Subject: Convex Analysis	Code: MO259	

Semester	 Hours: C+S+L	Type	 Section	
1	2+1+1	compulsory	Real and Compex Analysis - in English	

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	
Selective 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 during the seminars (20% of the final grade), and an oral exam at the end of the semester (80% of the final grade).