{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,22]],"date-time":"2026-08-22T07:04:52Z","timestamp":1787382292483,"version":"build-2736575974"},"reference-count":37,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Optim."],"published-print":{"date-parts":[[2002,1]]},"abstract":"<jats:p>In this paper, an interior point trust region algorithm for the solution of a class of nonlinear semidefinite programming (SDP) problems is described and analyzed. Such nonlinear and nonconvex programs arise, e.g., in the design of optimal static or reduced order output feedback control laws and have the structure of abstract optimal control problems in a finite dimensional Hilbert space. The algorithm treats the abstract states and controls as independent variables. In particular, an algorithm for minimizing a nonlinear matrix objective functional subject to a nonlinear SDP-condition, a positive definiteness condition, and a nonlinear matrix equation is considered. The algorithm is designed to take advantage of the structure of the problem. It is an extension of an interior point trust region method to nonlinear and nonconvex SDPs, with a special structure which applies sequential quadratic programming techniques to a sequence of barrier problems and uses trust regions to ensure robustness of the iteration. Some convergence results are given, and, finally, several numerical examples demonstrate the applicability of the considered algorithm.<\/jats:p>","DOI":"10.1137\/s1052623400375865","type":"journal-article","created":{"date-parts":[[2003,6,11]],"date-time":"2003-06-11T11:12:06Z","timestamp":1055329926000},"page":"1048-1074","source":"Crossref","is-referenced-by-count":57,"title":["An Interior Point Constrained Trust Region Method for a Special Class of Nonlinear Semidefinite Programming Problems"],"prefix":"10.1137","volume":"12","author":[{"given":"F.","family":"Leibfritz","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"E. M. E.","family":"Mostafa","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,7,28]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1137\/0805002"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1137\/S1052623496304700"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1115\/1.3426525"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1145\/200979.201043"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611970777"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1007\/PL00011391"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1109\/9.486637"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1007\/s101070050112"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898719857"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1137\/S1052623492238881"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1137\/S036012995279031"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1137\/0728015"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1007\/BF02275347"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1137\/S1052623496305560"},{"key":"R15","doi-asserted-by":"crossref","unstructured":"DavidGay, MichaelOverton, MargaretWright, A primal\u2010dual interior method for nonconvex nonlinear programming, Appl. Optim., Vol. 14, Kluwer Acad. Publ., Dordrecht, 1998, 31\u20135699h:90096","DOI":"10.1007\/978-1-4613-3335-7_2"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1109\/9.599979"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0043756"},{"key":"R18","unstructured":"F. Jarre,\n                      A QQP\u2010Minimization Method for Semidefinite and Smooth Nonconvex Programs\n                      , Technical report, University of Notre Dame, Notre Dame, IN, 2000."},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1109\/9.85062"},{"key":"R20","unstructured":"F. S. Kupfer,\n                      Reduced SQP in Hilbert Space with Applications to Optimal Control\n                      , Ph.D. thesis, Universit\u00e4t Trier, FB IV\u2010Mathematik, Trier, Germany, 1992."},{"key":"R21","doi-asserted-by":"crossref","unstructured":"F.\u2010S.Kupfer, E.Sachs, A prospective look at SQP methods for semilinear parabolic control problems, Lecture Notes in Control and Inform. Sci., Vol. 149, Springer, Berlin, 1991, 143\u20131571178297","DOI":"10.1007\/BFb0043221"},{"key":"R22","unstructured":"F. Leibfritz,\n                      Static Output Feedback Design Problems\n                      , Shaker Verlag, Aachen, Germany, 1998."},{"key":"R23","unstructured":"F. Leibfritz,\n                      Computational Design of Stabilizing Static Output Feedback Controllers\n                      , Technical report 99\u201301, Universit\u00e4t Trier, Germany, 1999."},{"key":"R24","unstructured":"F. Leibfritz,\n                      Static Output Feedback Design by Using a Newton\u2010SQP Interior Point Method\n                      , Technical report 99\u201303, Universit\u00e4t Trier, Trier, Germany, 1999."},{"key":"R25","unstructured":"F. Leibfritz,\n                      A Collection of Test Examples for a Special Class of Nonlinear SDP\u2010Problems\n                      , Technical report, Universit\u00e4t Trier, Trier, Germany, 2000."},{"key":"R26","doi-asserted-by":"publisher","DOI":"10.1137\/S0363012999349553"},{"key":"R27","unstructured":"F. Leibfritz and E. M. E. Mostafa,\n                      Optimal Static Output Feedback Design by Using a Trust Region Interior Point Method\n                      , Technical report 00\u201303, Universit\u00e4t Trier, Trier, Germany, 2000."},{"key":"R28","unstructured":"F. Leibfritz and E. M. E. Mostafa,\n                      Trust Region Methods for Solving the Optimal Output Feedback Design Problem\n                      , Technical report 00\u201301, Universit\u00e4t Trier, Trier, Germany, 2000."},{"key":"R29","unstructured":"El\u2010S. M. E. Mostafa,\n                      Efficient Trust\u2010Region Methods in Numerical Optimization\n                      , Ph.D. thesis, Department of Mathematics, Faculty of Science, Alexandria University, Alexandria, Egypt, 2000."},{"key":"R30","doi-asserted-by":"crossref","unstructured":"Y. Nesterov and A. Nemirovskii,\n                      Interior\u2010Point Polynomial Algorithms in Convex Programming\n                      , SIAM Stud. Appl. Math. 13, SIAM, Philadelphia, 1994.","DOI":"10.1137\/1.9781611970791"},{"key":"R31","doi-asserted-by":"publisher","DOI":"10.1137\/S1052623495290441"},{"key":"R32","doi-asserted-by":"publisher","DOI":"10.1080\/10556789508805599"},{"key":"R33","doi-asserted-by":"publisher","DOI":"10.1137\/0720042"},{"key":"R34","doi-asserted-by":"publisher","DOI":"10.1080\/00207178708547402"},{"key":"R35","doi-asserted-by":"publisher","DOI":"10.1137\/1038003"},{"key":"R36","doi-asserted-by":"publisher","DOI":"10.1023\/A:1008677427361"},{"key":"R37","doi-asserted-by":"publisher","DOI":"10.1080\/02331939008843578"}],"container-title":["SIAM Journal on Optimization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/S1052623400375865","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:28:28Z","timestamp":1787333308000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/S1052623400375865"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2002,1]]},"references-count":37,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2002,1]]}},"alternative-id":["10.1137\/S1052623400375865"],"URL":"https:\/\/doi.org\/10.1137\/s1052623400375865","relation":{},"ISSN":["1052-6234","1095-7189"],"issn-type":[{"value":"1052-6234","type":"print"},{"value":"1095-7189","type":"electronic"}],"subject":[],"published":{"date-parts":[[2002,1]]}}}