Babes-Bolyai University of Cluj-Napoca
Faculty of Mathematics and Computer Science
Study Cycle: Graduate
SUBJECT

Code	Subject	
MMA0012	Convex Functions	

Section	Semester	Hours: C+S+L	Category	Type	
Mathematics - in Romanian	6	2+1+0	speciality	optional	
Applied Mathematics	6	2+1+0	speciality	optional	

Teaching Staff in Charge	Assoc.Prof. TRIF Tiberiu Vasile, Ph.D., ttrif@math.ubbcluj.ro	
Aims	The main notions and results concerning the convex functions are presented. By including such topics as convex functions on normed linear spaces, Fenchel's conjugate and biconjugate, necessary and sufficient optimality conditions, a gentle introduction to Convex Analysis (a master level course) is ensured for those students that will take this course.	
Content	1. Convex functions of a real variable
The modern definition of convexity. Characterizations of convex functions of a real variable. Regularity properties of convex functions of a real variable: side differentiability, continuity, Lipschitz-continuity. Means and their inequalities: weighted quasiarithmetic means and their comparison, Rado-Popoviciu-type inequalities. Majorization theorem of Hardy-Littlewood-Polya, Popoviciu's inequality and Petrovic's inequality. Jensen-convex functions, logarithmically-convex functions and multiplicatively-convex functions.
2. Convex functions on normed linear spaces
Examples of convex functions on linear spaces: indicatrice functions, sublinear functions, support functions, affine functions, quadratic forms. Characterizations of convex functions. Continuity of convex functions on normed linear spaces: relationship between continuity, Lipschitz-continuity and local boundedness, continuity of convex functions on finite dimensional normed linear spaces. Directional differentiability and algebraic subdifferentiability of convex functions on linear spaces. Subdifferentiability of convex functions on normed linear spaces. Differentiability of convex functions of several variables.
3. Convex optimization
Necessary and sufficient optimality conditions in convex optimization. Fenchel's conjugate and Fenchel's duality theorem. Fenchel's biconjugate. Closed convex functions, equality between a closed convex function and its biconjugate. Lagrangian duality.	
Bibliography	1. BORWEIN J. M., LEWIS A. S.: Convex Analysis and Nonlinear Optimization. Theory and Examples. CMS Books in Mathematics, Springer-Verlag, 2000.
2. BRECKNER W. W.: Introducere in teoria problemelor de optimizare convexa cu restrictii. Editura Dacia, Cluj, 1974.
3. BRECKNER W. W., TRIF T.: Convex Functions and Related Functional Equations. Selected Topics. Cluj University Press, Cluj-Napoca, 2008.
4. HIRIART-URRUTY J. B., LEMARECHAL C.: Convex Analysis and Minimization Algorithms. Springer-Verlag, 1993.
5. KUCZMA M.: An Introduction to the Theory of Functional Equations and Inequalities. Panstwowe Wydawnictwo Naukowe, Warszawa-Krakow-Katowice, 1985.
6. NICULESCU C. P., PERSSON L.-E.: Convex Functions and Their Applications. A Contemporary Approach. Springer-Verlag, New York, 2006.
7. PRECUPANU T.: Spatii liniare topologice si elemente de analiza convexa. Editura Academiei Romane, Bucuresti, 1992.
8. ROBERTS A. W., VARBERG D. E.: Convex Functions. Academic Press, 1973.
9. ROCKAFELLAR R. T.: Convex Analysis. Princeton University Press, 1970.	
Assessment	Three test papers during the semester.