{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:24:57Z","timestamp":1787329497104,"version":"build-2736575974"},"reference-count":25,"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":[[2011,1]]},"abstract":"<jats:p>The feasible set M in generalized semi-infinite programming (GSIP) need not be closed. Under the so-called symmetric Mangasarian\u2013Fromovitz constraint qualification (Sym-MFCQ), its closure $\\overline{M}$ can be described by means of infinitely many inequality constraints of maximum type. In this paper we introduce the nonsmooth symmetric reduction ansatz (NSRA). Under NSRA we prove that the set $\\overline{M}$ can locally be described as the feasible set of a so-called disjunctive optimization problem defined by finitely many inequality constraints of maximum type. This also shows the appearance of re-entrant corners in $\\overline{M}$. Under Sym-MFCQ all local minimizers of GSIP are KKT points for GSIP. We show that NSRA is generic and stable at all KKT points and that all KKT points are nondegenerate. The concept of (nondegenerate) KKT points as well as a corresponding GSIP-index are introduced in this paper. In particular, a nondegenerate KKT-point is a local minimizer if and only if its GSIP-index vanishes. At local minimizers NSRA coincides with the symmetric reduction ansatz (SRA) as introduced in [H. G\u00fcnzel, H. Th. Jongen, and O. Stein, Optim. Lett., 2 (2008), pp. 415\u2013424]. In comparison with SRA, the main new issue in NSRA is the following: At KKT points different from local minimizers, the Lagrange polytope at the lower level generically need not be a singleton anymore. In fact, it will be a full-dimensional simplex. This fact is crucial to provide the above-mentioned local reduction to a disjunctive optimization problem. Finally, we establish a local cell-attachment theorem which will be the basis for the development of a global critical point theory for GSIP.<\/jats:p>","DOI":"10.1137\/100786757","type":"journal-article","created":{"date-parts":[[2011,1,18]],"date-time":"2011-01-18T18:06:43Z","timestamp":1295374003000},"page":"193-211","source":"Crossref","is-referenced-by-count":4,"title":["Generalized Semi-Infinite Programming: The Nonsmooth Symmetric Reduction Ansatz"],"prefix":"10.1137","volume":"21","author":[{"given":"H. TH.","family":"Jongen","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"V.","family":"Shikhman","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2011,1,18]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1137\/090751025"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1137\/090775294"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2007.02.012"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1080\/02331930701779039"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1007\/s10100-007-0030-2"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1007\/s11590-007-0069-y"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1007\/s10898-008-9302-1"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1137\/S1052623403434735"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1080\/02331939508844106"},{"key":"R10","unstructured":"M. W. Hirsch,\n                      Differential Topology,\n                      Springer, Berlin, Heidelberg, New York, 1976."},{"key":"R11","doi-asserted-by":"crossref","unstructured":"H. Th. Jongen, P. Jonker, and F. Twilt,\n                      Nonlinear Optimization in Finite Dimensions,\n                      Kluwer Academic Publishers, Dordrecht, the Netherlands, 2000.","DOI":"10.1007\/978-1-4615-0017-9"},{"key":"R12","unstructured":"H. Th. Jongen, K. Meer, and E. Triesch,\n                      Optimization Theory,\n                      Kluwer Academic Publishers, Dordrecht, the Netherlands, 2004."},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1023\/A:1008282216306"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1023\/A:1022650006477"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1007\/BF02680555"},{"key":"R16","unstructured":"H. Th. Jongen and V. Shikhman,\n                      General Semi-Infinite Programming: Critical Point Theory,\n                      Preprint 136, Lehrstuhl C f\u00fcr Mathematik, RWTH Aachen University, Aachen, Germany, 2010."},{"key":"R17","unstructured":"D. Klatte and B. Kummer,\n                      Nonsmooth Equations in Optimization: Regularity, Calculus, Methods and Applications,\n                      Nonconvex Optim. Appl. 60, Kluwer Academic Publishers, Dordrecht, the Netherlands, 2002."},{"key":"R18","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":"R19","doi-asserted-by":"publisher","DOI":"10.1023\/A:1021746305759"},{"key":"R20","doi-asserted-by":"crossref","unstructured":"E. H. Spanier,\n                      Algebraic Topology,\n                      McGraw-Hill, New York, 1966.","DOI":"10.1007\/978-1-4684-9322-1_5"},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1080\/02331930008844493"},{"key":"R22","doi-asserted-by":"crossref","unstructured":"O. Stein,\n                      Bi-level Strategies in Semi-infinite Programming,\n                      Kluwer Academic Publishers, Boston, 2003.","DOI":"10.1007\/978-1-4419-9164-5"},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1080\/02331930108844531"},{"key":"R24","unstructured":"G.W. Weber,\n                      Generalized Semi-Infinite Optimization and Related Topics,\n                      HeldermannVerlag, Lemgo, Germany, 2003."},{"key":"R25","doi-asserted-by":"publisher","DOI":"10.1007\/s10957-008-9352-z"}],"container-title":["SIAM Journal on Optimization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/100786757","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T15:19:37Z","timestamp":1787325577000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/100786757"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,1]]},"references-count":25,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2011,1]]}},"alternative-id":["10.1137\/100786757"],"URL":"https:\/\/doi.org\/10.1137\/100786757","relation":{},"ISSN":["1052-6234","1095-7189"],"issn-type":[{"value":"1052-6234","type":"print"},{"value":"1095-7189","type":"electronic"}],"subject":[],"published":{"date-parts":[[2011,1]]}}}