Babes-Bolyai University of Cluj-Napoca
Faculty of Mathematics and Computer Science
Study Cycle: Graduate
SUBJECT

Code	Subject	
MMA0006	Linear Optimization	

Section	Semester	Hours: C+S+L	Category	Type	
Applied Mathematics	5	2+2+0	speciality	compulsory	

Teaching Staff in Charge	Assoc.Prof. BRECKNER Brigitte Erika, Ph.D., brigitte@math.ubbcluj.ro	
Aims	An introduction in linear optimization in R^n is offered, including its theoretical foundations (properties of polyhedra and consequences of these properties in the case of linear optimization problems), numerical algorithms for solving linear optimization problems, elements of matrix game theory	
Content	1. Introduction in linear optimization; mathematical models; the general optimization problem in R^n; the special case of linear optimization problem in R^n. 
2. Elements of convex analysis in the n-dimensional euclidean space (convex sets; cones; polyhedra; extremal points of convex sets; extremal points and polyhedra faces) 
3. Alternative theorems for linear systems of equations and inequations in R^n 
4. The study of linear optimization problems in R^n (solutions set properties; characterization of solutions; linear optimization problems in canonical form and linear optimization problems in standard form) 
5. Numerical methods for solving linear optimization problems (graphical method for linear optimization problems in R^2; simplex algorithm for linear optimization problems in canonical and standard form; the two phases method for linear optimization problems in standard form) 
6. The duality of linear optimization problems (economy interpretations; duality theorems) 
7. Applications of linear optimization to the study of matrix games 

Bibliography	1) BRECKNER, B.E., De la poliedre la jocuri matriceale. O introducere in optimizarea liniara, EFES, Cluj-Napoca, 2007.
2) BRECKNER, B.E.,  POPOVICI, N., Convexity and Optimization. An Introduction, EFES, Cluj- Napoca, 2006.
3) BRECKNER, B.E., POPOVICI, N., Probleme de cercetare operationala, EFES, Cluj-Napoca, 2006.
4) Breckner W.W., Cercetare operationala, Cluj-Napoca: Universitatea $Babes-Bolyai$, Facultatea de matematica, 1981.
5) Breckner W.W., Duca I. D., Culegere de probleme de cercetare operationala, Cluj-Napoca: Universitatea $Babes-Bolyai$, Facultatea de Matematica si Informatica, 1983.
6) VANDERBEI, R. J., Linear Programming. Foundations and extensions, International Series in Operations Research&Management Science 37, Kluwer Academic Publishers, Boston, 2001.
7) Webster, R., Convexity, Oxford University Press, New York, 1994.	
Assessment	Written and oral examination (the last one is not compulsory). The written exam evaluates the ability to apply the theory in problems solving (both of computational nature, by applying one of the algorithms explained in the course, and theoretical).  At the end of the semestre, each student has to write project about a certain real linear optimization problem occuring in economy. This problem has to be determined by documentation in a company.

The evaluation of this project is 20% of the final grade, the exam is 80%.