Universitatea "Babeş-Bolyai" din Cluj-Napoca

Facultatea de Matematică şi Informatică
FISA DISCIPLINEI

Analiză convexă

Convex analysis

Cod	Semes- trul	Ore: C+S+L	Credite	Tipul	Sectia
MO259	1	2+1+1	8	obligatorie	Analiză Reală şi Complexă (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.

Continut

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.

Bibliografie

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

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