{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:02:14Z","timestamp":1787320934414,"version":"build-2736575974"},"reference-count":24,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Control Optim."],"published-print":{"date-parts":[[2015,1]]},"abstract":"<jats:p>In this paper we study local convergence of inexact Newton methods of the form $ {\\big(}f(x_k)+ A_k(x_{k+1}-x_k)+F(x_{k+1}){\\big)}\\cap R_k(x_k) \\neq \\emptyset \\ \\ {with} \\ \\ A_k \\in {\\cal H}(x_k)$ for solving the generalized equation $f(x) +F(x) \\ni 0$ in Banach spaces, where the function $f$ is continuous but not necessarily smooth and $F$ is a set-valued mapping with closed graph. The mapping $\\cal H$ plays the role of a generalized set-valued derivative of $f$ which in finite dimensions may be represented by Clarke's generalized Jacobian, while in Banach spaces it may be identified with Ioffe's strict prederivative. The set-valued mappings $R_k$ represent inexactness. We utilize conditions divided into three groups: the first concerns the kind of nonsmoothness of the function $f$, the second involves metric regularity properties of an approximation of the mapping $f+F$, and the third is about the sequence of mappings $R_k$. Under various combinations of these conditions we show linear, superlinear, or quadratic convergence of the method. In the second part of the paper we give two generalizations of the Dennis--Mor\u00e9 theorem. As corollaries, we obtain results regarding convergence of inexact semismooth quasi-Newton-type methods and Dennis--Mor\u00e9 theorems for semismooth equations.<\/jats:p>","DOI":"10.1137\/140969476","type":"journal-article","created":{"date-parts":[[2015,4,21]],"date-time":"2015-04-21T12:28:52Z","timestamp":1429619332000},"page":"1003-1019","source":"Crossref","is-referenced-by-count":23,"title":["Inexact Newton Methods and Dennis--Mor\u00e9 Theorems for Nonsmooth Generalized Equations"],"prefix":"10.1137","volume":"53","author":[{"given":"R.","family":"Cibulka","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"A.","family":"Dontchev","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"M. H.","family":"Geoffroy","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2015,4,21]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1137\/130926730"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-04-01646-1"},{"key":"atypb3","unstructured":"F. H. Clarke,\n                      Optimization and Nonsmooth Analysis\n                      , Wiley, New York, 1983."},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1974-0343581-1"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1137\/0719025"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1137\/110833567"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-013-0664-x"},{"key":"atypb8","doi-asserted-by":"crossref","unstructured":"A. L. Dontchev and R. T. Rockafellar,\n                      Implicit Functions and Solution Mappings: A View from Variational Analysis,\n                      2nd ed., Springer, New York, 2014.","DOI":"10.1007\/978-1-4939-1037-3"},{"key":"atypb9","first-page":"345","volume":"20","author":"Fabian M.","year":"1979","journal-title":"Comment. Math. Univ. Carolin."},{"key":"atypb10","doi-asserted-by":"crossref","unstructured":"F. Facchinei and J.S. Pang,\n                      Finite-Dimensional Variational Inequalities and Complementarity Problems,\n                      Springer, New York, 2003.","DOI":"10.1007\/b97544"},{"key":"atypb11","unstructured":"M. Hintermu\u0308ller,\n                      Semismooth Newton Methods and Applications\n                      , Department of Mathematics, Humboldt-University, Berlin, 2010."},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1981-0613784-7"},{"key":"atypb13","doi-asserted-by":"crossref","unstructured":"K. Ito and K. Kunisch,\n                      Lagrange Multiplier Approach to Variational Problems and Applications,\n                      SIAM, Philadelphia, 2008.","DOI":"10.1137\/1.9780898718614"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1007\/s11228-012-0218-z"},{"key":"atypb15","doi-asserted-by":"crossref","unstructured":"A. F. Izmailov and M. V. Solodov,\n                      Newton-Type Methods for Optimization and Variational Problems\n                      , Springer, New York, 2014.","DOI":"10.1007\/978-3-319-04247-3"},{"key":"atypb16","doi-asserted-by":"crossref","unstructured":"C. T. Kelley,\n                      Solving Nonlinear Equations with Newton's Method,\n                      Fundam. Algorithms, SIAM, Philadelphia, 2003.","DOI":"10.1137\/1.9780898718898"},{"key":"atypb17","unstructured":"D. Klatte and B. Kummer,\n                      Nonsmooth Equations in Optimization: Regularity\n                      , Calculus, Methods and Applications, Kluwer, New York, 2002."},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1007\/s11228-007-0043-y"},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2008.02.044"},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1137\/0803021"},{"key":"atypb21","doi-asserted-by":"crossref","unstructured":"J.P. Penot,\n                      Calculus Without Derivatives,\n                      Springer, New York, 2013.","DOI":"10.1007\/978-1-4614-4538-8"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1007\/BF01581275"},{"key":"atypb23","doi-asserted-by":"crossref","unstructured":"M. Ulbrich,\n                      Semismooth Newton Methods for Variational Inequalities and Constrained Optimization Problems in Function Spaces\n                      , SIAM, Philadelphia, 2011.","DOI":"10.1137\/1.9781611970692"},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1007\/s101070050028"}],"container-title":["SIAM Journal on Control and Optimization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/140969476","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:01:59Z","timestamp":1787317319000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/140969476"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,1]]},"references-count":24,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2015,1]]}},"alternative-id":["10.1137\/140969476"],"URL":"https:\/\/doi.org\/10.1137\/140969476","relation":{},"ISSN":["0363-0129","1095-7138"],"issn-type":[{"value":"0363-0129","type":"print"},{"value":"1095-7138","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,1]]}}}