{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:25:19Z","timestamp":1787336719200,"version":"build-2736575974"},"reference-count":16,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Optim."],"published-print":{"date-parts":[[2009,1]]},"abstract":"<jats:p>We study mathematical programs with complementarity constraints (MPCC) from a topological point of view. Special focus will be on C-stationary points. Under the linear independence constraint qualification (LICQ) we derive an equivariant Morse lemma at nondegenerate C-stationary points. Then, two basic theorems from Morse theory (deformation theorem and cell-attachment theorem) are proved. Outside the C-stationary point set, continuous deformation of lower level sets can be performed. As a consequence, the topological data (such as the number of connected components) then remain invariant. However, when passing a C-stationary level, the topology of the lower level set changes via the attachment of a q-dimensional cell. The dimension q equals the stationary C-index of the (nondegenerate) C-stationary point. The stationary C-index depends on both the restricted Hessian of the Lagrangian and the Lagrange multipliers related to biactive complementarity constraints. Finally, some relations with other stationarity concepts, such as W-, A-, M-, S-, and B-stationarity, are discussed.<\/jats:p>","DOI":"10.1137\/080733693","type":"journal-article","created":{"date-parts":[[2009,4,29]],"date-time":"2009-04-29T18:05:09Z","timestamp":1241028309000},"page":"473-484","source":"Crossref","is-referenced-by-count":18,"title":["MPCC: Critical Point Theory"],"prefix":"10.1137","volume":"20","author":[{"given":"H. Th.","family":"Jongen","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jan.-J.","family":"R\u00fcckmann","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"V.","family":"Shikhman","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2009,4,29]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1007\/s101070050047"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1007\/s10957-004-1176-x"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1007\/s10898-004-0865-1"},{"key":"R4","doi-asserted-by":"crossref","unstructured":"M. Goresky and R. MacPherson,\n                      Stratified Morse Theory,\n                      Springer, Berlin-Heidelberg-New York, 1988.","DOI":"10.1007\/978-3-642-71714-7"},{"key":"R5","unstructured":"M.W. Hirsch,\n                      Differential Topology,\n                      Springer, Berlin-Heidelberg-New York, 1976."},{"key":"R6","unstructured":"H.Th. Jongen, K. Meer, and E. Triesch,\n                      Optimization Theory,\n                      Kluwer Academic Publishers, Dordrecht, 2004."},{"key":"R7","doi-asserted-by":"crossref","unstructured":"H.Th. Jongen, P. Jonker, and F. Twilt,\n                      Nonlinear Optimization in Finite Dimensions,\n                      Kluwer Academic Publishers, Dordrecht, 2000.","DOI":"10.1007\/978-1-4615-0017-9"},{"key":"R8","doi-asserted-by":"crossref","unstructured":"M. Kojima,\n                      Strongly stable stationary solutions in nonlinear programs,\n                      in Analysis and Computation of Fixed Points, S.M. Robinson, ed., Academic Press, New York, 1980, pp. 93\u2013138.","DOI":"10.1016\/B978-0-12-590240-3.50009-4"},{"key":"R9","unstructured":"S. Leyffer and T.S. Munson,\n                      A globally convergent filter method for MPECs,\n                      preprint ANL\/MCS-P1457-0907, Argonne National Laboratory, Mathematics and Computer Science Division, September 2007."},{"key":"R10","first-page":"1","volume":"51","author":"Milnor J.","year":"1963","journal-title":"Annals of Mathematic Studies, Princeton University Press, Princeton, NJ"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1287\/moor.24.3.627"},{"key":"R12","unstructured":"D. Ralph and O. Stein,\n                      Homotopy methods for quadratic programs with complementarity constraints,\n                      preprint No. 120, Department of Mathematics - C, RWTH Aachen University, 2006."},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1287\/moor.25.1.1.15213"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1287\/moor.26.4.851.10007"},{"key":"R15","doi-asserted-by":"crossref","unstructured":"E.H. Spanier,\n                      Algebraic Topology,\n                      McGraw-Hill Book Company, New York, 1966.","DOI":"10.1007\/978-1-4684-9322-1_5"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2004.10.032"}],"container-title":["SIAM Journal on Optimization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/080733693","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:28:10Z","timestamp":1787333290000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/080733693"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,1]]},"references-count":16,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2009,1]]}},"alternative-id":["10.1137\/080733693"],"URL":"https:\/\/doi.org\/10.1137\/080733693","relation":{},"ISSN":["1052-6234","1095-7189"],"issn-type":[{"value":"1052-6234","type":"print"},{"value":"1095-7189","type":"electronic"}],"subject":[],"published":{"date-parts":[[2009,1]]}}}