{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T19:34:12Z","timestamp":1787340852569,"version":"build-2736575974"},"reference-count":36,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","funder":[{"DOI":"10.13039\/100000015","name":"U.S. Department of Energy","doi-asserted-by":"publisher","award":["DE-AC02-06CH11357"],"award-info":[{"award-number":["DE-AC02-06CH11357"]}],"id":[{"id":"10.13039\/100000015","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Optim."],"published-print":{"date-parts":[[2018,1]]},"abstract":"<jats:p>We develop a manifold sampling algorithm for the minimization of a nonsmooth composite function $f \\triangleq \\psi + h \\circ F$ when $\\psi$ is smooth with known derivatives, $h$ is a known, nonsmooth, piecewise linear function, and $F$ is smooth but expensive to evaluate. The trust-region algorithm classifies points in the domain of $h$ as belonging to different manifolds and uses this knowledge when computing search directions. Since $h$ is known, classifying objective manifolds using only the values of $F$ is simple. We prove that all cluster points of the sequence of the manifold sampling algorithm iterates are Clarke stationary; this holds although points evaluated by the algorithm are not assumed to be differentiable and when only approximate derivatives of $F$ are available. Numerical results show that manifold sampling using zeroth-order information about $F$ is competitive with algorithms that employ exact subgradient values from $\\partial f$.<\/jats:p>","DOI":"10.1137\/17m114741x","type":"journal-article","created":{"date-parts":[[2018,10,25]],"date-time":"2018-10-25T13:05:04Z","timestamp":1540472704000},"page":"3001-3024","source":"Crossref","is-referenced-by-count":13,"title":["Manifold Sampling for Optimization of Nonconvex Functions That Are Piecewise Linear Compositions of Smooth Components"],"prefix":"10.1137","volume":"28","author":[{"given":"Kamil A.","family":"Khan","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9924-2082","authenticated-orcid":true,"given":"Jeffrey","family":"Larson","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Stefan M.","family":"Wild","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2018,10,25]]},"reference":[{"key":"atypb1","first-page":"217","volume":"15","author":"Aravkin A.","year":"2014","journal-title":"J. Mach. Learn. Res."},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1007\/s10957-007-9335-5"},{"key":"atypb3","first-page":"321","author":"Bengio Y.","year":"2007","journal-title":"MA"},{"key":"atypb4","first-page":"530","author":"Bhojanapalli S.","year":"2016","journal-title":"New York"},{"key":"atypb5","unstructured":"F. H. Clarke,\n                      Optimization and Nonsmooth Analysis\n                      , John Wiley & Sons, New York, 1983."},{"key":"atypb6","first-page":"201","author":"Collobert R.","year":"2006","journal-title":"New York"},{"key":"atypb7","doi-asserted-by":"crossref","unstructured":"A. R. Conn, K. Scheinberg, and L. N. Vicente,\n                      Introduction to Derivative-Free Optimization\n                      , MOS-SIAM Ser. Optim. 8, SIAM, Philadelphia, 2009,https:\/\/doi.org\/10.1137\/1.9780898718768.","DOI":"10.1137\/1.9780898718768"},{"key":"atypb8","unstructured":"F. E. Curtis,\n                      \\textsfSLQP-GS\n                      , 2017,http:\/\/coral.ise.lehigh.edu\/frankecurtis\/software."},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1080\/10556788.2016.1208749"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1137\/090780201"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1007\/s12532-015-0086-2"},{"key":"atypb12","unstructured":"D. Drusvyatskiy, A. D. Ioffe, and A. S. Lewis,\n                      Nonsmooth Optimization Using Taylor-Like Models: Error Bounds, Convergence, and Termination Criteria\n                      , Tech. report,https:\/\/arxiv.org\/abs\/1610.03446, 2016."},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1287\/moor.2017.0889"},{"key":"atypb14","doi-asserted-by":"crossref","unstructured":"F. Facchinei and J.S. Pang,\n                      Finite-Dimensional Variational Inequalities and Complementarity Problems, Vol.II\n                      , Springer-Verlag, New York, 2003.","DOI":"10.1007\/b97543"},{"key":"atypb15","first-page":"67","author":"Fletcher R.","year":"1982","journal-title":"Heidelberg"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1137\/151005683"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1007\/s11590-017-1117-x"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1007\/s10589-013-9547-6"},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1137\/050639673"},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1137\/090748408"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1137\/15M1042097"},{"key":"atypb22","unstructured":"J. Larson and S. M. Wild,\n                      Censored L1 Problem Specifications For Benchmarking Manifold Sampling\n                      , 2018,http:\/\/www.mcs.anl.gov\/ jlarson\/MS4PL."},{"key":"atypb23","unstructured":"Y. LeCun,\n                      Who is afraid of non-convex loss functions?\n                      , presented at The NIPS Workshop on Efficient Machine Learning, 2007,https:\/\/www.cs.nyu.edu\/~yann\/talks\/lecun-20071207-nonconvex.pdf."},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1214\/16-AOS1471"},{"key":"atypb25","unstructured":"T. Mitchell,\n                      \\textsfGRANSO\n                      , 2017,https:\/\/gitlab.com\/timmitchell\/GRANSO."},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.1137\/0904038"},{"key":"atypb27","unstructured":"J. J. Mor\u00e9 and S. M. Wild,\n                      Derivative-free optimization test functions\n                      ,http:\/\/www.mcs.anl.gov\/~more\/dfo\/matlab\/dfovec.m(accessed: 2018-06-25)."},{"key":"atypb28","doi-asserted-by":"publisher","DOI":"10.1137\/080724083"},{"key":"atypb29","unstructured":"A. Neumaier,\n                      MINQ\n                      , 2017,http:\/\/www.mat.univie.ac.at\/~neum\/software\/minq."},{"key":"atypb30","doi-asserted-by":"publisher","DOI":"10.1287\/moor.18.1.227"},{"key":"atypb31","doi-asserted-by":"crossref","unstructured":"S. Scholtes,\n                      Introduction to Piecewise Differentiable Equations\n                      , Springer, New York, 2012,https:\/\/doi.org\/10.1007\/978-1-4614-4340-7.","DOI":"10.1007\/978-1-4614-4340-7"},{"key":"atypb32","doi-asserted-by":"publisher","DOI":"10.1007\/s10589-017-9912-y"},{"key":"atypb33","first-page":"529","author":"Wild S. M.","year":"2017","journal-title":"Philadelphia"},{"key":"atypb34","doi-asserted-by":"publisher","DOI":"10.1137\/0907008"},{"key":"atypb35","doi-asserted-by":"publisher","DOI":"10.1007\/BF00940574"},{"key":"atypb36","doi-asserted-by":"publisher","DOI":"10.1007\/BF02591750"}],"container-title":["SIAM Journal on Optimization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/17M114741X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:52:11Z","timestamp":1787338331000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/17M114741X"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,1]]},"references-count":36,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2018,1]]}},"alternative-id":["10.1137\/17M114741X"],"URL":"https:\/\/doi.org\/10.1137\/17m114741x","relation":{},"ISSN":["1052-6234","1095-7189"],"issn-type":[{"value":"1052-6234","type":"print"},{"value":"1095-7189","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,1]]}}}