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

SUBJECT

Code
Subject
MO044 Convex Functions
Section
Semester
Hours: C+S+L
Category
Type
Mathematics - in Hungarian
7
2+2+0
optional
Mathematics-Computer Science - in Hungarian
7
2+2+0
optional
Teaching Staff in Charge
Lect. TRIF Tiberiu Vasile, Ph.D.,  ttrifmath.ubbcluj.ro
Prof. KASSAY Gabor, Ph.D.,  kassaymath.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.
References
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. HIRIART-URRUTY J. B., LEMARECHAL C.: Convex Analysis and Minimization Algorithms. Springer-Verlag, 1993.
4. KUCZMA M.: An Introduction to the Theory of Functional Equations and Inequalities. Panstwowe Wydawnictwo Naukowe, Warszawa-Krakow-Katowice, 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.
Links: Syllabus for all subjects
Romanian version for this subject
Rtf format for this subject