol. 4729 of Lecture Notes in Computer Science, pp. 490\u2013498. Springer, Heidelberg (2007)","DOI":"10.1007\/978-3-540-75555-5_47"},{"key":"874_CR30","unstructured":"Sloane, N.: Challenge problems: independent sets in graphs. http:\/\/neilsloane.com\/doc\/graphs.html"},{"key":"874_CR31","doi-asserted-by":"crossref","first-page":"625","DOI":"10.1080\/10556789908805766","volume":"11 & 12","author":"JF Strum","year":"1999","unstructured":"Strum, J.F.: SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optim. Methods Softw. 11 & 12, 625\u2013653 (1999)","journal-title":"Optim. Methods Softw."},{"key":"874_CR32","doi-asserted-by":"crossref","first-page":"545","DOI":"10.1080\/10556789908805762","volume":"11 & 12","author":"K Toh","year":"1999","unstructured":"Toh, K., Todd, M.J., T\u00fct\u00fcnt\u00fc, R.H.: SDPT3\u2014a MATLAB software package for semidefinite programming. Optim. Methods Softw. 11 & 12, 545\u2013581 (1999)","journal-title":"Optim. Methods Softw."},{"key":"874_CR33","doi-asserted-by":"crossref","unstructured":"Tucker, A.W.: Dual systems of homogeneous linear relations. In: Kuhn, Tucker (eds.) Linear Inequalities and Related Systems, Annals of Mathematics Studies, No. 38. Princeton University Press, Princeton (1956)","DOI":"10.1515\/9781400881987-002"},{"key":"874_CR34","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1007\/s12532-010-0017-1","volume":"2","author":"Z Wen","year":"2010","unstructured":"Wen, Z., Goldfarb, D., Yin, W.: Alternating direction augmented Lagrangian methods for semidefinite programming. Math. Program. Comput. 2, 203\u2013230 (2010)","journal-title":"Math. Program. Comput."},{"key":"874_CR35","unstructured":"Sun, D.F., Toh, K.C., Yang, L.Q.: A convergent proximal alternating direction method of multipliers for conic programming with 4-block constraints (2014). arXiv:1404.5378"},{"key":"874_CR36","doi-asserted-by":"crossref","unstructured":"Yoshise, A., Matsukawa, Y.: On optimization over the doubly nonnegative cone. In: Proceedings of 2010 IEEE Multi-conference on Systems and Control, pp. 13\u201319 (2010)","DOI":"10.1109\/CACSD.2010.5612811"},{"key":"874_CR37","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\/content\/pdf\/10.1007\/s10107-015-0874-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10107-015-0874-5\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10107-015-0874-5","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,21]],"date-time":"2019-08-21T03:22:07Z","timestamp":1566357727000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10107-015-0874-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,2,21]]},"references-count":37,"journal-issue":{"issue":"1-2","published-print":{"date-parts":[[2016,3]]}},"alternative-id":["874"],"URL":"https:\/\/doi.org\/10.1007\/s10107-015-0874-5","relation":{},"ISSN":["0025-5610","1436-4646"],"issn-type":[{"value":"0025-5610","type":"print"},{"value":"1436-4646","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,2,21]]}}}