{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T03:41:04Z","timestamp":1740109264922,"version":"3.37.3"},"reference-count":29,"publisher":"Springer Science and Business Media LLC","issue":"1-2","license":[{"start":{"date-parts":[[2019,4,27]],"date-time":"2019-04-27T00:00:00Z","timestamp":1556323200000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2019,4,27]],"date-time":"2019-04-27T00:00:00Z","timestamp":1556323200000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/100000086","name":"Directorate for Mathematical and Physical Sciences","doi-asserted-by":"publisher","award":["1720237"],"award-info":[{"award-number":["1720237"]}],"id":[{"id":"10.13039\/100000086","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100007297","name":"Office of Naval Research Global","doi-asserted-by":"publisher","award":["000141712162"],"award-info":[{"award-number":["000141712162"]}],"id":[{"id":"10.13039\/100007297","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004147","name":"Tsinghua University","doi-asserted-by":"publisher","award":["Xuetang Mathematics Program and Top Open Program"],"award-info":[{"award-number":["Xuetang Mathematics Program and Top Open Program"]}],"id":[{"id":"10.13039\/501100004147","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Math. Program."],"published-print":{"date-parts":[[2019,7]]},"DOI":"10.1007\/s10107-019-01397-w","type":"journal-article","created":{"date-parts":[[2019,4,27]],"date-time":"2019-04-27T07:30:09Z","timestamp":1556350209000},"page":"39-67","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Run-and-Inspect Method for nonconvex optimization and global optimality bounds for R-local minimizers"],"prefix":"10.1007","volume":"176","author":[{"given":"Yifan","family":"Chen","sequence":"first","affiliation":[]},{"given":"Yuejiao","family":"Sun","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6697-9731","authenticated-orcid":false,"given":"Wotao","family":"Yin","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2019,4,27]]},"reference":[{"key":"1397_CR1","unstructured":"Chaudhari, P., Choromanska, A., Soatto, S., LeCun, Y.: Entropy-SGD: biasing gradient descent into wide valleys. arXiv preprint arXiv:1611.01838 (2016)"},{"issue":"3","key":"1397_CR2","doi-asserted-by":"publisher","first-page":"30","DOI":"10.1007\/s40687-018-0148-y","volume":"5","author":"P Chaudhari","year":"2018","unstructured":"Chaudhari, P., Oberman, A., Osher, S., Soatto, S., Carlier, G.: Deep relaxation: partial differential equations for optimizing deep neural networks. Res. Math. Sci. 5(3), 30 (2018)","journal-title":"Res. Math. Sci."},{"key":"1397_CR3","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898719857","volume-title":"Trust Region Methods","author":"AR Conn","year":"2000","unstructured":"Conn, A.R., Gould, N.I., Toint, P.L.: Trust Region Methods. SIAM, Philadelphia (2000)"},{"key":"1397_CR4","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898718768","volume-title":"Introduction to Derivative-Free Optimization. No. 8 in MPS-SIAM Series on Optimization","author":"AR Conn","year":"2009","unstructured":"Conn, A.R., Scheinberg, K., Vicente, L.N.: Introduction to Derivative-Free Optimization. No. 8 in MPS-SIAM Series on Optimization. Society for Industrial and Applied Mathematics\/Mathematical Programming Society, Philadelphia (2009)"},{"key":"1397_CR5","volume-title":"An R and S-Plus Companion to Applied Regression","author":"J Fox","year":"2002","unstructured":"Fox, J.: An R and S-Plus Companion to Applied Regression. Sage Publications, Thousand Oaks (2002)"},{"key":"1397_CR6","unstructured":"Ge, R., Huang, F., Jin, C., Yuan, Y.: Escaping from saddle points\u2014online stochastic gradient for tensor decomposition. In: Conference on Learning Theory, pp. 797\u2013842 (2015)"},{"key":"1397_CR7","unstructured":"Ge, R., Lee, J.D., Ma, T.: Matrix completion has no spurious local minimum. In: Lee, D.D., Sugiyama, M., Luxburg, U.V., Guyon, I., Garnett, R. (eds.) Advances in Neural Information Processing Systems, pp. 2973\u20132981. Curran Associates, Inc. (2016)"},{"issue":"1\u20132","key":"1397_CR8","doi-asserted-by":"publisher","first-page":"59","DOI":"10.1007\/s10107-015-0871-8","volume":"156","author":"S Ghadimi","year":"2016","unstructured":"Ghadimi, S., Lan, G.: Accelerated gradient methods for nonconvex nonlinear and stochastic programming. Math. Program. 156(1\u20132), 59\u201399 (2016)","journal-title":"Math. Program."},{"key":"1397_CR9","unstructured":"Jin, C., Ge, R., Netrapalli, P., Kakade, S.M., Jordan, M.I.: How to escape saddle points efficiently. arXiv preprint arXiv:1703.00887 (2017)"},{"key":"1397_CR10","doi-asserted-by":"publisher","first-page":"795","DOI":"10.1007\/978-3-319-46128-1_50","volume-title":"Machine Learning and Knowledge Discovery in Databases","author":"H Karimi","year":"2016","unstructured":"Karimi, H., Nutini, J., Schmidt, M.: Linear convergence of gradient and proximal-gradient methods under the Polyak\u2013\u0141ojasiewicz condition. In: Frasconi, P., Landwehr, N., Manco, G., Vreeken, J. (eds.) Machine Learning and Knowledge Discovery in Databases, vol. 9851, pp. 795\u2013811. Springer, Cham (2016)"},{"issue":"4598","key":"1397_CR11","doi-asserted-by":"publisher","first-page":"671","DOI":"10.1126\/science.220.4598.671","volume":"220","author":"S Kirkpatrick","year":"1983","unstructured":"Kirkpatrick, S., Gelatt Jr., C.D., Vecchi, M.P.: Optimization by simulated annealing. Science 220(4598), 671\u2013680 (1983). https:\/\/doi.org\/10.1126\/science.220.4598.671","journal-title":"Science"},{"issue":"2","key":"1397_CR12","doi-asserted-by":"publisher","first-page":"367","DOI":"10.1007\/s10898-016-0475-8","volume":"68","author":"JM Mart\u00ednez","year":"2017","unstructured":"Mart\u00ednez, J.M., Raydan, M.: Cubic-regularization counterpart of a variable-norm trust-region method for unconstrained minimization. J. Glob. Optim. 68(2), 367\u2013385 (2017)","journal-title":"J. Glob. Optim."},{"issue":"1","key":"1397_CR13","doi-asserted-by":"publisher","first-page":"177","DOI":"10.1007\/s10107-006-0706-8","volume":"108","author":"Y Nesterov","year":"2006","unstructured":"Nesterov, Y., Polyak, B.T.: Cubic regularization of Newton method and its global performance. Math. Program. 108(1), 177\u2013205 (2006)","journal-title":"Math. Program."},{"key":"1397_CR14","unstructured":"Panageas, I., Piliouras, G.: Gradient descent converges to minimizers: the case of non-isolated critical points. CoRR arXiv:1605.00405 (2016)"},{"key":"1397_CR15","unstructured":"Pascanu, R., Dauphin, Y.N., Ganguli, S., Bengio, Y.: On the saddle point problem for non-convex optimization. arXiv preprint arXiv:1405.4604 (2014)"},{"issue":"1","key":"1397_CR16","doi-asserted-by":"crossref","first-page":"57","DOI":"10.4310\/AMSA.2016.v1.n1.a2","volume":"1","author":"Z Peng","year":"2016","unstructured":"Peng, Z., Wu, T., Xu, Y., Yan, M., Yin, W.: Coordinate friendly structures, algorithms and applications. Ann. Math. Sci. Appl. 1(1), 57\u2013119 (2016)","journal-title":"Ann. Math. Sci. Appl."},{"issue":"4","key":"1397_CR17","first-page":"643","volume":"3","author":"BT Polyak","year":"1963","unstructured":"Polyak, B.T.: Gradient methods for minimizing functionals. Zhurnal Vychislitel\u2019noi Matematiki i Matematicheskoi Fiziki 3(4), 643\u2013653 (1963)","journal-title":"Zhurnal Vychislitel\u2019noi Matematiki i Matematicheskoi Fiziki"},{"key":"1397_CR18","doi-asserted-by":"crossref","unstructured":"Reddi, S.J., Hefny, A., Sra, S., Poczos, B., Smola, A.: Stochastic variance reduction for nonconvex optimization. In: International Conference on Machine Learning, pp. 314\u2013323 (2016)","DOI":"10.1109\/ALLERTON.2016.7852377"},{"key":"1397_CR19","unstructured":"Sagun, L., Bottou, L., LeCun, Y.: Singularity of the Hessian in deep learning. arXiv preprint arXiv:1611.07476 (2016)"},{"issue":"12","key":"1397_CR20","doi-asserted-by":"publisher","first-page":"3199","DOI":"10.1109\/TSP.2018.2824289","volume":"66","author":"X Shen","year":"2018","unstructured":"Shen, X., Gu, Y.: Nonconvex sparse logistic regression with weakly convex regularization. IEEE Trans. Signal Process. 66(12), 3199\u20133211 (2018)","journal-title":"IEEE Trans. Signal Process."},{"key":"1397_CR21","doi-asserted-by":"crossref","unstructured":"Sun, J., Qu, Q., Wright, J.: Complete dictionary recovery over the sphere. In: 2015 International Conference on Sampling Theory and Applications (SampTA), pp. 407\u2013410. IEEE (2015)","DOI":"10.1109\/SAMPTA.2015.7148922"},{"key":"1397_CR22","doi-asserted-by":"crossref","unstructured":"Sun, J., Qu, Q., Wright, J.: A geometric analysis of phase retrieval. In: 2016 IEEE International Symposium on Information Theory (ISIT), pp. 2379\u20132383. IEEE (2016)","DOI":"10.1109\/ISIT.2016.7541725"},{"key":"1397_CR23","doi-asserted-by":"publisher","first-page":"29","DOI":"10.1007\/s10915-018-0757-z","volume":"78","author":"Y Wang","year":"2018","unstructured":"Wang, Y., Yin, W., Zeng, J.: Global convergence of ADMM in nonconvex nonsmooth optimization. J. Sci. Comput. 78, 29\u201363 (2018)","journal-title":"J. Sci. Comput."},{"key":"1397_CR24","unstructured":"Wu, L., Zhu, Z., Weinan, E.: Towards understanding generalization of deep learning: perspective of loss landscapes. arXiv preprint arXiv:1706.10239 (2017)"},{"issue":"3","key":"1397_CR25","doi-asserted-by":"publisher","first-page":"1758","DOI":"10.1137\/120887795","volume":"6","author":"Y Xu","year":"2013","unstructured":"Xu, Y., Yin, W.: A block coordinate descent method for regularized multiconvex optimization with applications to nonnegative tensor factorization and completion. SIAM J. Imaging Sci. 6(3), 1758\u20131789 (2013)","journal-title":"SIAM J. Imaging Sci."},{"issue":"7","key":"1397_CR26","doi-asserted-by":"publisher","first-page":"1013","DOI":"10.1109\/TNNLS.2012.2197412","volume":"23","author":"Z Xu","year":"2012","unstructured":"Xu, Z., Chang, X., Xu, F., Zhang, H.: $$l_{1\/2}$$ regularization: a thresholding representation theory and a fast solver. IEEE Trans. Neural Netw. Learn. Syst. 23(7), 1013\u20131027 (2012)","journal-title":"IEEE Trans. Neural Netw. Learn. Syst."},{"issue":"2","key":"1397_CR27","doi-asserted-by":"publisher","first-page":"1133","DOI":"10.1007\/s10915-018-0744-4","volume":"77","author":"P Yin","year":"2018","unstructured":"Yin, P., Pham, M., Oberman, A., Osher, S.: Stochastic backward Euler: an implicit gradient descent algorithm for k-means clustering. J. Sci. Comput. 77(2), 1133\u20131146 (2018)","journal-title":"J. Sci. Comput."},{"key":"1397_CR28","doi-asserted-by":"publisher","first-page":"220","DOI":"10.1016\/j.cam.2017.01.010","volume":"319","author":"J Zeng","year":"2017","unstructured":"Zeng, J., Peng, Z., Lin, S.: GAITA: a Gauss\u2013Seidel iterative thresholding algorithm for $$\\ell _q$$ regularized least squares regression. J. Comput. Appl. Math. 319, 220\u2013235 (2017)","journal-title":"J. Comput. Appl. Math."},{"issue":"2","key":"1397_CR29","doi-asserted-by":"publisher","first-page":"894","DOI":"10.1214\/09-AOS729","volume":"38","author":"CH Zhang","year":"2010","unstructured":"Zhang, C.H.: Nearly unbiased variable selection under minimax concave penalty. Ann. Stat. 38(2), 894\u2013942 (2010)","journal-title":"Ann. Stat."}],"container-title":["Mathematical Programming"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10107-019-01397-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10107-019-01397-w\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10107-019-01397-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,9,17]],"date-time":"2022-09-17T00:11:24Z","timestamp":1663373484000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10107-019-01397-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,4,27]]},"references-count":29,"journal-issue":{"issue":"1-2","published-print":{"date-parts":[[2019,7]]}},"alternative-id":["1397"],"URL":"https:\/\/doi.org\/10.1007\/s10107-019-01397-w","relation":{},"ISSN":["0025-5610","1436-4646"],"issn-type":[{"type":"print","value":"0025-5610"},{"type":"electronic","value":"1436-4646"}],"subject":[],"published":{"date-parts":[[2019,4,27]]},"assertion":[{"value":"18 November 2017","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 April 2019","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"27 April 2019","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}