{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,18]],"date-time":"2025-10-18T15:05:58Z","timestamp":1760799958427},"reference-count":39,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2017,2,14]],"date-time":"2017-02-14T00:00:00Z","timestamp":1487030400000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"name":"Hong Kong Research Grants Council","award":["PolyU253008\/15p"],"award-info":[{"award-number":["PolyU253008\/15p"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comput Optim Appl"],"published-print":{"date-parts":[[2017,7]]},"DOI":"10.1007\/s10589-017-9900-2","type":"journal-article","created":{"date-parts":[[2017,2,14]],"date-time":"2017-02-14T12:59:21Z","timestamp":1487077161000},"page":"489-520","update-policy":"http:\/\/dx.doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":38,"title":["Further properties of the forward\u2013backward envelope with applications to difference-of-convex programming"],"prefix":"10.1007","volume":"67","author":[{"given":"Tianxiang","family":"Liu","sequence":"first","affiliation":[]},{"given":"Ting Kei","family":"Pong","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2017,2,14]]},"reference":[{"key":"9900_CR1","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1007\/s10107-007-0133-5","volume":"116","author":"H Attouch","year":"2009","unstructured":"Attouch, H., Bolte, J.: On the convergence of the proximal algorithm for nonsmooth functions involving analytic features. Math. Program. 116, 5\u201316 (2009)","journal-title":"Math. Program."},{"key":"9900_CR2","doi-asserted-by":"crossref","first-page":"438","DOI":"10.1287\/moor.1100.0449","volume":"35","author":"H Attouch","year":"2010","unstructured":"Attouch, H., Bolte, J., Redont, P., Soubeyran, A.: Proximal alternating minimization and projection methods for nonconvex problems: an approach based on the Kurdyka\u2013\u0141ojasiewicz inequality. Math. Oper. Res. 35, 438\u2013457 (2010)","journal-title":"Math. Oper. Res."},{"key":"9900_CR3","doi-asserted-by":"crossref","first-page":"91","DOI":"10.1007\/s10107-011-0484-9","volume":"137","author":"H Attouch","year":"2013","unstructured":"Attouch, H., Bolte, J., Svaiter, B.F.: Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward\u2013backward splitting, and regularized Gauss\u2013Seidel methods. Math. Program. 137, 91\u2013129 (2013)","journal-title":"Math. Program."},{"key":"9900_CR4","volume-title":"Asymptotic Cones and Functions in Optimization and Variational Inequalities","author":"A Auslender","year":"2003","unstructured":"Auslender, A., Teboulle, M.: Asymptotic Cones and Functions in Optimization and Variational Inequalities. Springer, Berlin (2003)"},{"key":"9900_CR5","doi-asserted-by":"crossref","first-page":"1205","DOI":"10.1137\/050644641","volume":"17","author":"J Bolte","year":"2006","unstructured":"Bolte, J., Daniilidis, A., Lewis, A.: The \u0141ojasiewicz inequality for nonsmooth subanalytic functions with applications to subgradient dynamical systems. SIAM J. Optim. 17, 1205\u20131223 (2006)","journal-title":"SIAM J. Optim."},{"key":"9900_CR6","doi-asserted-by":"crossref","first-page":"596","DOI":"10.1137\/S0363012902407120","volume":"42","author":"HH Bauschke","year":"2003","unstructured":"Bauschke, H.H., Borwein, J.M., Combettes, P.L.: Bregman monotone optimization algorithms. SIAM J. Control Optim. 42, 596\u2013636 (2003)","journal-title":"SIAM J. Control Optim."},{"key":"9900_CR7","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, Berlin (2011)"},{"key":"9900_CR8","doi-asserted-by":"crossref","first-page":"435","DOI":"10.1007\/s10107-015-0965-3","volume":"159","author":"RH Byrd","year":"2016","unstructured":"Byrd, R.H., Chin, G.M., Nocedal, J., Oztoprak, F.: A family of second-order methods for convex $$\\ell _1$$ \u2113 1 -regularized optimization. Math. Program. 159, 435\u2013467 (2016)","journal-title":"Math. Program."},{"key":"9900_CR9","doi-asserted-by":"crossref","first-page":"4203","DOI":"10.1109\/TIT.2005.858979","volume":"51","author":"EJ Cand\u00e8s","year":"2005","unstructured":"Cand\u00e8s, E.J., Tao, T.: Decoding by linear programming. IEEE Trans. Inf. Theory 51, 4203\u20134215 (2005)","journal-title":"IEEE Trans. Inf. Theory"},{"key":"9900_CR10","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1023\/B:JMIV.0000011320.81911.38","volume":"20","author":"A Chambolle","year":"2004","unstructured":"Chambolle, A.: An algorithm for total variation minimization and applications. J. Math. Imaging Vis. 20, 89\u201397 (2004)","journal-title":"J. Math. Imaging Vis."},{"key":"9900_CR11","doi-asserted-by":"crossref","first-page":"1465","DOI":"10.1137\/15M1028054","volume":"26","author":"X Chen","year":"2016","unstructured":"Chen, X., Lu, Z., Pong, T.K.: Penalty methods for a class of non-Lipschitz optimization problems. SIAM J. Optim. 26, 1465\u20131492 (2016)","journal-title":"SIAM J. Optim."},{"key":"9900_CR12","doi-asserted-by":"crossref","first-page":"1289","DOI":"10.1109\/TIT.2006.871582","volume":"52","author":"DL Donoho","year":"2006","unstructured":"Donoho, D.L.: Compressed sensing. IEEE Trans. Inf. Theory 52, 1289\u20131306 (2006)","journal-title":"IEEE Trans. Inf. Theory"},{"key":"9900_CR13","volume-title":"Finite-Dimensional Variational Inequalities and Complementarity Problems Volume I\/II","author":"F Facchinei","year":"2003","unstructured":"Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems Volume I\/II. Springer, Berlin (2003)"},{"key":"9900_CR14","doi-asserted-by":"crossref","first-page":"1348","DOI":"10.1198\/016214501753382273","volume":"96","author":"J Fan","year":"2011","unstructured":"Fan, J., Li, R.: Variable selection via nonconvex penalized likelihood and its oracle properties. J. Am. Stat. Assoc. 96, 1348\u20131360 (2011)","journal-title":"J. Am. Stat. Assoc."},{"key":"9900_CR15","unstructured":"Friedlander, M., Goh, G.: Efficient evaluation of scaled proximal operators. Preprint arXiv:1603.05719 (2016)"},{"key":"9900_CR16","unstructured":"Gong, P., Zhang, C., Lu, Z., Huang, J.Z., Ye, J.: A general iterative Shrinkage and thresholding algorithm for non-convex regularized optimization problems. Proc. Int. Conf. Mach. Learn. 28, 37\u201345 (2013)"},{"key":"9900_CR17","doi-asserted-by":"crossref","first-page":"035007","DOI":"10.1088\/0266-5611\/24\/3\/035007","volume":"24","author":"R Griesse","year":"2008","unstructured":"Griesse, R., Lorenz, D.A.: A semismooth Newton method for Tikhonov functionals with sparsity constraints. Inverse Probl. 24, 035007 (2008)","journal-title":"Inverse Probl."},{"key":"9900_CR18","doi-asserted-by":"crossref","first-page":"1385","DOI":"10.1016\/j.na.2011.07.031","volume":"75","author":"C Kan","year":"2012","unstructured":"Kan, C., Song, W.: The Moreau envelope function and proximal mapping in the sense of the Bregman distance. Nonlinear Anal. Theory Methods Appl. 75, 1385\u20131399 (2012)","journal-title":"Nonlinear Anal. Theory Methods Appl."},{"key":"9900_CR19","doi-asserted-by":"crossref","first-page":"1420","DOI":"10.1137\/130921428","volume":"24","author":"JD Lee","year":"2014","unstructured":"Lee, J.D., Sun, Y., Saunders, M.A.: Proximal Newton-type methods for minimizing composite functions. SIAM J. Optim. 24, 1420\u20131443 (2014)","journal-title":"SIAM J. Optim."},{"key":"9900_CR20","doi-asserted-by":"crossref","unstructured":"Li, G., Pong, T.K.: Calculus of the exponent of Kurdyka\u2013\u0141ojasiewicz inequality and its applications to linear convergence of first-order methods. Preprint arXiv:1602.02915 (2016)","DOI":"10.1007\/s10208-017-9366-8"},{"key":"9900_CR21","doi-asserted-by":"crossref","first-page":"4037","DOI":"10.1016\/j.csda.2012.04.019","volume":"56","author":"Z Lu","year":"2012","unstructured":"Lu, Z., Pong, T.K., Zhang, Y.: An alternating direction method for finding Dantzig selectors. Comput. Stat. Data Anal. 56, 4037\u20134946 (2012)","journal-title":"Comput. Stat. Data Anal."},{"key":"9900_CR22","doi-asserted-by":"crossref","first-page":"408","DOI":"10.1137\/0330025","volume":"30","author":"ZQ Luo","year":"1992","unstructured":"Luo, Z.Q., Tseng, P.: On the linear convergence of descent methods for convex essentially smooth minimization. SIAM J. Control Optim. 30, 408\u2013425 (1992)","journal-title":"SIAM J. Control Optim."},{"key":"9900_CR23","first-page":"157","volume":"46","author":"ZQ 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":"9900_CR24","doi-asserted-by":"crossref","first-page":"846","DOI":"10.1287\/moor.18.4.846","volume":"18","author":"ZQ Luo","year":"1993","unstructured":"Luo, Z.Q., Tseng, P.: On the convergence rate of dual ascent methods for linearly constrained convex minimization. Math. Oper. Res. 18, 846\u2013867 (1993)","journal-title":"Math. Oper. Res."},{"key":"9900_CR25","doi-asserted-by":"crossref","first-page":"298","DOI":"10.1137\/120892167","volume":"24","author":"A Milzarek","year":"2014","unstructured":"Milzarek, A., Ulbrich, M.: A semismooth Newton method with multidimensional filter globalization for $$\\ell _1$$ \u2113 1 -optimization. SIAM J. Optim. 24, 298\u2013333 (2014)","journal-title":"SIAM J. Optim."},{"key":"9900_CR26","doi-asserted-by":"crossref","DOI":"10.1007\/b98874","volume-title":"Numerical Optimization","author":"J Nocedal","year":"1999","unstructured":"Nocedal, J., Wright, S.J.: Numerical Optimization, 1st edn. Springer, Berlin (1999)","edition":"1"},{"key":"9900_CR27","volume-title":"Computational and Analytical Mathematics","author":"D Noll","year":"2013","unstructured":"Noll, D., Rondepierre, A.: Convergence of linesearch and trust-region methods using the Kurdyka\u2013\u0141ojasiewicz inequality. In: Bailey, D.H., Bauschke, H.H., Borwein, P., Garvan, F., Th\u00e9ra, M., Vanderwerff, J.D., Wolkowicz, H. (eds.) Computational and Analytical Mathematics. Springer, Berlin (2013)"},{"key":"9900_CR28","doi-asserted-by":"crossref","unstructured":"Patrinos, P., Bemporad, A.: Proximal Newton methods for convex composite optimization. In: Proceedings of the IEEE Conference on Decision and Control, pp. 2358\u20132363. (2013)","DOI":"10.1109\/CDC.2013.6760233"},{"key":"9900_CR29","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":"9900_CR30","doi-asserted-by":"publisher","unstructured":"Stella, L., Themelis, A., Patrinos, P.: Forward\u2013backward quasi-Newton methods for nonsmooth optimization problems. Comput. Optim. Appl. (2017). doi: 10.1007\/s10589-017-9912-y","DOI":"10.1007\/s10589-017-9912-y"},{"key":"9900_CR31","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1007\/s10589-008-9215-4","volume":"47","author":"P Tseng","year":"2010","unstructured":"Tseng, P., Yun, S.: A coordinate gradient descent method for linearly constrained smooth optimization and support vector machines training. Comput. Optim. Appl. 47, 179\u2013206 (2010)","journal-title":"Comput. Optim. Appl."},{"key":"9900_CR32","doi-asserted-by":"crossref","first-page":"387","DOI":"10.1007\/s10107-007-0170-0","volume":"117","author":"P Tseng","year":"2009","unstructured":"Tseng, P., Yun, S.: A coordinate gradient descent method for nonsmooth separable minimization. Math. Program. Ser. B 117, 387\u2013423 (2009)","journal-title":"Math. Program. Ser. B"},{"key":"9900_CR33","doi-asserted-by":"crossref","first-page":"263","DOI":"10.1007\/s10107-010-0394-2","volume":"125","author":"P Tseng","year":"2010","unstructured":"Tseng, P.: Approximation accuracy, gradient methods, and error bound for structured convex optimization. Math. Program. Ser. B 125, 263\u2013295 (2010)","journal-title":"Math. Program. Ser. B"},{"key":"9900_CR34","doi-asserted-by":"publisher","unstructured":"Wang, Y., Luo, Z., Zhang, X.: New improved penalty methods for sparse reconstruction based on difference of two norms. Preprint. doi: 10.13140\/RG.2.1.3256.3369 (2015)","DOI":"10.13140\/RG.2.1.3256.3369"},{"key":"9900_CR35","doi-asserted-by":"crossref","first-page":"2479","DOI":"10.1109\/TSP.2009.2016892","volume":"57","author":"SJ Wright","year":"2009","unstructured":"Wright, S.J., Nowak, R., Figueiredo, M.A.T.: Sparse reconstruction by separable approximation. IEEE T. Signal Process. 57, 2479\u20132493 (2009)","journal-title":"IEEE T. Signal Process."},{"key":"9900_CR36","unstructured":"Xiao, X., Li, Y., Wen, Z., Zhang, L.: Semi-smooth second-order type methods for composite convex programs. Preprint arXiv:1603.07870 (2016)"},{"key":"9900_CR37","doi-asserted-by":"crossref","first-page":"A536","DOI":"10.1137\/140952363","volume":"37","author":"P Yin","year":"2015","unstructured":"Yin, P., Lou, Y., He, Q., Xin, J.: Minimization of $$\\ell _{1-2}$$ \u2113 1 - 2 for compressed sensing. SIAM J. Sci. Comput. 37, A536\u2013A563 (2015)","journal-title":"SIAM J. Sci. Comput."},{"key":"9900_CR38","doi-asserted-by":"crossref","first-page":"894","DOI":"10.1214\/09-AOS729","volume":"38","author":"C-H Zhang","year":"2010","unstructured":"Zhang, C.-H.: Nearby unbiased variable selection under minimax concave penalty. Ann. Stat. 38, 894\u2013942 (2010)","journal-title":"Ann. Stat."},{"key":"9900_CR39","unstructured":"Zhou, Z., So, A.M.-C.: A unified approach to error bounds for structured convex optimization problems. Preprint arXiv:1512.03518 (2015)"}],"container-title":["Computational Optimization and Applications"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10589-017-9900-2\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10589-017-9900-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10589-017-9900-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,9,18]],"date-time":"2019-09-18T17:18:55Z","timestamp":1568827135000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10589-017-9900-2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,2,14]]},"references-count":39,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2017,7]]}},"alternative-id":["9900"],"URL":"https:\/\/doi.org\/10.1007\/s10589-017-9900-2","relation":{},"ISSN":["0926-6003","1573-2894"],"issn-type":[{"value":"0926-6003","type":"print"},{"value":"1573-2894","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,2,14]]}}}