Convex functions 
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Teaching Staff in Charge 
 

Aims 
The main notions and results concerning the convex functions are presented. By including such topics as convex functions on topological linear spaces, Fenchel conjugate and biconjugate, necessary and sufficient optimality conditions, a milder introduction to Convex Analysis (a master level course) is ensured for those students that will take this course. 
Content 
The 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. BORWEIN J. M., LEWIS A. S.: Convex Analysis and Nonlinear Optimization. Theory and Examples. CMS Books in Mathematics, SpringerVerlag, 2000.
2. BRECKNER W. W.: Introducere in teoria problemelor de optimizare convexa cu restrictii. Editura Dacia, Cluj, 1974. 3. HIRIARTURRUTY J. B., LEMARECHAL C.: Convex Analysis and Minimization Algorithms. SpringerVerlag, 1993. 4. KUCZMA M.: An Introduction to the Theory of Functional Equations and Inequalities. Panstwowe Wydawnictwo Naukowe, WarszawaKrakowKatowice, 1985. 5. PRECUPANU T.: Spatii liniare topologice si elemente de analiza convexa. Editura Academiei Romane, Bucuresti, 1992. 6. ROBERTS A. W., VARBERG D. E.: Convex Functions. Academic Press, 1973. 7. ROCKAFELLAR R. T.: Convex Analysis. Princeton University Press, 1970. 
Assessment 
Three test papers during the semester. 