{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,13]],"date-time":"2026-01-13T22:00:43Z","timestamp":1768341643625,"version":"3.49.0"},"reference-count":7,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2023,12,1]],"date-time":"2023-12-01T00:00:00Z","timestamp":1701388800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,12,1]],"date-time":"2023-12-01T00:00:00Z","timestamp":1701388800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["441468770"],"award-info":[{"award-number":["441468770"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100005626","name":"Universit\u00e4t Regensburg","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100005626","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Optim Theory Appl"],"published-print":{"date-parts":[[2024,2]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Clarke\u2019s inverse function theorem for Lipschitz mappings states that a bi-Lipschitz mapping <jats:italic>f<\/jats:italic> is locally invertible about a point <jats:inline-formula><jats:alternatives><jats:tex-math>$$x_0$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>x<\/mml:mi>\n                    <mml:mn>0<\/mml:mn>\n                  <\/mml:msub>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> if the generalized Jacobian <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\partial f(x_0)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>\u2202<\/mml:mi>\n                    <mml:mi>f<\/mml:mi>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:msub>\n                      <mml:mi>x<\/mml:mi>\n                      <mml:mn>0<\/mml:mn>\n                    <\/mml:msub>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> does not contain singular matrices. It is shown that under these assumptions the generalized Jacobian of the inverse mapping at <jats:inline-formula><jats:alternatives><jats:tex-math>$$f(x_0)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>f<\/mml:mi>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:msub>\n                      <mml:mi>x<\/mml:mi>\n                      <mml:mn>0<\/mml:mn>\n                    <\/mml:msub>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is the convex hull of the set of matrices that can be obtained as limits of sequences <jats:inline-formula><jats:alternatives><jats:tex-math>$$J_f(x_k)^{-1}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:msub>\n                      <mml:mi>J<\/mml:mi>\n                      <mml:mi>f<\/mml:mi>\n                    <\/mml:msub>\n                    <mml:msup>\n                      <mml:mrow>\n                        <mml:mo>(<\/mml:mo>\n                        <mml:msub>\n                          <mml:mi>x<\/mml:mi>\n                          <mml:mi>k<\/mml:mi>\n                        <\/mml:msub>\n                        <mml:mo>)<\/mml:mo>\n                      <\/mml:mrow>\n                      <mml:mrow>\n                        <mml:mo>-<\/mml:mo>\n                        <mml:mn>1<\/mml:mn>\n                      <\/mml:mrow>\n                    <\/mml:msup>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> with <jats:italic>f<\/jats:italic> differentiable in <jats:inline-formula><jats:alternatives><jats:tex-math>$$x_k$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>x<\/mml:mi>\n                    <mml:mi>k<\/mml:mi>\n                  <\/mml:msub>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:tex-math>$$x_k$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>x<\/mml:mi>\n                    <mml:mi>k<\/mml:mi>\n                  <\/mml:msub>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> converging to <jats:inline-formula><jats:alternatives><jats:tex-math>$$x_0$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>x<\/mml:mi>\n                    <mml:mn>0<\/mml:mn>\n                  <\/mml:msub>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. This identity holds as well if <jats:italic>f<\/jats:italic> is assumed to be locally bi-Lipschitz at <jats:inline-formula><jats:alternatives><jats:tex-math>$$x_0$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>x<\/mml:mi>\n                    <mml:mn>0<\/mml:mn>\n                  <\/mml:msub>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>.\n<\/jats:p>","DOI":"10.1007\/s10957-023-02333-x","type":"journal-article","created":{"date-parts":[[2023,12,1]],"date-time":"2023-12-01T08:03:12Z","timestamp":1701417792000},"page":"852-857","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["A Note on Clarke\u2019s Generalized Jacobian for the Inverse of Bi-Lipschitz Maps"],"prefix":"10.1007","volume":"200","author":[{"given":"Florian","family":"Behr","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7622-1174","authenticated-orcid":false,"given":"Georg","family":"Dolzmann","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2023,12,1]]},"reference":[{"key":"2333_CR1","doi-asserted-by":"publisher","first-page":"247","DOI":"10.1090\/S0002-9947-1975-0367131-6","volume":"205","author":"FH Clarke","year":"1975","unstructured":"Clarke, F.H.: Generalized gradients and applications. 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