{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,22]],"date-time":"2026-08-22T07:03:02Z","timestamp":1787382182017,"version":"build-2736575974"},"reference-count":9,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Optim."],"published-print":{"date-parts":[[1998,5]]},"abstract":"<jats:p>\n                    In this paper we introduce a new class of real-valued locally Lipschitz functions (that are similar in nature and definition to Valadier's saine functions), which we call arcwise essentially smooth, and we show that if g : R\n                    <jats:sup>m<\/jats:sup>\n                    \\rightarrow R$ is arcwise essentially smooth on R\n                    <jats:sup>m<\/jats:sup>\n                    and each function $f_j : R^n \\rightarrow R,\\ 1 \\leq j \\leq m$, is strictly differentiable almost everywhere in R\n                    <jats:sup>n<\/jats:sup>\n                    , then $g \\circ f$ is strictly differentiable almost everywhere in R\n                    <jats:sup>n<\/jats:sup>\n                    , where $f \\equiv (f_1,f_2,\\ldots,f_m)$. We also show that all the semismooth and all the pseudoregular functions are arcwise essentially smooth. Thus, we provide a large and robust lattice algebra of Lipschitz functions whose generalized derivatives are well behaved.\n                  <\/jats:p>","DOI":"10.1137\/s1052623496297838","type":"journal-article","created":{"date-parts":[[2003,6,11]],"date-time":"2003-06-11T11:12:06Z","timestamp":1055329926000},"page":"300-308","source":"Crossref","is-referenced-by-count":12,"title":["A Chain Rule for Essentially Smooth Lipschitz Functions"],"prefix":"10.1137","volume":"8","author":[{"given":"Jonathan M.","family":"Borwein","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Warren B.","family":"Moors","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,7,31]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1006\/jmaa.1994.1104"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1016\/0022-247X(87)90203-4"},{"key":"R3","doi-asserted-by":"crossref","unstructured":"JonathanBorwein, WarrenMoors, Essentially smooth Lipschitz functions: compositions and chain rules, Springer, Berlin, 1997, 16\u20132299k:49031","DOI":"10.1007\/978-3-642-60539-0_2"},{"key":"R4","unstructured":"J. M. Borwein and W. B. Moors,\n                      Haar\u2010null sets and essentially smooth Lipschitz functions\n                      ; SIAM J. Optim., to appear."},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1007\/BF02762799"},{"key":"R6","volume-title":"Pontryagin Conference. 2. Nonsmooth Analysis and Optimization","author":"Sakharova V.","year":"2000"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1007\/BF00940840"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1007\/BF01038595"},{"key":"R9","first-page":"241","volume":"308","author":"Valadier Michel","year":"1989","journal-title":"C. R. Acad. Sci. Paris S\u00e9r. I Math."}],"container-title":["SIAM Journal on Optimization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/S1052623496297838","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:16:45Z","timestamp":1787332605000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/S1052623496297838"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1998,5]]},"references-count":9,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1998,5]]}},"alternative-id":["10.1137\/S1052623496297838"],"URL":"https:\/\/doi.org\/10.1137\/s1052623496297838","relation":{},"ISSN":["1052-6234","1095-7189"],"issn-type":[{"value":"1052-6234","type":"print"},{"value":"1095-7189","type":"electronic"}],"subject":[],"published":{"date-parts":[[1998,5]]}}}