{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:29:12Z","timestamp":1787336952743,"version":"build-2736575974"},"reference-count":31,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Matrix Anal. Appl."],"published-print":{"date-parts":[[2003,1]]},"abstract":"<jats:p>\n                    Any spectral function can be written as a composition of a symmetric function $f: \\rn \\mapsto \\Re$ and the eigenvalue function $\\lambda(\\cdot): \\s \\mapsto \\rn$, often denoted by $(f \\circ \\lambda)$, where $\\s$ is the subspace of n \u00d7 n symmetric matrices. In this paper, we present some nonsmooth analysis for such spectral functions. Our main results are (a) $(f \\circ \\lambda)$ is directionally differentiable if f is semidifferentiable, (b) $(f \\circ \\lambda)$ is LC\n                    <jats:sup>1<\/jats:sup>\n                    if and only if f is LC\n                    <jats:sup>1<\/jats:sup>\n                    , and (c) $(f \\circ \\lambda)$ is SC\n                    <jats:sup>1<\/jats:sup>\n                    if and only if f is SC\n                    <jats:sup>1<\/jats:sup>\n                    . Result (a) is complementary to a known (negative) fact that $(f \\circ \\lambda)$ might not be directionally differentiable if f is directionally differentiable only. Results (b) and (c) are particularly useful for the solution of LC\n                    <jats:sup>1<\/jats:sup>\n                    and SC\n                    <jats:sup>1<\/jats:sup>\n                    minimization problems which often can be solved by fast (generalized) Newton methods. Our analysis makes use of recent results on continuously differentiable spectral functions as well as on nonsmooth symmetric--matrix-valued functions.\n                  <\/jats:p>","DOI":"10.1137\/s0895479802417921","type":"journal-article","created":{"date-parts":[[2004,3,5]],"date-time":"2004-03-05T21:00:50Z","timestamp":1078520450000},"page":"766-783","source":"Crossref","is-referenced-by-count":12,"title":["Semismoothness of Spectral Functions"],"prefix":"10.1137","volume":"25","author":[{"given":"Houduo","family":"Qi","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xiaoqi","family":"Yang","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":"10.1007\/978-1-4612-0653-8"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-002-0306-1"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1137\/S1052623400380584"},{"key":"R4","volume-title":"Optimization and nonsmooth analysis","author":"Clarke Frank","year":"1983"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1007\/BF01898787"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1016\/0167-6377(94)00059-F"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1016\/S0025-5610(96)00059-7"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1023\/A:1008644520093"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1007\/BF01442169"},{"key":"R10","doi-asserted-by":"crossref","unstructured":"Houyuan Jiang, Daniel Ralph, Global and local superlinear convergence analysis of Newton\u2010type methods for semismooth equations with smooth least squares, Appl. 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Vol. IV: Continuous geometry and other topics","author":"von Neumann John","year":"1962"},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1007\/BF02193060"},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1287\/moor.18.1.227"},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1007\/BF01582577"},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1007\/BF01581275"},{"key":"R24","unstructured":"L. Qi and P. Tseng,\n                      An analysis of piecewise smooth functions and almost smooth functions\n                      , Math. Oper. Res., submitted."},{"key":"R25","unstructured":"F. Rellich,\n                      Perturbation Theory of Eigenvalue Problems\n                      , Gordon and Breach, New York, 1969."},{"key":"R26","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-02431-3"},{"key":"R27","doi-asserted-by":"publisher","DOI":"10.1287\/moor.27.1.150.342"},{"key":"R28","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142901393814"},{"key":"R29","doi-asserted-by":"publisher","DOI":"10.1016\/S0025-5610(98)00004-5"},{"key":"R30","doi-asserted-by":"publisher","DOI":"10.1137\/S1052623400371806"},{"key":"R31","doi-asserted-by":"publisher","DOI":"10.1080\/02331939208843851"}],"container-title":["SIAM Journal on Matrix Analysis and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/S0895479802417921","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:37:59Z","timestamp":1787333879000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/S0895479802417921"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2003,1]]},"references-count":31,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2003,1]]}},"alternative-id":["10.1137\/S0895479802417921"],"URL":"https:\/\/doi.org\/10.1137\/s0895479802417921","relation":{},"ISSN":["0895-4798","1095-7162"],"issn-type":[{"value":"0895-4798","type":"print"},{"value":"1095-7162","type":"electronic"}],"subject":[],"published":{"date-parts":[[2003,1]]}}}