Analiză convexă | Convex analysis |
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(Real and Compex Analysis) |
Cadre didactice indrumatoare | Teaching Staff in Charge |
Conf. Dr. KASSAY Gabor, kassay@math.ubbcluj.ro |
Obiective | Aims |
Predarea elementelor de baza ale analizei convexe care sunt esentiale in formarea studentilor de la studii aprofundate, care se specializeaza in teoria optimizarii. |
Getting some knowledges in convex analysis, especially those considered to be essential in the education of students at the post-graduate level. |
1. Multimi studiate in analiza convexa.
Proprieti algebrice ale multimilor: subspatii liniare, multimi afine, multimi convexe, semispatii, multimi regulat convexe, conuri. Invelitoare liniara, afina, convexa, conica. Proprietati. Proprietati topologice ale multimilor: aderenta, interior, interior relativ. Proprietati. Separarea multimilor convexe prin hiperplane. Reprezentarea duala a multimilor convexe. Teorema bipolarei. 2. Functii studiate in analiza convexa. Proprietati algebrice ale functiilor: epigraf, epigraf strict, functii convexe, functii cvaziconvexe. Invelitoare convexa, invelitoare cvaziconvexa. Proprietati topologice ale functiilor: inferior/superior semicontinuitate. Inferior semicontinuitatea functiilor convexe. Reprezentarea duala a functiilor convexe. Teoremele lui Minkowski si Fenchel-Moreau. Reprezentarea duala a functiilor cvaziconvexe. 3. Aplicatii: teoreme de minimax si teoria jocurilor, dualitatea in optimizare. |
1. J.P.Aubin: Optima and Equilibria: An Introduction to Nonliniar Analysis, Springer-Verlag, Berlin Heidelberg, 1993
2. J.P.Aubin, I.Ekeland: Applied Nonliniar Analysis, John Wiley and Sohns, 1984 3. V. Barbu, T.Precupanu: Convexity and Optimization in Banach Spaces, Publ.House of Roum. Acad. and Reidel Publishing Comp.,1986 4. L.Danzer, B. Grunbaum, V.Klee: Helly's Theorem and its Relatives, Convexity, Proceedings of Symposia in Pure Mathematics, vol VII, A.M.S.,Providence, Rhode Island, 1963 5. J.-B.Hiriart-Urruty, C. Lemarechal: Convex Analysis and Minimization Algoritms, I,II,Springer-Verlag, Berlin Heidelberg, 1993 6. R. Holmes: Geometric Functional Analysis and its Applicatons, Springer Verlag, Berlin, 1975 7. J. Kolumban: Convex Analysis , I, Babes-Bolyai University Cluj-Napoca, 1997 8. T. Precupanu: Spatii liniare topologice si elemente de analiza convexa, Ed. Acad. Romane, 1992 8. R.T.Rockafellar: Convex Analysis, Princepton Univ.Press,1970 |
Evaluare | Assessment |
Examen |
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