Convex functions 
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Teaching Staff in Charge 

Aims 
Presentation of recent results related to the theory of convex functions taking values in ordered topological linear spaces. The school of mathematics of ClujNapoca has brought outstanding contributions to the development of this field. The course continues the courses of Functional Analysis (1) and (2). 
Content 
he basic notions and results concerning the convex functions are presented. The main topics included are: convex functions of a real variable (side differentiability, continuity, Lipschitz continuity), means and their inequalities, Jensenconvex, logarithmicallyconvex, and multiplicativelyconvex functions, convex functions on topological linear spaces (relationship between continuity, Lipschitz continuity, and local boundedness, directional differentiability, algebraic subdifferentiability and subdifferentiability of convex functions on topological linear spaces, differentiability of convex functions of several real variables), necessary and sufficient optimality conditions in convex programming, Fenchel conjugate and Fenchel biconjugate, Lagrangian duality. 
References 
1. BRECKNER W. W., GOEPFERT A., TRIF T.: Characterizations of ultrabarrelledness and barrelledness involving the singularities of families of convex mappings. Manuscripta Math. 91, 1734 (1996).
2. BRECKNER W. W., ORBAN G.: Continuity properties of rationally sconvex mappings with values in an ordered topological linear space. Universitatea "BabesBolyai", ClujNapoca, 1978. 3. BRECKNER W. W., TRIF T.: On the singularities of certain families of nonlinear mappings. Pure Math. Appl. 6, 121137 (1995). 4. JAHN J.: Mathematical vector optimization in partially ordered linear spaces. Verlag Peter Lang, Frankfurt am Main, 1986. 5. KOSMOL P.: Optimierung und Approximation. W. de Gruyter, Berlin, 1991. 
Assessment 
Exam. 