Babes-Bolyai University of Cluj-Napoca
Faculty of Mathematics and Computer Science
Study Cycle: Master
SUBJECT
| MMA1003 | Convex Analysis |
| Teaching Staff in Charge |
Prof. DUCA Dorel, Ph.D., dduca math.ubbcluj.ro |
| Aims |
|
Getting some knowledges in convex analysis, especially those considered to be essential in the education of students at the post-graduate level. |
| Content |
|
1. Linear subspaces, algebraic properties, linear hull
2. Affine sets, algebraic properties, affine hull 3. Convex sets, convex hull, algebraic and topological properties of convex sets 4. Separation theorems for convex sets 5. Cones, convex cones, polyhedral cones, the convex cone hull 6. The dual (polar) of a set, algebraic and topological properties of the polar of a set 7. Convex functions, characterizations of the convex functions 8. Algebraic and topological properties of the convex functions 9. The dual representation of the convex functions 10. Generalizations of convex functions 11. Theorems of the alternative 12. Minimax theorems, saddle points 13. Optimization problems, necessary and sufficient conditions 14. Duality theory in optimization |
| References |
|
1. J.-P. AUBIN: Optima and Equilibria. An Introduction to Nonlinear Analysis, Springer-Verlag, Berlin/Heidelberg 1993
2. J.-P. AUBIN and I. EKELAND: Applied Nonlinear Analysis, John Wiley and Sons, New York, 1984 3. V. BARBU and T. PRECUPANU: Convexity and Optimization in Banach Spaces, Publ. House of Roum. Acad. and Reidel Publishing Comp., Bucureşti, 1986 4. D.I. DUCA: Multicriteria Optimization in Complex Space, Casa Cartii de Stiinta, Cluj-Napoca, 2005 5. J.-B. HIRIART-URRUTY and C. LEMARÉCHAL: Convex Analysis and Minimization Algorithms, I, II, Springer-Verlag, Berlin - Heidelberg - New York, 1993 6. J. KOLUMBÁN: Convex Analysis I, Babes-Bolyai University, Cluj-Napoca, 1997 7. T. PRECUPANU: Spatii liniare topologice si elemente de analiza convexa, Editura Academiei Române, Bucuresti, 1992 8. T.R. ROCKAFELLAR: Convex Analysis, Princeton University Press, Princeton, 1970 |
| Assessment |
|
The activity ends with a written final exam (50%). During the semester, the students will have to prepare two reports (25%). The students’ activity during the semester will be also considered (25%).
All university official rules with respect to students’ attendance of academic activities, as well as to cheating and plagiarism, are valid and enforced. |
| Links: | Syllabus for all subjects Romanian version for this subject Rtf format for this subject |