Universitatea Babeş-Bolyai Cluj-Napoca
Facultatea de Matematică şi Informatică
Ciclul de studii: Masterat
Domeniul: Informatica
Programul de studii: Programare bazată pe componente - în limba engleză
Limba de predare: Engleză

SYLLABUS

I. General data
Code	Subject	
MMA1011	Mathematical Foundations of Decision-Making Process	

Semester	Hours: C+S+L+P	Category	Status	
1	2+1+0+1	speciality	compulsory	

II. Full status faculty members
Name and	Scientific	Didactic	Chair	Type of activity	
surname	title	title		C	S	L	
POPOVICI Nicolae	Ph.D.	Assoc.Prof.	Algebra, Mathematical Analysis and Geometry	*	*		

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

III. Course objectives
The aim of the course is to introduce fundamental mathematical concepts and practical methods relevant to the study of decision-making processes as optimization problems.	

IV. Course contents
- Partial ordered sets; linear preorder relations and their characterizations in terms of ordering cones.
- Preference relations induced by utility functions; optimality concepts; decisional processes as scalar or vector optimization problems. 
- Convex sets and polyhedral sets; extreme points.
- Convex functions and the properties of their minimum points. 
- Necessary and sufficient optimality conditions for unconstrained and constrained optimization problems. 
- Saddle point theorems. Duality theorems in linear optimization. 
- Numerical methods for solving optimization problems: the Simplex algorithm in primal and dual form; the cutting planes method.
- Conflict and competition problems, two-player matrix games and their solution by means of linear programming.	

V. Bibliography
1. ANDERSON, D.R., SWEENEY, D.J., WILLIAMS, T.A., An Introduction to Management Science. Quantitative Approaches to Decision Making, South-Western College Publishing, Cincinnati, 2000.
2. BRECKNER, B.E., POPOVICI, N.: Probleme de cercetare operaţională, EFES, Cluj-Napoca, 2006.
3. BRECKNER, B.E., POPOVICI, N.: Convexity and Optimization. An Introduction, EFES, Cluj-Napoca, 2006. 
4. BRECKNER, W.W., DUCA, D.: Culegere de probleme de cercetare operationala,  Universitatea Babes-Bolyai, Facultatea de Matematica, Cluj-Napoca, 1983.
5. EHRGOT, M.: Multicriteria Optimization, Springer, Berlin Heidelberg New York, 2005.
6. POPOVICI, N.: Optimizare vectoriala, Casa Cartii de Stiinta, Cluj-Napoca, 2005.
7. YU, P.L.: Multiple Criteria Decision Making: Concepts, Techniques and Extensions, Plenum Press, New York - London, 1985.	

VI. Thematic of didactic activities per weeks
Lecture 1: Partial ordered sets: binary relations, preorder and order relations, and related concepts of minimality and maximality.
Lecture 2: Partially ordered linear spaces: linear order relations; fundamental examples (the componetwise domination-type relations and the lexicographic order in Rn); geometric characterization.
Lectures 3 and 4: Preference relations induced by utility functions; decisional processes as scalar or vector (multicriteria) optimization problems; optimality concepts (ideal efficiency, Pareto strong efficiency and Slater weak efficiency).
Lecture 5: Level sets and their role in the study of optimal solutions; the existence and the unicity of optimal solutions.
Lecture 6: Convex sets and cones; some special classes of cones; the characterization of convex cones; the relationship between linear order relations and convex cones.
Lecture 7: Convex real-valued functions and their characterizations by means of the epigraph; once or twice differentiable convex functions and their characterization by means of the gradient and hessian matrix, respectively.
Lecture 8: Local versus global minimal points of real-valued convex functions; the convexity of the level sets (the set of optimal solutions as a particular instance); the boundedness of level sets and coercivity of convex functions.
Lecture 9: Necessary and sufficient optimality conditions for unconstrained and constrained smooth optimization problems. Saddle point theorems.
Lecture 10: Cone-convex vector-valued functions and their characterization; the fundamental property of the image of a cone-convex function and its applications to the weighting scalarization of multicriteria optimization problems.
Lecture 11: Mathematical foundations of linear programming; duality theorems.
Lecture 12: Numerical methods for solving optimization problems: the Simplex algorithm and the cutting planes method.
Lectures 13 and 14: Conflict and competition problems, two-player matrix games and their solution by means of linear programming.

Seminar 1: Mathematical modeling of some practical decision problems.
Seminar 2: Exercises involving binary relations and related topics.
Seminars 3 and 4: The study of several elementary theoretical results concerning multicriteria optimization problems.
Seminars 5: The geometric description of the level sets of some important classes of functions; applications to the solution of some particular scalar and bicriteria optimization problems.
Seminar 6: Exercises involving convex sets and cones.
Seminars 7 and 8: The study of the convexity and other regularity properties of certain real-valued functions.
Seminar 9: The solution of certain optimization problems by using necessary and sufficient optimality conditions.
Seminar 10: Exercises involving cone-convex vector-valued functions.
Seminars 11, 12, and 13: The solution of different linear optimization problems by using the Simplex algorithm.
Seminar 14: Matrix games.

Project: Each student will elaborate a project on a topic of Seminars 1-14.  	

VII. Didactic methods used
Interactive lectures, modeling and case study, individual study and discovery.	

VIII. Assessment
Continuous evaluation (contributes 20% to the assesment; consists of attendance and active class participation); individual project (contributes 20% to the assesment); written and oral exam (contributes 60% to the assesment).	

IX. Additional references
1.	BRECKNER, W.W.: Cercetare operaţională, Universitatea “Babes-Bolyai”, Facultatea de Matematică, Cluj-Napoca, 1981.
2.	RUSZCZYNSKI, A.: Nonlinear Optimization, Princeton University Press, Princeton and Oxford, 2006.
3.	VANDERBEI, R.: Linear Programming. Foundations and Extensions, Springer, New York, 2008.	

Date,							Course responsible,
October 18, 2010					Assoc. Prof. Nicolae POPOVICI, Ph.D.

Dean,							Chair Head,
Prof. Leon ŢÂMBULEA, Ph.D.			Assoc. Prof. Simion BREAZ, Ph.D.