{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,31]],"date-time":"2025-10-31T07:55:47Z","timestamp":1761897347564,"version":"3.37.3"},"reference-count":82,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2020,9,23]],"date-time":"2020-09-23T00:00:00Z","timestamp":1600819200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2020,9,23]],"date-time":"2020-09-23T00:00:00Z","timestamp":1600819200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100000848","name":"University of Edinburgh","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100000848","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Math. Prog. Comp."],"published-print":{"date-parts":[[2021,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>The Alternating Direction Method of Multipliers (ADMM) has gained a lot of attention for solving large-scale and objective-separable constrained optimization. However, the two-block variable structure of the ADMM still limits the practical computational efficiency of the method, because one big matrix factorization is needed at least once even for linear and convex quadratic programming. This drawback may be overcome by enforcing a multi-block structure of the decision variables in the original optimization problem. Unfortunately, the multi-block ADMM, with more than two blocks, is not guaranteed to be convergent. On the other hand, two positive developments have been made: first, if in each cyclic loop one randomly permutes the updating order of the multiple blocks, then the method converges in expectation for solving any system of linear equations with any number of blocks. Secondly, such a randomly permuted ADMM also works for equality-constrained convex quadratic programming even when the objective function is not separable. The goal of this paper is twofold. First, we add more randomness into the ADMM by developing a randomly assembled cyclic ADMM (RAC-ADMM) where the decision variables in each block are randomly assembled. We discuss the theoretical properties of RAC-ADMM and show when random assembling helps and when it hurts, and develop a criterion to guarantee that it converges almost surely. Secondly, using the theoretical guidance on RAC-ADMM, we conduct multiple numerical tests on solving both randomly generated and large-scale benchmark quadratic optimization problems, which include continuous, and binary graph-partition and quadratic assignment, and selected machine learning problems. Our numerical tests show that the RAC-ADMM, with a variable-grouping strategy, could significantly improve the computation efficiency on solving most quadratic optimization problems.<\/jats:p>","DOI":"10.1007\/s12532-020-00192-5","type":"journal-article","created":{"date-parts":[[2020,9,23]],"date-time":"2020-09-23T12:02:52Z","timestamp":1600862572000},"page":"339-413","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Managing randomization in the multi-block alternating direction method of multipliers for quadratic optimization"],"prefix":"10.1007","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7896-2427","authenticated-orcid":false,"given":"Kre\u0161imir","family":"Mihi\u0107","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5513-0038","authenticated-orcid":false,"given":"Mingxi","family":"Zhu","sequence":"additional","affiliation":[]},{"given":"Yinyu","family":"Ye","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2020,9,23]]},"reference":[{"issue":"4","key":"192_CR1","doi-asserted-by":"crossref","first-page":"1067","DOI":"10.1007\/s11081-019-09450-5","volume":"20","author":"A Allman","year":"2019","unstructured":"Allman, A., Tang, W., Daoutidis, P.: Decode: a community-based algorithm for generating high-quality decompositions of optimization problems. Optim. Eng. 20(4), 1067\u20131084 (2019)","journal-title":"Optim. Eng."},{"unstructured":"Baingana, B., Traganitis, P., Giannakis, G., Mateos, G.: Big data analytics for social networks (2015)","key":"192_CR2"},{"doi-asserted-by":"crossref","unstructured":"Bastani, H., Bayati, M.: Online decision-making with high-dimensional covariates. SSRN 2661896 (2015)","key":"192_CR3","DOI":"10.2139\/ssrn.2661896"},{"issue":"3","key":"192_CR4","doi-asserted-by":"crossref","first-page":"1162","DOI":"10.1016\/j.engappai.2012.09.001","volume":"26","author":"U Benlic","year":"2013","unstructured":"Benlic, U., Hao, J.K.: Breakout local search for the max-cutproblem. Eng. Appl. Artif. Intell. 26(3), 1162\u20131173 (2013)","journal-title":"Eng. Appl. Artif. Intell."},{"unstructured":"Bertsekas, D.P.: Incremental aggregated proximal and augmented Lagrangian algorithms. CoRR arXiv:1509.09257 (2015)","key":"192_CR5"},{"key":"192_CR6","volume-title":"Parallel and Distributed Computation: Numerical Methods","author":"DP Bertsekas","year":"1989","unstructured":"Bertsekas, D.P., Tsitsiklis, J.N.: Parallel and Distributed Computation: Numerical Methods, vol. 23. Prentice Hall Englewood Cliffs, NJ (1989)"},{"issue":"10","key":"192_CR7","doi-asserted-by":"crossref","first-page":"P10008","DOI":"10.1088\/1742-5468\/2008\/10\/P10008","volume":"2008","author":"VD Blondel","year":"2008","unstructured":"Blondel, V.D., Guillaume, J.L., Lambiotte, R., Lefebvre, E.: Fast unfolding of communities in large networks. J. Stat. Mech. Theory Exp. 2008(10), P10008 (2008)","journal-title":"J. Stat. Mech. Theory Exp."},{"doi-asserted-by":"crossref","unstructured":"Boser, B.E., Guyon, I.M., Vapnik, V.N.: A training algorithm for optimal margin classifiers. In: Proceedings of the Fifth Annual Workshop on Computational Learning Theory, pp. 144\u2013152. ACM (1992)","key":"192_CR8","DOI":"10.1145\/130385.130401"},{"issue":"1","key":"192_CR9","first-page":"301","volume":"3","author":"L Bottou","year":"2007","unstructured":"Bottou, L., Lin, C.J.: Support vector machine solvers. Large Scale Kernel Machines 3(1), 301\u2013320 (2007)","journal-title":"Large Scale Kernel Machines"},{"issue":"1","key":"192_CR10","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1561\/2200000016","volume":"3","author":"S Boyd","year":"2011","unstructured":"Boyd, S., Parikh, N., Chu, E., Peleato, B., Eckstein, J., et al.: Distributed optimization and statistical learning via the alternating direction method of multipliers. Found. Trends Mach. Learn. 3(1), 1\u2013122 (2011)","journal-title":"Found. Trends Mach. Learn."},{"unstructured":"Burkard, R.E., Karisch, S.E., Rendl, F.: QAPLIB\u2014a quadratic assignment problem library. J. Global Optim. 10(4), 391\u2013403 (1997). Revised 02.04.2003 (electronic update): http:\/\/www.seas.upenn.edu\/qaplib\/","key":"192_CR11"},{"issue":"2","key":"192_CR12","doi-asserted-by":"crossref","first-page":"1008","DOI":"10.1137\/140954362","volume":"26","author":"RH Byrd","year":"2016","unstructured":"Byrd, R.H., Hansen, S.L., Nocedal, J., Singer, Y.: A stochastic quasi-newton method for large-scale optimization. SIAM J. Optim. 26(2), 1008\u20131031 (2016)","journal-title":"SIAM J. Optim."},{"key":"192_CR13","first-page":"230","volume":"229","author":"X Cai","year":"2014","unstructured":"Cai, X., Han, D., Yuan, X.: The direct extension of ADMM for three-block separable convex minimization models is convergent when one function is strongly convex. Optim. Online 229, 230 (2014)","journal-title":"Optim. Online"},{"doi-asserted-by":"crossref","unstructured":"Chang, C.C., Lin, C.J.: LIBSVM: a library for support vector machines. ACM Trans. Intell. Syst. Technol. 2, 27:1\u201327:27 (2011). Software available at http:\/\/www.csie.ntu.edu.tw\/~cjlin\/libsvm","key":"192_CR14","DOI":"10.1145\/1961189.1961199"},{"issue":"1","key":"192_CR15","doi-asserted-by":"publisher","first-page":"57","DOI":"10.1007\/s10107-014-0826-5","volume":"155","author":"C Chen","year":"2016","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. 155(1), 57\u201379 (2016). https:\/\/doi.org\/10.1007\/s10107-014-0826-5","journal-title":"Math. Program."},{"unstructured":"Chen, C., Li, M., Liu, X., Ye, Y.: On the convergence of multi-block alternating direction method of multipliers and block coordinate descent method (2015). http:\/\/www.optimization-online.org\/DB_HTML\/2015\/08\/5046.html","key":"192_CR16"},{"key":"192_CR17","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-017-1205-9","author":"C Chen","year":"2017","unstructured":"Chen, C., Li, M., Liu, X., Ye, Y.: Extended ADMM and BCD for nonseparable convex minimization models with quadratic coupling terms: convergence analysis and insights. Math. Program. (2017). https:\/\/doi.org\/10.1007\/s10107-017-1205-9","journal-title":"Math. Program."},{"issue":"3","key":"192_CR18","first-page":"273","volume":"20","author":"C Cortes","year":"1995","unstructured":"Cortes, C., Vapnik, V.: Support-vector networks. Mach. Learn. 20(3), 273\u2013297 (1995)","journal-title":"Mach. Learn."},{"key":"192_CR19","volume-title":"Discrete-Time Markov Jump Linear Systems","author":"OLV Costa","year":"2006","unstructured":"Costa, O.L.V., Fragoso, M.D., Marques, R.P.: Discrete-Time Markov Jump Linear Systems. Springer, Berlin (2006)"},{"issue":"3","key":"192_CR20","doi-asserted-by":"crossref","first-page":"889","DOI":"10.1007\/s10915-015-0048-x","volume":"66","author":"W Deng","year":"2016","unstructured":"Deng, W., Yin, W.: On the global and linear convergence of the generalized alternating direction method of multipliers. J. Sci. Comput. 66(3), 889\u2013916 (2016)","journal-title":"J. Sci. Comput."},{"issue":"1","key":"192_CR21","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1007\/s10479-005-3444-z","volume":"139","author":"Z Drezner","year":"2005","unstructured":"Drezner, Z., Hahn, P., Taillard, \u00c9.D.: Recent advances for the quadratic assignment problem with special emphasis on instances that are difficult for meta-heuristic methods. Ann. Oper. Res. 139(1), 65\u201394 (2005)","journal-title":"Ann. Oper. Res."},{"issue":"1\u20133","key":"192_CR22","doi-asserted-by":"crossref","first-page":"293","DOI":"10.1007\/BF01581204","volume":"55","author":"J Eckstein","year":"1992","unstructured":"Eckstein, J., Bertsekas, D.P.: On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators. Math. Program. 55(1\u20133), 293\u2013318 (1992)","journal-title":"Math. Program."},{"issue":"4","key":"192_CR23","first-page":"619","volume":"11","author":"J Eckstein","year":"2015","unstructured":"Eckstein, J., Yao, W.: Understanding the convergence of the alternating direction method of multipliers: theoretical and computational perspectives. Pac. J. Optim. 11(4), 619\u2013644 (2015)","journal-title":"Pac. J. Optim."},{"issue":"4","key":"192_CR24","doi-asserted-by":"crossref","first-page":"1015","DOI":"10.1137\/09076934X","volume":"3","author":"E Esser","year":"2010","unstructured":"Esser, E., Zhang, X., Chan, T.F.: A general framework for a class of first order primal-dual algorithms for convex optimization in imaging science. SIAM J. Imaging Sci. 3(4), 1015\u20131046 (2010)","journal-title":"SIAM J. Imaging Sci."},{"doi-asserted-by":"crossref","unstructured":"Evstigneev, I.V., Hens, T., Schenk-Hopp\u00e9, K.R.: Mean\u2013variance portfolio analysis: the markowitz model. In: Mathematical Financial Economics, pp. 11\u201318. Springer, Berlin (2015)","key":"192_CR25","DOI":"10.1007\/978-3-319-16571-4_2"},{"key":"192_CR26","first-page":"1889","volume":"6","author":"RE Fan","year":"2005","unstructured":"Fan, R.E., Chen, P.H., Lin, C.J.: Working set selection using second order information for training support vector machines. J. Mach. Learn. Res. 6, 1889\u20131918 (2005)","journal-title":"J. Mach. Learn. Res."},{"issue":"1","key":"192_CR27","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1007\/BF01582130","volume":"80","author":"MC Ferris","year":"1998","unstructured":"Ferris, M.C., Horn, J.D.: Partitioning mathematical programs for parallel solution. Math. Program. 80(1), 35\u201361 (1998)","journal-title":"Math. Program."},{"issue":"4","key":"192_CR28","doi-asserted-by":"crossref","first-page":"707","DOI":"10.1109\/JSTSP.2011.2114324","volume":"5","author":"PA Forero","year":"2011","unstructured":"Forero, P.A., Cano, A., Giannakis, G.B.: Distributed clustering using wireless sensor networks. IEEE J. Sel. Top. Signal Process. 5(4), 707\u2013724 (2011)","journal-title":"IEEE J. Sel. Top. Signal Process."},{"issue":"2","key":"192_CR29","doi-asserted-by":"crossref","first-page":"302","DOI":"10.1214\/07-AOAS131","volume":"1","author":"J Friedman","year":"2007","unstructured":"Friedman, J., Hastie, T., H\u00f6fling, H., Tibshirani, R., et al.: Pathwise coordinate optimization. Ann. Appl. Stat. 1(2), 302\u2013332 (2007)","journal-title":"Ann. Appl. Stat."},{"issue":"1","key":"192_CR30","doi-asserted-by":"crossref","first-page":"1","DOI":"10.18637\/jss.v033.i01","volume":"33","author":"J Friedman","year":"2010","unstructured":"Friedman, J., Hastie, T., Tibshirani, R.: Regularization paths for generalized linear models via coordinate descent. J. Stat. Softw. 33(1), 1 (2010)","journal-title":"J. Stat. Softw."},{"issue":"1","key":"192_CR31","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 approximation. Comput. Math. Appl. 2(1), 17\u201340 (1976)","journal-title":"Comput. Math. Appl."},{"key":"192_CR32","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1007\/978-94-017-9054-3_4","volume-title":"Modeling, Simulation and Optimization for Science and Technology","author":"R Glowinski","year":"2014","unstructured":"Glowinski, R.: On alternating direction methods of multipliers: a historical perspective. In: Ditzgibbon, W., Kuznetsov, Y., Neittaanm\u00e4ki, P., Pironneau, O. (eds.) Modeling, Simulation and Optimization for Science and Technology, pp. 59\u201382. Springer, Berlin (2014)"},{"unstructured":"Gset http:\/\/web.stanford.edu\/yyye\/yyye\/Gset\/","key":"192_CR33"},{"unstructured":"Gurobi Optimizer 8.1.1 http:\/\/gurobi.com\/ (2018)","key":"192_CR34"},{"issue":"2","key":"192_CR35","doi-asserted-by":"crossref","first-page":"313","DOI":"10.1137\/110822347","volume":"22","author":"B He","year":"2012","unstructured":"He, B., Tao, M., Yuan, X.: Alternating direction method with Gaussian back substitution for separable convex programming. SIAM J. Optim. 22(2), 313\u2013340 (2012)","journal-title":"SIAM J. Optim."},{"issue":"5","key":"192_CR36","doi-asserted-by":"crossref","first-page":"303","DOI":"10.1007\/BF00927673","volume":"4","author":"MR Hestenes","year":"1969","unstructured":"Hestenes, M.R.: Multiplier and gradient methods. J. Optim. Theory Appl. 4(5), 303\u2013320 (1969)","journal-title":"J. Optim. Theory Appl."},{"issue":"1\u20132","key":"192_CR37","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1007\/s10107-016-1034-2","volume":"162","author":"M Hong","year":"2017","unstructured":"Hong, M., Luo, Z.Q.: On the linear convergence of the alternating direction method of multipliers. Math. Program. 162(1\u20132), 165\u2013199 (2017)","journal-title":"Math. Program."},{"issue":"1","key":"192_CR38","doi-asserted-by":"crossref","first-page":"337","DOI":"10.1137\/140990309","volume":"26","author":"M Hong","year":"2016","unstructured":"Hong, M., Luo, Z.Q., Razaviyayn, M.: Convergence analysis of alternating direction method of multipliers for a family of nonconvex problems. SIAM J. Optim. 26(1), 337\u2013364 (2016)","journal-title":"SIAM J. Optim."},{"issue":"20","key":"192_CR39","doi-asserted-by":"crossref","first-page":"5297","DOI":"10.1109\/TSP.2016.2593681","volume":"64","author":"K Huang","year":"2016","unstructured":"Huang, K., Sidiropoulos, N.D.: Consensus-ADMM for general quadratically constrained quadratic programming. IEEE Trans. Signal Process. 64(20), 5297\u20135310 (2016)","journal-title":"IEEE Trans. Signal Process."},{"doi-asserted-by":"crossref","unstructured":"Jiang, B., Lin, T., Ma, S., Zhang, S.: Structured nonconvex and nonsmooth optimization: algorithms and iteration complexity analysis. arXiv e-prints arXiv:1605.02408 (2018)","key":"192_CR40","DOI":"10.1007\/s10589-018-0034-y"},{"issue":"2","key":"192_CR41","doi-asserted-by":"crossref","first-page":"423","DOI":"10.1007\/s10107-014-0774-0","volume":"150","author":"B Jiang","year":"2015","unstructured":"Jiang, B., Ma, S., Zhang, S.: Tensor principal component analysis via convex optimization. Math. Program. 150(2), 423\u2013457 (2015)","journal-title":"Math. Program."},{"unstructured":"Joachims, T.: Making large-scale SVM learning practical. Technical report, SFB 475: Komplexit\u00e4tsreduktion in Multivariaten\u00a0...\u00a0(1998)","key":"192_CR42"},{"issue":"1","key":"192_CR43","doi-asserted-by":"crossref","first-page":"359","DOI":"10.1137\/S1064827595287997","volume":"20","author":"G Karypis","year":"1998","unstructured":"Karypis, G., Kumar, V.: A fast and high quality multilevel scheme for partitioning irregular graphs. SIAM J. Sci. Comput. 20(1), 359\u2013392 (1998)","journal-title":"SIAM J. Sci. Comput."},{"issue":"2","key":"192_CR44","doi-asserted-by":"crossref","first-page":"431","DOI":"10.1007\/s10915-013-9740-x","volume":"58","author":"R Lai","year":"2014","unstructured":"Lai, R., Osher, S.: A splitting method for orthogonality constrained problems. J. Sci. Comput. 58(2), 431\u2013449 (2014)","journal-title":"J. Sci. Comput."},{"unstructured":"Lin, T., Ma, S., Ye, Y., Zhang, S.: An ADMM-based interior-point method for large-scale linear programming. arXiv e-prints arXiv:1805.12344 (2017)","key":"192_CR45"},{"issue":"3","key":"192_CR46","doi-asserted-by":"crossref","first-page":"1478","DOI":"10.1137\/140971178","volume":"25","author":"T Lin","year":"2015","unstructured":"Lin, T., Ma, S., Zhang, S.: On the global linear convergence of the ADMM with multiblock variables. SIAM J. Optim. 25(3), 1478\u20131497 (2015)","journal-title":"SIAM J. Optim."},{"issue":"1","key":"192_CR47","doi-asserted-by":"crossref","first-page":"52","DOI":"10.1007\/s10915-016-0182-0","volume":"69","author":"T Lin","year":"2016","unstructured":"Lin, T., Ma, S., Zhang, S.: Iteration complexity analysis of multi-block ADMM for a family of convex minimization without strong convexity. J. Sci. Comput. 69(1), 52\u201381 (2016)","journal-title":"J. Sci. Comput."},{"key":"192_CR48","doi-asserted-by":"crossref","first-page":"78","DOI":"10.1016\/j.cor.2016.12.012","volume":"81","author":"F Ma","year":"2017","unstructured":"Ma, F., Hao, J.K., Wang, Y.: An effective iterated tabu search for the maximum bisection problem. Comput. Oper. Res. 81, 78\u201389 (2017)","journal-title":"Comput. Oper. Res."},{"issue":"1\u20134","key":"192_CR49","doi-asserted-by":"crossref","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)","journal-title":"Optim. Methods Softw."},{"unstructured":"Matlab R2018b https:\/\/www.mathworks.com\/ (2018)","key":"192_CR50"},{"issue":"2","key":"192_CR51","doi-asserted-by":"crossref","first-page":"295","DOI":"10.1287\/ijoc.2017.0781","volume":"30","author":"K Mihic","year":"2018","unstructured":"Mihic, K., Ryan, K., Wood, A.: Randomized decomposition solver with the quadratic assignment problem as a case study. INFORMS J. Comput. 30(2), 295\u2013308 (2018)","journal-title":"INFORMS J. Comput."},{"issue":"3","key":"192_CR52","doi-asserted-by":"crossref","first-page":"331","DOI":"10.1007\/s12293-019-00289-y","volume":"11","author":"A Misevi\u010dius","year":"2019","unstructured":"Misevi\u010dius, A.: New best known solution for the most difficult qap instance \u201ctai100a\u201d. Memet. Comput. 11(3), 331\u2013332 (2019)","journal-title":"Memet. Comput."},{"issue":"1","key":"192_CR53","first-page":"445","volume":"15","author":"K Mohan","year":"2014","unstructured":"Mohan, K., London, P., Fazel, M., Witten, D., Lee, S.I.: Node-based learning of multiple Gaussian graphical models. J. Mach. Learn. Res. 15(1), 445\u2013488 (2014)","journal-title":"J. Mach. Learn. Res."},{"issue":"1","key":"192_CR54","doi-asserted-by":"crossref","first-page":"475","DOI":"10.1137\/110849468","volume":"23","author":"RD Monteiro","year":"2013","unstructured":"Monteiro, R.D., Svaiter, B.F.: Iteration-complexity of block-decomposition algorithms and the alternating direction method of multipliers. SIAM J. Optim. 23(1), 475\u2013507 (2013)","journal-title":"SIAM J. Optim."},{"unstructured":"MOSEK version 8.1.0.49 https:\/\/www.mosek.com\/ (2018)","key":"192_CR55"},{"issue":"23","key":"192_CR56","doi-asserted-by":"crossref","first-page":"8577","DOI":"10.1073\/pnas.0601602103","volume":"103","author":"ME Newman","year":"2006","unstructured":"Newman, M.E.: Modularity and community structure in networks. Proc. Natl. Acad. Sci. 103(23), 8577\u20138582 (2006)","journal-title":"Proc. Natl. Acad. Sci."},{"unstructured":"Ohlsson, H., Yang, A., Dong, R., Sastry, S.: Cprl\u2014an extension of compressive sensing to the phase retrieval problem. In: Advances in Neural Information Processing Systems, pp. 1367\u20131375 (2012)","key":"192_CR57"},{"issue":"11","key":"192_CR58","doi-asserted-by":"crossref","first-page":"2233","DOI":"10.1109\/TPAMI.2011.282","volume":"34","author":"Y Peng","year":"2012","unstructured":"Peng, Y., Ganesh, A., Wright, J., Xu, W., Ma, Y.: Rasl: robust alignment by sparse and low-rank decomposition for linearly correlated images. IEEE Trans. Pattern Anal. Mach. Intell. 34(11), 2233\u20132246 (2012)","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"192_CR59","doi-asserted-by":"crossref","first-page":"224","DOI":"10.1007\/BF01588967","volume":"14","author":"MJD Powell","year":"1978","unstructured":"Powell, M.J.D.: Algorithms for nonlinear constraints that use Lagrangian functions. Math. Program. 14, 224\u2013248 (1978)","journal-title":"Math. Program."},{"unstructured":"QAPLIB http:\/\/anjos.mgi.polymtl.ca\/qaplib\/","key":"192_CR60"},{"doi-asserted-by":"publisher","unstructured":"RACQP https:\/\/github.com\/kmihic\/RACQP. https:\/\/doi.org\/10.5281\/zenodo.4008720","key":"192_CR61","DOI":"10.5281\/zenodo.4008720"},{"unstructured":"Rudi, A., Carratino, L., Rosasco, L.: Falkon: an optimal large scale kernel method. In: Advances in Neural Information Processing Systems, pp. 3888\u20133898 (2017)","key":"192_CR62"},{"unstructured":"SCIP Optimization Suite https:\/\/www.scipopt.org\/","key":"192_CR63"},{"issue":"5","key":"192_CR64","doi-asserted-by":"crossref","first-page":"1","DOI":"10.18637\/jss.v039.i05","volume":"39","author":"N Simon","year":"2011","unstructured":"Simon, N., Friedman, J., Hastie, T., Tibshirani, R.: Regularization paths for Cox\u2019s proportional hazards model via coordinate descent. J Stat Softw 39(5), 1\u201313 (2011)","journal-title":"J Stat Softw"},{"doi-asserted-by":"crossref","unstructured":"Stellato, B., Banjac, G., Goulart, P., Bemporad, A., Boyd, S.: OSQP: an operator splitting solver for quadratic programs. In: 2018 UKACC 12th International Conference on Control (CONTROL), pp. 339\u2013339. IEEE (2018)","key":"192_CR65","DOI":"10.1109\/CONTROL.2018.8516834"},{"doi-asserted-by":"crossref","unstructured":"Stellato, B., Naik, V.V., Bemporad, A., Goulart, P., Boyd, S.: Embedded mixed-integer quadratic optimization using the OSQP solver. In: 2018 European Control Conference (ECC), pp. 1536\u20131541. IEEE (2018)","key":"192_CR66","DOI":"10.23919\/ECC.2018.8550136"},{"doi-asserted-by":"crossref","unstructured":"Sun, D.L., Fevotte, C.: Alternating direction method of multipliers for non-negative matrix factorization with the beta-divergence. In: 2014 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 6201\u20136205. IEEE (2014)","key":"192_CR67","DOI":"10.1109\/ICASSP.2014.6854796"},{"unstructured":"Sun, R., Luo, Z.Q., Ye, Y.: On the expected convergence of randomly permuted ADMM. Optimization for Machine Learning, OPT2015 (2015)","key":"192_CR68"},{"issue":"1","key":"192_CR69","doi-asserted-by":"crossref","first-page":"233","DOI":"10.1287\/moor.2019.0990","volume":"45","author":"R Sun","year":"2020","unstructured":"Sun, R., Luo, Z.Q., Ye, Y.: On the efficiency of random permutation for ADMM and coordinate descent. Math. Oper. Res. 45(1), 233\u2013271 (2020)","journal-title":"Math. Oper. Res."},{"issue":"1","key":"192_CR70","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1137\/100781894","volume":"21","author":"M Tao","year":"2011","unstructured":"Tao, M., Yuan, X.: Recovering low-rank and sparse components of matrices from incomplete and noisy observations. SIAM J. Optim. 21(1), 57\u201381 (2011)","journal-title":"SIAM J. Optim."},{"unstructured":"Taylor, G., Burmeister, R., Xu, Z., Singh, B., Patel, A., Goldstein, T.: Training neural networks without gradients: a scalable ADMM approach. In: International Conference on Machine Learning, pp. 2722\u20132731 (2016)","key":"192_CR71"},{"unstructured":"The Maros and Meszaros Convex QP Test Problem Set. http:\/\/www.cuter.rl.ac.uk\/Problems\/marmes.html","key":"192_CR72"},{"unstructured":"The Mittelmann LP test set. http:\/\/plato.asu.edu\/ftp\/lptestset\/","key":"192_CR73"},{"key":"192_CR74","first-page":"156","volume-title":"Statistical Learning Theory","author":"V Vapnik","year":"1998","unstructured":"Vapnik, V., Vapnik, V.: Statistical Learning Theory, pp. 156\u2013160. Wiley, New York (1998)"},{"key":"192_CR75","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4757-2440-0","volume-title":"The Nature of Statistical Learning Theory","author":"VN Vapnik","year":"1995","unstructured":"Vapnik, V.N.: The Nature of Statistical Learning Theory. Springer, Berlin (1995)"},{"unstructured":"Wang, F., Cao, W., Xu, Z.: Convergence of multi-block Bregman ADMM for nonconvex composite problems. arXiv preprint arXiv:1505.03063 (2015)","key":"192_CR76"},{"unstructured":"Wang, Y., Yin, W., Zeng, J.: Global convergence of ADMM in nonconvex nonsmooth optimization. arXiv e-prints arXiv:1511.06324 (2017)","key":"192_CR77"},{"unstructured":"Wharton Research Data Services https:\/\/wrds-web.wharton.upenn.edu\/wrds\/index.cfm (2018)","key":"192_CR78"},{"issue":"2","key":"192_CR79","doi-asserted-by":"crossref","first-page":"241","DOI":"10.1007\/s10589-009-9296-8","volume":"49","author":"K Woodsend","year":"2011","unstructured":"Woodsend, K., Gondzio, J.: Exploiting separability in large-scale linear support vector machine training. Comput. Optim. Appl. 49(2), 241\u2013269 (2011)","journal-title":"Comput. Optim. Appl."},{"issue":"1","key":"192_CR80","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1007\/s10107-015-0892-3","volume":"151","author":"SJ Wright","year":"2015","unstructured":"Wright, S.J.: Coordinate descent algorithms. Math. Program. 151(1), 3\u201334 (2015)","journal-title":"Math. Program."},{"issue":"1","key":"192_CR81","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1007\/s11590-017-1116-y","volume":"12","author":"M Zarepisheh","year":"2018","unstructured":"Zarepisheh, M., Xing, L., Ye, Y.: A computation study on an integrated alternating direction method of multipliers for large scale optimization. Optim. Lett. 12(1), 3\u201315 (2018)","journal-title":"Optim. Lett."},{"unstructured":"Zhang, J., Ma, S., Zhang, S.: Primal-dual optimization algorithms over Riemannian manifolds: an iteration complexity analysis. arXiv e-prints arXiv:1710.02236 (2017)","key":"192_CR82"}],"container-title":["Mathematical Programming Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12532-020-00192-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s12532-020-00192-5\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12532-020-00192-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,8]],"date-time":"2023-10-08T01:38:43Z","timestamp":1696729123000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s12532-020-00192-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,9,23]]},"references-count":82,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2021,6]]}},"alternative-id":["192"],"URL":"https:\/\/doi.org\/10.1007\/s12532-020-00192-5","relation":{},"ISSN":["1867-2949","1867-2957"],"issn-type":[{"type":"print","value":"1867-2949"},{"type":"electronic","value":"1867-2957"}],"subject":[],"published":{"date-parts":[[2020,9,23]]},"assertion":[{"value":"17 April 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"24 August 2020","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 September 2020","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Compliance with ethical standards"}},{"value":"The authors declare that they have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}