{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,16]],"date-time":"2026-04-16T14:01:26Z","timestamp":1776348086702,"version":"3.51.2"},"reference-count":53,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2022,3,25]],"date-time":"2022-03-25T00:00:00Z","timestamp":1648166400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2022,3,25]],"date-time":"2022-03-25T00:00:00Z","timestamp":1648166400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100000923","name":"Australian Research Council","doi-asserted-by":"publisher","award":["DP190100555"],"award-info":[{"award-number":["DP190100555"]}],"id":[{"id":"10.13039\/501100000923","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["1816449"],"award-info":[{"award-number":["1816449"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["1816386"],"award-info":[{"award-number":["1816386"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Optim Theory Appl"],"published-print":{"date-parts":[[2022,7]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In the previous paper Bello-Cruz et al. (J Optim Theory Appl 188:378\u2013401, 2021), we showed that the quadratic growth condition plays a key role in obtaining Q-linear convergence of the widely used forward\u2013backward splitting method with Beck\u2013Teboulle\u2019s line search. In this paper, we analyze the property of quadratic growth condition via second-order variational analysis for various structured optimization problems that arise in machine learning and signal processing. This includes, for example, the Poisson linear inverse problem as well as the<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\ell _1$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:msub><mml:mi>\u2113<\/mml:mi><mml:mn>1<\/mml:mn><\/mml:msub><\/mml:math><\/jats:alternatives><\/jats:inline-formula>-regularized optimization problems. As a by-product of this approach, we also obtain several full characterizations for the uniqueness of optimal solution to Lasso problem, which complements and extends recent important results in this direction.<\/jats:p>","DOI":"10.1007\/s10957-022-02013-2","type":"journal-article","created":{"date-parts":[[2022,3,25]],"date-time":"2022-03-25T16:30:21Z","timestamp":1648225821000},"page":"167-190","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Quadratic Growth Conditions and Uniqueness of Optimal Solution to Lasso"],"prefix":"10.1007","volume":"194","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7877-5688","authenticated-orcid":false,"given":"Yunier","family":"Bello-Cruz","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2099-7974","authenticated-orcid":false,"given":"Guoyin","family":"Li","sequence":"additional","affiliation":[]},{"given":"Tran Thai An","family":"Nghia","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,3,25]]},"reference":[{"key":"2013_CR1","first-page":"365","volume":"15","author":"FJ Arag\u00f3n Artacho","year":"2008","unstructured":"Arag\u00f3n Artacho, F.J., Geoffroy, M.H.: Characterizations of metric regularity of subdifferentials. J. Convex Anal. 15, 365\u2013380 (2008)","journal-title":"J. Convex Anal."},{"key":"2013_CR2","first-page":"35","volume":"15","author":"FJ Arag\u00f3n Artacho","year":"2014","unstructured":"Arag\u00f3n Artacho, F.J., Geoffroy, M.H.: Metric subregularity of the convex subdifferential in Banach spaces. J. Nonlinear Convex Anal. 15, 35\u201347 (2014)","journal-title":"J. Nonlinear Convex Anal."},{"key":"2013_CR3","first-page":"251","volume":"16","author":"D Az\u00e9","year":"2014","unstructured":"Az\u00e9, D., Corvellec, J.-N.: Nonlinear local error bounds via a change of metric. J. Fixed Point Theory Appl. 16, 251\u2013372 (2014)","journal-title":"J. Fixed Point Theory Appl."},{"key":"2013_CR4","doi-asserted-by":"crossref","first-page":"330","DOI":"10.1287\/moor.2016.0817","volume":"42","author":"HH Bauschke","year":"2017","unstructured":"Bauschke, H.H., Bolte, J., Teboulle, M.: A descent lemma beyond Lipschitz gradient continuity: first-order methods revisited and applications. Math. Oper. Res. 42, 330\u2013348 (2017)","journal-title":"Math. Oper. Res."},{"key":"2013_CR5","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4419-9467-7","volume-title":"Convex Analysis and Monotone Operator Theory in Hilbert Spaces","author":"HH Bauschke","year":"2011","unstructured":"Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, New York (2011)"},{"key":"2013_CR6","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.jmaa.2014.06.075","volume":"421","author":"HH Bauschke","year":"2015","unstructured":"Bauschke, H.H., Noll, D., Phan, H.M.: Linear and strong convergence of algorithms involving averaged nonexpansive operators. J. Math. Anal. Appl. 421, 1\u201320 (2015)","journal-title":"J. Math. Anal. Appl."},{"key":"2013_CR7","doi-asserted-by":"crossref","unstructured":"Beck, A., Teboulle, M.: Gradient-based algorithms with applications to signal recovery problems. in Convex Optimization in Signal Processing and Communications, (D. Palomar and Y. Eldar, eds.) 42\u201388 University Press, Cambribge (2010)","DOI":"10.1017\/CBO9780511804458.003"},{"key":"2013_CR8","doi-asserted-by":"crossref","unstructured":"Bello-Cruz, J.Y., Li, G., Nghia, T. T.A.: On the Q-linear convergence of forward-backward splitting method. Part I: Convergence analysis. J. Optim. Theory Appl. 188, 378\u2013401 (2021)","DOI":"10.1007\/s10957-020-01787-7"},{"key":"2013_CR9","doi-asserted-by":"crossref","first-page":"1209","DOI":"10.1080\/10556788.2016.1214959","volume":"31","author":"JY Bello Cruz","year":"2016","unstructured":"Bello Cruz, J.Y., Nghia, T.T.A.: On the convergence of the proximal forward-backward splitting method with linesearches. Optim. Method Softw. 31, 1209\u20131238 (2016)","journal-title":"Optim. Method Softw."},{"key":"2013_CR10","doi-asserted-by":"crossref","first-page":"471","DOI":"10.1007\/s10107-016-1091-6","volume":"165","author":"J Bolte","year":"2017","unstructured":"Bolte, J., Nguyen, T.P., Peypouquet, J., Suter, B.W.: From error bounds to the complexity of first-order descent methods for convex functions. Math. Program. 165, 471\u2013507 (2017)","journal-title":"Math. Program."},{"key":"2013_CR11","unstructured":"Borwein, J., Lewis, A.S.: Convex analysis and nonlinear optimization: Theory and Examples. Springer Science & Business Media (2010)"},{"key":"2013_CR12","doi-asserted-by":"crossref","first-page":"813","DOI":"10.1007\/s00041-008-9041-1","volume":"14","author":"K Bredies","year":"2008","unstructured":"Bredies, K., Lorenz, D.A.: Linear convergence of iterative soft-thresholding. J. Fourier Anal. Appl. 14, 813\u2013837 (2008)","journal-title":"J. Fourier Anal. Appl."},{"key":"2013_CR13","doi-asserted-by":"crossref","first-page":"805","DOI":"10.1007\/s10208-012-9135-7","volume":"12","author":"V Chandrasekaran","year":"2012","unstructured":"Chandrasekaran, V., Recht, B., Parrilo, P.A., Willsky, A.S.: The convex geometry of linear inverse problems. Found Comput Math 12, 805\u2013849 (2012)","journal-title":"Found Comput Math"},{"key":"2013_CR14","doi-asserted-by":"crossref","unstructured":"Combettes, P. L., Pesquet, J.-C.: Proximal splitting methods in signal processing. in Fixed-Point Algorithms for Inverse Problems. Science and Engineering. Springer Optimization and Its Applications 49, 185\u2013212 Springer, New York, (2011)","DOI":"10.1007\/978-1-4419-9569-8_10"},{"key":"2013_CR15","doi-asserted-by":"crossref","first-page":"1168","DOI":"10.1137\/050626090","volume":"4","author":"PL Combettes","year":"2005","unstructured":"Combettes, P.L., Wajs, V.R.: Signal recovery by proximal forward-backward splitting. Multiscale Model. Simul. 4, 1168\u20131200 (2005)","journal-title":"Multiscale Model. Simul."},{"key":"2013_CR16","doi-asserted-by":"crossref","first-page":"2032","DOI":"10.1214\/aos\/1176348385","volume":"19","author":"I Csisz\u00e1r","year":"1991","unstructured":"Csisz\u00e1r, I.: Why least squares and maximum entropy? An axiomatic approach to inference for linear inverse problems. Ann. Statist. 19, 2032\u20132066 (1991)","journal-title":"Ann. Statist."},{"issue":"4","key":"2013_CR17","doi-asserted-by":"crossref","first-page":"2332","DOI":"10.1137\/17M1116325","volume":"27","author":"Y Cui","year":"2017","unstructured":"Cui, Y., Ding, C., Zhao, X.: Quadratic growth conditions for convex matrix optimization problems associated with spectral functions. SIAM Journal on Optimization 27(4), 2332\u20132355 (2017)","journal-title":"SIAM Journal on Optimization"},{"key":"2013_CR18","doi-asserted-by":"crossref","first-page":"1413","DOI":"10.1002\/cpa.20042","volume":"57","author":"I Daubechies","year":"2004","unstructured":"Daubechies, I., Defrise, M., De Mol, D.: An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Comm. Pure Appl. Math. 57, 1413\u20131457 (2004)","journal-title":"Comm. Pure Appl. Math."},{"key":"2013_CR19","doi-asserted-by":"crossref","unstructured":"Davis, D., Yin, W.: Convergence rate analysis of several splitting schemes. Splitting Methods in Communications, Image Science, and Engineering. Scientific Computation, Springer, Cham, 2016","DOI":"10.1007\/978-3-319-41589-5_4"},{"key":"2013_CR20","doi-asserted-by":"crossref","DOI":"10.1007\/978-0-387-87821-8","volume-title":"Implicit functions and solution mappings","author":"AL Dontchev","year":"2009","unstructured":"Dontchev, A.L., Rockafellar, R.T.: Implicit functions and solution mappings. A View from Variational Analysis, Springer, Dordrecht (2009)"},{"key":"2013_CR21","doi-asserted-by":"crossref","first-page":"693","DOI":"10.1287\/moor.2017.0889","volume":"43","author":"D Drusvyatskiy","year":"2018","unstructured":"Drusvyatskiy, D., Lewis, A.: Error bounds, quadratic growth, and linear convergence of proximal methods. Math. Oper. Res. 43, 693\u20131050 (2018)","journal-title":"Math. Oper. Res."},{"key":"2013_CR22","first-page":"1165","volume":"21","author":"D Drusvyatskiy","year":"2014","unstructured":"Drusvyatskiy, D., Mordukhovich, B.S., Nghia, T.T.A.: Second-order growth, tilt stability, and metric regularity of the subdifferential. J. Convex Anal. 21, 1165\u20131192 (2014)","journal-title":"J. Convex Anal."},{"key":"2013_CR23","doi-asserted-by":"crossref","first-page":"1341","DOI":"10.1109\/TIT.2004.828141","volume":"50","author":"J-J Fuchs","year":"2004","unstructured":"Fuchs, J.-J.: On sparse representations in arbitrary redundant bases. IEEE Trans. Inform. Theory. 50, 1341\u20131344 (2004)","journal-title":"IEEE Trans. Inform. Theory."},{"key":"2013_CR24","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1002\/cpa.20350","volume":"64","author":"M Grasmair","year":"2011","unstructured":"Grasmair, M., Haltmeier, M., Scherzer, O.: Necessary and sufficient conditions for linear convergence of $$\\ell _1$$ regularization. Comm. Pure Applied Math. 64, 161\u2013182 (2011)","journal-title":"Comm. Pure Applied Math."},{"key":"2013_CR25","unstructured":"Garrigos, G., Rosasco, L., and Villa, S.: Convergence of the forward-backward algorithm: beyond the worst case with the help of geometry, arXiv:1703.09477 (2017)"},{"key":"2013_CR26","doi-asserted-by":"publisher","unstructured":"Garrigos, G., Rosasco, L., and Villa, S.: Thresholding gradient methods in Hilbert spaces: support identification and linear convergence, ESAIM: COCV 26 (2020), https:\/\/doi.org\/10.1051\/cocv\/2019011","DOI":"10.1051\/cocv\/2019011"},{"key":"2013_CR27","doi-asserted-by":"crossref","first-page":"70","DOI":"10.1007\/s10957-016-1004-0","volume":"172","author":"JC Gilbert","year":"2017","unstructured":"Gilbert, J.C.: On the solution uniqueness characterization in the $$\\ell _1$$ norm and polyhedral gauge recovery. J. Optim. Theory Appl. 172, 70\u2013101 (2017)","journal-title":"J. Optim. Theory Appl."},{"key":"2013_CR28","doi-asserted-by":"crossref","first-page":"1107","DOI":"10.1137\/070698920","volume":"19","author":"ET Hale","year":"2008","unstructured":"Hale, E.T., Yin, W., Zhang, Y.: Fixed-point continuation for $$\\ell _1$$-minimization: methodology and convergence. SIAM J. Optim. 19, 1107\u20131130 (2008)","journal-title":"SIAM J. Optim."},{"key":"2013_CR29","doi-asserted-by":"crossref","first-page":"702","DOI":"10.1137\/S1052623401387623","volume":"23","author":"AS Lewis","year":"2002","unstructured":"Lewis, A.S.: Active sets, nonsmoothness, and sensitivity. SIAM J. Optim. 23, 702\u2013725 (2002)","journal-title":"SIAM J. Optim."},{"key":"2013_CR30","doi-asserted-by":"crossref","first-page":"74","DOI":"10.1137\/110852103","volume":"23","author":"AS Lewis","year":"2013","unstructured":"Lewis, A.S., Zhang, S.: Partial smoothness, tilt stability, and generalized Hessians. SIAM J. Optim. 23, 74\u201394 (2013)","journal-title":"SIAM J. Optim."},{"key":"2013_CR31","doi-asserted-by":"crossref","first-page":"1199","DOI":"10.1007\/s10208-017-9366-8","volume":"18","author":"G Li","year":"2018","unstructured":"Li, G., Pong, T.K.: Calculus of the exponent of Kurdyka-\u0141ojasiewicz inequality and its applications to linear convergence of first-order methods. Found. Comp. Math. 18, 1199\u20131232 (2018)","journal-title":"Found. Comp. Math."},{"key":"2013_CR32","volume-title":"Local linear convergence of forward-backward under partial smoothness","author":"J Liang","year":"2014","unstructured":"Liang, J., Fadili, J., Peyr\u00e9, G.: Local linear convergence of forward-backward under partial smoothness. Adv. Neural Inf, Process Syst (2014)"},{"key":"2013_CR33","doi-asserted-by":"crossref","first-page":"408","DOI":"10.1137\/16M106340X","volume":"27","author":"J Liang","year":"2017","unstructured":"Liang, J., Fadili, J., Peyr\u00e9, G.: Activity identification and local linear convergence of forward-backward type methods. SIAM J. Optim. 27, 408\u2013437 (2017)","journal-title":"SIAM J. Optim."},{"key":"2013_CR34","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1007\/BF02096261","volume":"46","author":"Z-Q Luo","year":"1993","unstructured":"Luo, Z.-Q., Tseng, P.: Error bounds and convergence analysis of feasible descent methods: a general approach. Ann. Oper. Res. 46, 157\u2013178 (1993)","journal-title":"Ann. Oper. Res."},{"key":"2013_CR35","volume-title":"Variational Analysis and Generalized Differentiation, I: Basic Theory, II: Applications","author":"BS Mordukhovich","year":"2006","unstructured":"Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, I: Basic Theory, II: Applications. Springer, Berlin (2006)"},{"key":"2013_CR36","doi-asserted-by":"publisher","unstructured":"Mousavi, S. and Shen, J.: Solution uniqueness of convex piecewise affine functions based optimization with applications to constrained $$\\ell _1$$-minimization, ESAIM: Control Optim. Cal. Variations, 25 (2019), https:\/\/doi.org\/10.1051\/cocv\/2018061","DOI":"10.1051\/cocv\/2018061"},{"key":"2013_CR37","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1561\/2400000003","volume":"1","author":"P Neal","year":"2014","unstructured":"Neal, P., Boyd, S.: Proximal algorithms. Found. Trends in Optim. 1, 127\u2013239 (2014)","journal-title":"Found. Trends in Optim."},{"key":"2013_CR38","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1007\/s10107-018-1232-1","volume":"175","author":"I Necoara","year":"2019","unstructured":"Necoara, I., Nesterov, Yu., Glineur, F.: Linear convergence of first order methods for non-strongly convex optimization. Math. Program. 175, 69\u2013107 (2019)","journal-title":"Math. Program."},{"key":"2013_CR39","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-02431-3","volume-title":"Variational analysis","author":"RT Rockafellar","year":"1998","unstructured":"Rockafellar, R.T., Wets, R.J.-B.: Variational analysis. Springer, Berlin (1998)"},{"key":"2013_CR40","doi-asserted-by":"crossref","first-page":"2153","DOI":"10.1137\/16M1073741","volume":"27","author":"S Salzo","year":"2017","unstructured":"Salzo, S.: The variable metric forward-backward splitting algorithm under mild differentiability assumptions. SIAM J. Optim. 27, 2153\u20132181 (2017)","journal-title":"SIAM J. Optim."},{"key":"2013_CR41","doi-asserted-by":"crossref","first-page":"313","DOI":"10.1137\/151004549","volume":"26","author":"S Tao","year":"2016","unstructured":"Tao, S., Boley, D., Zhang, S.: Local linear convergence of ISTA and FISTA on the Lasso problem. SIAM J. Optim. 26, 313\u2013336 (2016)","journal-title":"SIAM J. Optim."},{"key":"2013_CR42","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1111\/j.2517-6161.1996.tb02080.x","volume":"58","author":"R Tibshirani","year":"1996","unstructured":"Tibshirani, R.: Regression shrinkage and selection via the Lasso. J. R. Stat. Soc. 58, 267\u2013288 (1996)","journal-title":"J. R. Stat. Soc."},{"key":"2013_CR43","doi-asserted-by":"crossref","first-page":"1456","DOI":"10.1214\/13-EJS815","volume":"7","author":"RJ Tibshirani","year":"2013","unstructured":"Tibshirani, R.J.: The Lasso problem and uniqueness. Electron. J. Stat. 7, 1456\u20131490 (2013)","journal-title":"Electron. J. Stat."},{"key":"2013_CR44","doi-asserted-by":"crossref","first-page":"1030","DOI":"10.1109\/TIT.2005.864420","volume":"52","author":"J Tropp","year":"2006","unstructured":"Tropp, J.: Just relax: Convex programming methods for identifying sparse signals in noise. IEEE Trans. Inform. Theory. 52, 1030\u20131051 (2006)","journal-title":"IEEE Trans. Inform. Theory."},{"key":"2013_CR45","doi-asserted-by":"crossref","first-page":"431","DOI":"10.1137\/S0363012998338806","volume":"38","author":"P Tseng","year":"2000","unstructured":"Tseng, P.: A modified forward-backward splitting method for maximal monotone mappings. SIAM J. Control Optim. 38, 431\u2013446 (2000)","journal-title":"SIAM J. Control Optim."},{"key":"2013_CR46","doi-asserted-by":"crossref","first-page":"387","DOI":"10.1007\/s10107-007-0170-0","volume":"117","author":"P Tseng","year":"2000","unstructured":"Tseng, P., Yun, S.: A coordinate gradient descent method for nonsmooth separable minimization. Math. Program. 117, 387\u2013423 (2000)","journal-title":"Math. Program."},{"key":"2013_CR47","doi-asserted-by":"crossref","first-page":"8","DOI":"10.1080\/01621459.1985.10477119","volume":"80","author":"Y Vardi","year":"1985","unstructured":"Vardi, Y., Shepp, L.A., Kaufman, L.: A statistical model for positron emission tomography. J. Amer. Statist. Assoc. 80, 8\u201337 (1985)","journal-title":"J. Amer. Statist. Assoc."},{"key":"2013_CR48","doi-asserted-by":"crossref","first-page":"2183","DOI":"10.1109\/TIT.2009.2016018","volume":"55","author":"MJ Wainwright","year":"2009","unstructured":"Wainwright, M.J.: Sharp thresholds for high-dimensional and noisy sparsity recovery using $$\\ell _1$$-constrained quadratic programming (lasso). IEEE Trans. Inform. Theory. 55, 2183\u20132202 (2009)","journal-title":"IEEE Trans. Inform. Theory."},{"key":"2013_CR49","doi-asserted-by":"publisher","DOI":"10.1007\/s10208-021-09528-6","author":"P Yu","year":"2021","unstructured":"Yu, P., Li, G., Pong, T.K.: Kurdyka-\u0141ojasiewicz exponent via inf-projection, to appear in Found. Comput. Math. (2021). https:\/\/doi.org\/10.1007\/s10208-021-09528-6","journal-title":"Comput. Math."},{"key":"2013_CR50","doi-asserted-by":"crossref","first-page":"1381","DOI":"10.1007\/s10444-016-9467-y","volume":"42","author":"H Zhang","year":"2016","unstructured":"Zhang, H., Yan, M., Yin, W.: One condition for solution uniqueness and robustness of both $$\\ell _1$$-synthesis and $$\\ell _1$$-analysis minimizations. Adv. Comput. Math. 42, 1381\u20131399 (2016)","journal-title":"Adv. Comput. Math."},{"key":"2013_CR51","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1007\/s10957-014-0581-z","volume":"164","author":"H Zhang","year":"2015","unstructured":"Zhang, H., Yin, W., Cheng, L.: Necessary and sufficient conditions of solution uniqueness in 1-norm minimization. J. Optim. Theory Appl. 164, 109\u2013122 (2015)","journal-title":"J. Optim. Theory Appl."},{"issue":"3","key":"2013_CR52","doi-asserted-by":"crossref","first-page":"557","DOI":"10.1007\/s11228-013-0237-4","volume":"21","author":"L Zhang","year":"2013","unstructured":"Zhang, L., Zhang, N.: Xiao, X: On the second-order directional derivatives of singular values of matrices and symmetric matrix-valued functions. Set-Valued and Variational Analysis 21(3), 557\u2013586 (2013)","journal-title":"Set-Valued and Variational Analysis"},{"key":"2013_CR53","doi-asserted-by":"crossref","first-page":"689","DOI":"10.1007\/s10107-016-1100-9","volume":"165","author":"Z Zhou","year":"2017","unstructured":"Zhou, Z., So, A.M.-C.: A unified approach to error bounds for structured convex optimization. Math. Program. 165, 689\u2013728 (2017)","journal-title":"Math. Program."}],"container-title":["Journal of Optimization Theory and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-022-02013-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10957-022-02013-2\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-022-02013-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,9,21]],"date-time":"2024-09-21T00:53:01Z","timestamp":1726879981000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10957-022-02013-2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,3,25]]},"references-count":53,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2022,7]]}},"alternative-id":["2013"],"URL":"https:\/\/doi.org\/10.1007\/s10957-022-02013-2","relation":{},"ISSN":["0022-3239","1573-2878"],"issn-type":[{"value":"0022-3239","type":"print"},{"value":"1573-2878","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,3,25]]},"assertion":[{"value":"10 January 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"18 January 2022","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"25 March 2022","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}