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	
MMA1012	Optimization Models	

Semester	Hours: C+S+L+P	Category	Status	
2	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 this course is to present several classical and modern optimization models, from both theoretical and practical points of view.	

IV. Course contents
- Special instances of linear optimization: resource allocation problems, diet problems.
- Integer programming (the Gomory algorithm and the branch and bound method).
- Multicriteria linear optimization with respect to domination or lexicographic ordering.
- Transportation problems.
- Nonlinear optimization models; numerical methods for solving unconstrained or constrained optimization problems (gradient methods, penalty and barrier functions methods). 
- Dynamic programming models: the Bellman’s principle of dynamic programming; applications of its continuous and discrete versions to economics and network-type problems. 
- Optimization models via calculus of variations: the fundamental problem of the calculus of variations; the Euler equation and some of its special cases; transversality conditions; applications.	

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, W.W.: Cercetare operaţională, Universitatea “Babes-Bolyai”, Facultatea de Matematică, Cluj-Napoca, 1981.
4. CHIANG, A.C.: Elements of Dynamic Optimization. McGraw-Hill, New York, 1992.
5. EHRGOT, M.: Multicriteria Optimization. Springer, Berlin Heidelberg New York, 2005.
6. MARUŞCIAC, I.: Programare matematică, Universitatea Babeş-Bolyai, Cluj-Napoca, 1975.
7. POPOVICI, N.: Optimizare vectorială, Casa Cărţii de Stiinţă, Cluj-Napoca, 2005.
8. VANDERBEI, R.: Linear Programming. Foundations and Extensions, Springer, New York, 2008.	

VI. Thematic of didactic activities per weeks
Lecture 1: Special instances of linear optimization problems: resource allocation and diet-type problems including the economic interpretation of the optimization models.
Lecture 2: Multicriteria linear optimization with respect to domination or lexicographic ordering; numerical solution methods based on different scalarization methods and the Simplex algorithm.
Lectures 3 and 4: Integer programming; the Gomory algorithm; the branch and bound method.
Lectures 5 and 6: Transportation problems: theoretical background, numerical solution methods, and applications.
Lecture 7: Gradient methods for solving unconstrained nonlinear optimization problems.
Lecture 8: The penalty function method for solving constrained optimization problems.
Lecture 9: The barrier function method for solving constrained optimization problems.
Lectures 10 and 11: Dynamic programming models: the Bellman’s principle of dynamic programming; applications of its continuous and discrete versions to economics and network-type problems.
Lectures 12, 13, and 14: Optimization models via calculus of variations: preliminary results concerning integrals depending on parameters;the fundamental Lemma in variational calculus; the fundamental problem of the calculus of variations; the Euler equation and some of its special cases; transversality conditions; applications.

Seminar 1: Mathematical modeling of real-life problems as optimization problems.
Seminar 2: Special classes of matrix games arising from certain practical problems.
Seminar 3: Multicriteria linear optimization problems solved by different scalarization methods.
Seminars 4 and 5: Integer optimization problems solved by the Gomory algorithm.
Seminars 6 and 7: Numerical solution of certain transportation problems.
Seminars 8 and 9: Constrained optimization problems and their solution by both penalty function and barrier function methods.
Seminar 10: Discrete dynamic programming models.
Seminar 11: Continuous dynamic programming models.
Seminar 12: An overview of certain network-type optimization models.
Seminars 13 and 14: Optimization models teated by the calculus of variations: the solution of the Euler equation in different cases.

Project: Each student will elaborate a project within one 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); individual project (contributes 20% to the assesment); written and oral exam (contributes 60% to the assesment).	

IX. Additional references
1. DACOROGNA, B.: Introduction au calcul des variations. Presses Polytechniques et Universitaires Romandes, Laussane, 1992.
2. HILLERMEIER, C.: Nonlinear Multiobjective Optimization: A Generalized Homotopy Approach. Birkhäuser Verlag, Basel - Boston - Berlin, 2001.
3. ŞTEFĂNESCU, A., ZIDĂROIU, C.: Cercetări operaţionale, Editura Didactica şi Pedagogică, Bucureşti, 1981.	

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.