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, affine sets, algebraic properties
2.	Convex sets, convex hull, algebraic and topological properties of convex sets
3.	Separation theorems for convex sets
4.	Cones, convex cones, polyhedral cones, the convex cone hull
5.	The dual (polar) of a set, algebraic and topological properties of the polar of a set
6.	Convex functions, characterizations, algebraic and topological properties of the convex functions
7.	The dual representation of the convex functions
8.	Generalizations of convex functions 
9.	Theorems of the alternative 
10.	Minimax theorems, saddle points
11.	Optimization problems, necessary and sufficient conditions
12.	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
Course: 
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

Seminar:
1. Linear subspaces. Affine sets.
2 Convex sets. Convex hull
3 Separation theorems for convex sets
4 Cones, convex cones, polyhedral cones
5 The dual (polar) of a set
6 Convex functions
7 Reports on a given subject: Convex functions
8 Reports on a given subject: Convex functions
9 Reports on a given subject: Generalized convex functions
10 Reports on a given subject: Theorems of the alternative
11 Reports on a given subject: Minimax theorems, saddle-points
12 Reports on a given subject: Optimization problems
13 Reports on a given subject: Optimization Problems
14 Reports on a given subject: Duality

VII. Didactic methods used
Expositions: description, explanation, class lectures, dialog-based lectures, lectures with demonstrations, introductive lectures, current lectures, synthesis lectures. 
Conversations: dialog, introductive conversations, conversations for knowledge consolidation, conversations to systematize and synthesize knowledge 
Use of problems: use of problem questions 
Discovery: directed and independent rediscovery, creative discovery, discovery by documenting 
Other methods: individual study, exercise, homework study.	

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’ 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.	

IX. Additional references
1.	B.E. Breckner si N. Popovici: Convexity and Optimization an Introduction, Editura Fundatiei pentru Studii Europene, Cluj-Napoca, 2006
2.	W.W. Breckner si D.I. Duca: Culegere de probleme de cercetare operationala, Univ. Babes-Bolyai,  Cluj-Napoca, 1983
3.	P. –J. Laurent: Approximation et Optimization, Hermann, Paris, 1972
  4. A. Ruszczynski: Nonlinear Optimization, Princeton University Press, Princeton and Oxford, 2006	

	Dean,							Chair head,
  Prof. Leon TAMBULEA, PhD                                   Prof. Stefan COBZAS, PhD

	Date,							Course responsible,
   Sept 15 2008					        Prof. DUCA Dorel, Ph.D.