New insights into conjugate duality [Elektronische Ressource] / vorgelegt von Sorin-Mihai Grad

New insights into conjugate dualityVon der Fakult at fur Mathematik der Technischen Universit at ChemnitzgenehmigteD i s s e r t a t i o nzur Erlangung des akademischen GradesDoctor rerum naturalium(Dr. rer. nat.)vorgelegtvonDipl. - Math. Sorin - Mihai Gradgeboren am 02.03.1979 in Satu Mare (Rum anien)eingereicht am 26.01.06Gutachter: Prof. Dr. rer. nat. habil. Gert WankaProf. Dr. Juan Enrique Mart nez - LegazProf. Dr. rer. nat. habil. Winfried SchirotzekTag der Verteidigung: 13.07.2006Bibliographical descriptionSorin - Mihai GradNew insights into conjugate dualityDissertation, 116 pages, Chemnitz University of Technology, Faculty of Mathe-matics, 2006ReportWith this thesis we bring some new results and improve some existing ones inconjugate duality and some of the areas it is applied in.First we recall the way Lagrange, Fenchel and Fenchel - Lagrange dual problemsto a given primal optimization problem can be obtained via perturbations and wepresent some connections between them. For the Fenchel - Lagrange dual problemwe prove strong duality under more general conditions than known so far, whilefor the Fenchel duality we show that the convexity assumptions on the functionsinvolved can be weakened without altering the conclusion. In order to prove thelatter we prove also that some formulae concerning conjugate functions given so faronly for convex functions hold also for almost convex, respectively nearly convexfunctions.