{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,25]],"date-time":"2026-04-25T12:01:10Z","timestamp":1777118470753,"version":"3.51.4"},"reference-count":22,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2020,8,31]],"date-time":"2020-08-31T00:00:00Z","timestamp":1598832000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2020,8,31]],"date-time":"2020-08-31T00:00:00Z","timestamp":1598832000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Optim Theory Appl"],"published-print":{"date-parts":[[2020,10]]},"DOI":"10.1007\/s10957-020-01740-8","type":"journal-article","created":{"date-parts":[[2020,8,31]],"date-time":"2020-08-31T11:03:01Z","timestamp":1598871781000},"page":"181-204","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["A Gradient Sampling Method Based on Ideal Direction for Solving Nonsmooth Optimization Problems"],"prefix":"10.1007","volume":"187","author":[{"given":"Morteza","family":"Maleknia","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3806-9316","authenticated-orcid":false,"given":"Mostafa","family":"Shamsi","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2020,8,31]]},"reference":[{"issue":"3","key":"1740_CR1","doi-asserted-by":"publisher","first-page":"751","DOI":"10.1137\/030601296","volume":"15","author":"JV Burke","year":"2005","unstructured":"Burke, J.V., Lewis, A.S., Overton, M.L.: A robust gradient sampling algorithm for nonsmooth, nonconvex optimization. SIAM J. Optim. 15(3), 751\u2013779 (2005)","journal-title":"SIAM J. Optim."},{"issue":"2","key":"1740_CR2","doi-asserted-by":"publisher","first-page":"379","DOI":"10.1137\/050639673","volume":"18","author":"KC Kiwiel","year":"2007","unstructured":"Kiwiel, K.C.: Convergence of the gradient sampling algorithm for nonsmooth nonconvex optimization. SIAM J. Optim. 18(2), 379\u2013388 (2007)","journal-title":"SIAM J. Optim."},{"issue":"4","key":"1740_CR3","doi-asserted-by":"publisher","first-page":"1983","DOI":"10.1137\/090748408","volume":"20","author":"KC Kiwiel","year":"2010","unstructured":"Kiwiel, K.C.: A nonderivative version of the gradient sampling algorithm for nonsmooth nonconvex optimization. SIAM J. Optim. 20(4), 1983\u20131994 (2010)","journal-title":"SIAM J. Optim."},{"issue":"6","key":"1740_CR4","doi-asserted-by":"publisher","first-page":"1302","DOI":"10.1080\/10556788.2012.714781","volume":"28","author":"FE Curtis","year":"2013","unstructured":"Curtis, F.E., Que, X.: An adaptive gradient sampling algorithm for nonconvex nonsmooth optimization. Optim. Methods Softw. 28(6), 1302\u20131324 (2013)","journal-title":"Optim. Methods Softw."},{"issue":"2","key":"1740_CR5","doi-asserted-by":"publisher","first-page":"474","DOI":"10.1137\/090780201","volume":"22","author":"FE Curtis","year":"2012","unstructured":"Curtis, F.E., Overton, M.L.: A sequential quadratic programming algorithm for nonconvex, nonsmooth constrained optimization. SIAM J. Optim. 22(2), 474\u2013500 (2012)","journal-title":"SIAM J. Optim."},{"issue":"1","key":"1740_CR6","doi-asserted-by":"publisher","first-page":"137","DOI":"10.1007\/s10957-017-1160-x","volume":"175","author":"ES Helou","year":"2017","unstructured":"Helou, E.S., Santos, A.S., Sim\u00f5es, L.E.A.: On the local convergence analysis of the gradient sampling method for finite max-functions. J. Optim. Theory Appl. 175(1), 137\u2013157 (2017)","journal-title":"J. Optim. Theory Appl."},{"issue":"3","key":"1740_CR7","doi-asserted-by":"publisher","first-page":"673","DOI":"10.1007\/s10589-018-0030-2","volume":"71","author":"ES Helou","year":"2018","unstructured":"Helou, E.S., Santos, A.S., Sim\u00f5es, L.E.A.: A fast gradient and function sampling method for finite-max functions. Comput. Optim. Appl. 71(3), 673\u2013717 (2018)","journal-title":"Comput. Optim. Appl."},{"key":"1740_CR8","volume-title":"Measure Theory and Fine Properties of Functions","author":"LC Evans","year":"1992","unstructured":"Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions, Revised edn. CRC Press, Boca Raton (1992)","edition":"Revised"},{"key":"1740_CR9","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611971309","volume-title":"Optimization and Nonsmooth Analysis","author":"FH Clarke","year":"1990","unstructured":"Clarke, F.H.: Optimization and Nonsmooth Analysis. SIAM, Montreal (1990)"},{"key":"1740_CR10","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-08114-4","volume-title":"Introduction to Nonsmooth Optimization","author":"AM Bagirov","year":"2014","unstructured":"Bagirov, A.M., Karmitsa, N., M\u00e4kel\u00e4, M.M.: Introduction to Nonsmooth Optimization. Springer, Berlin (2014)"},{"key":"1740_CR11","volume-title":"Variational Analysis","author":"RT Rockafellar","year":"2004","unstructured":"Rockafellar, R.T., Wets, R.J.B.: Variational Analysis. Springer, Berlin (2004)"},{"key":"1740_CR12","doi-asserted-by":"publisher","first-page":"201","DOI":"10.1007\/978-3-030-34910-3_6","volume-title":"Numerical Nonsmooth Optimization: State of the Art Algorithms","author":"JV Burke","year":"2020","unstructured":"Burke, J.V., Curtis, F.E., Lewis, A.S., Overton, M.L., Sim\u00f5es, L.E.A.: Gradient sampling methods for nonsmooth optimization. In: Bagirov, A.M., et al. (eds.) Numerical Nonsmooth Optimization: State of the Art Algorithms, pp. 201\u2013225. Springer, Cham (2020)"},{"key":"1740_CR13","doi-asserted-by":"publisher","DOI":"10.1515\/9781400873173","volume-title":"Convex Analysis","author":"RT Rockafellar","year":"1970","unstructured":"Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)"},{"key":"1740_CR14","volume-title":"Multicriteria Optimization","author":"M Ehrgott","year":"2005","unstructured":"Ehrgott, M.: Multicriteria Optimization. Springer, Berlin (2005)"},{"key":"1740_CR15","doi-asserted-by":"publisher","DOI":"10.1142\/1493","volume-title":"Nonsmooth Optimization: Analysis and Algorithms with Applications to Optimal Control","author":"MM M\u00e4kel\u00e4","year":"1992","unstructured":"M\u00e4kel\u00e4, M.M., Neittaanm\u00e4ki, P.: Nonsmooth Optimization: Analysis and Algorithms with Applications to Optimal Control. World Scientific, Singapore (1992)"},{"key":"1740_CR16","unstructured":"Luk\u0161an, L., Vl\u010dek, J.: Test problems for nonsmooth unconstrained and linearly constrained optimization. Technical report, Institute of Computer Science, Academy of Sciences of the Czech Republic (2000)"},{"key":"1740_CR17","unstructured":"Skaaja, M.: Limited memory BFGS for nonsmooth optimization. Master\u2019s Thesis, New York University (2010)"},{"key":"1740_CR18","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0074500","volume-title":"Methods of Descent for Nondifferentiable Optimization","author":"KC Kiwiel","year":"1985","unstructured":"Kiwiel, K.C.: Methods of Descent for Nondifferentiable Optimization. Springer, Berlin (1985)"},{"key":"1740_CR19","unstructured":"Luk\u0161an, L., Tcma, M., Siska, M., Vl\u010dek, J., Ramesova, N.: Ufo 2002. Interactive system for universal functional optimization. Technical report, Institute of Computer Science, Academy of Sciences of the Czech Republic (2002)"},{"issue":"6","key":"1740_CR20","doi-asserted-by":"publisher","first-page":"673","DOI":"10.1080\/10556780410001689225","volume":"19","author":"M Haarala","year":"2004","unstructured":"Haarala, M., Miettinen, K., M\u00e4kel\u00e4, M.M.: New limited memory bundle method for large-scale nonsmooth optimization. Optim. Methods Softw. 19(6), 673\u2013692 (2004)","journal-title":"Optim. Methods Softw."},{"key":"1740_CR21","unstructured":"Grothey, A.: Decomposition methods for nonlinear nonconvex optimization problems. Ph.D. Thesis, University of Edinburgh (2001)"},{"issue":"2","key":"1740_CR22","doi-asserted-by":"publisher","first-page":"201","DOI":"10.1007\/s101070100263","volume":"91","author":"E Dolan","year":"2002","unstructured":"Dolan, E., Mor\u00e9, J.: Benchmarking optimization software with performance profiles. Math. Program. 91(2), 201\u2013213 (2002)","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\/s10957-020-01740-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10957-020-01740-8\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-020-01740-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,8,31]],"date-time":"2021-08-31T00:15:07Z","timestamp":1630368907000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10957-020-01740-8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,8,31]]},"references-count":22,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2020,10]]}},"alternative-id":["1740"],"URL":"https:\/\/doi.org\/10.1007\/s10957-020-01740-8","relation":{},"ISSN":["0022-3239","1573-2878"],"issn-type":[{"value":"0022-3239","type":"print"},{"value":"1573-2878","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,8,31]]},"assertion":[{"value":"29 May 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 August 2020","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"31 August 2020","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}