Universitatea Babeş-Bolyai Cluj-Napoca
Facultatea de Matematică şi Informatică
Ciclul de studii: Masterat
Domeniul: Matematica
Programul de studii: Analiză reală şi complexă (în limba engleză)
Limba de predare: Engleză

SYLLABUS

I. General data
Code	Subject	
MO259	Convex Analysis	

Semester	Hours: C+S+L	Category	Status	
2	2+2+0		compulsory	

II. Full status faculty members
Name and	Scientific	Didactic	Chair	Type of activity	
surname	title	title		C	S	L	
DUCA Dorel	Ph.D.	Prof.	Mathematical Analysis and Optimization	*			

Associated faculty members
Name and	Scientific	Institution	Type of	Type of activity	
surname	title		position	C	S	L	

III. Course objectives
Getting some knowledges in convex analysis, especially those considered to be essential in the education of students at the post-graduate level. 	

IV. Course contents

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

V. Bibliography
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

VI. Thematic of didactic activities per weeks

VII. Didactic methods used

VIII. Assessment
The activity ends with a written final exam (50%). During the semester, the students will have to prepare two reports (25%).  The students&#8217; activity during the semester will be also considered (25%).  
All university official rules with respect to students&#8217; attendance of academic activities, as well as to cheating and plagiarism, are valid and enforced.

IX. Additional references

	Dean,							Chair head,

	Date,							Course responsible,
								Prof. DUCA Dorel, Ph.D.