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	Department	Type of activity	
surname	title	title		C	S	L	
POPOVICI Nicolae	Ph.D.	Assoc.Prof.	Mathematics	*			

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 operationala, EFES, Cluj-Napoca, 2006.
3. BRECKNER, W.W.: Cercetare operationala, Universitatea “Babes-Bolyai”, Facultatea de Matematica, 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. MARUSCIAC, I.: Programare matematica, Universitatea Babes-Bolyai, Cluj-Napoca, 1975.
7. POPOVICI, N.: Optimizare vectoriala, Casa Cartii de Stiinta, 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

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

	Dean,							Department head,

	Date,							Course responsible,
								Assoc.Prof. POPOVICI Nicolae, Ph.D.