syllabus
1. Information regarding the programme
1.1 Higher education institution 	Babeş Bolyai University	
1.2 Faculty	Faculty of Mathematics and Computer Science	
1.3 Department	Department of Computer Science	
1.4 Field of study	Computer Science	
1.5 Study cycle	Master	
1.6 Study programme / Qualification 	High Performance Computing and Big Data Analytics	

2. Information regarding the discipline 
2.1 Name of the discipline	Optimization models	
2.2 Course coordinator 	Assoc. Prof. Nicolae Popovici, Ph.D.	
2.3 Seminar coordinator	Assoc. Prof. Nicolae Popovici, Ph.D.	
2.4. Year of study	1	2.5 Semester	2	2.6. Type of evaluation	Exam	2.7 Type of discipline	Optional	

3. Total estimated time (hours/semester of didactic activities) 
3.1 Hours per week 	3	Of which: 3.2 course	2	3.3 seminar	1	
3.4 Total hours in the curriculum 	42	Of which: 3.5 course	28	3.6 seminar	14	
Time allotment:	hours	
Learning using manual, course support, bibliography, course notes	56	
Additional documentation (in libraries, on electronic platforms, field documentation) 	7	
Preparation for seminars/labs, homework, papers, portfolios and essays	28	
Tutorship	7	
Evaluations	35	
Other activities: ..................	-	
3.7 Total individual study hours 	133	
3.8 Total hours per semester	175	
3.9 Number of ECTS credits	7	

4. Prerequisites (if necessary)
4.1. curriculum	•	Mathematical foundations of decision-making process
•	Mathematical Analysis 
•	Dynamical Systems	
4.2. competencies	Basic knowledge of linear optimization, convex analysis, differential calculus, and dynamical systems 	

5. Conditions (if necessary)
5.1. for the course		
5.2. for the seminar /lab activities		

6. Specific competencies acquired 
Professional competencies	•	Knowledge of basic notions and fundamental results from linear and nonlinear optimization as well as dynamic optimization	
Transversal competencies	•	Ability to model different (theoretical or real-life) problems as optimization problems and to solve them by implementable numerical methods	

7. Objectives of the discipline (outcome of the acquired competencies)
7.1 General objective of the discipline	The aim of this course is to present classical as well as modern optimization models, from both theoretical and practical points of view	
7.2 Specific objective of the discipline	Students should acquire knowledge about:
•	Integer programming
•	Multicriteria optimization
•	Transportation problems.
•	Nonlinear optimization models; numerical methods for solving unconstrained or constrained optimization problems (gradient methods, penalty and barrier functions methods). 
•	Dynamic programming models; applications to economics and network-type problems. 
•	Optimization models via calculus of variations.	

8. Content
8.1 Course	Teaching methods	Remarks	
1.	Classical models of optimization	Direct instruction, mathematical proof, exemplification		
2.	Special instances of linear optimization; integer programming; the Gomory algorithm	Direct instruction, mathematical proof, exemplification		
3.	Multicriteria linear optimization; the weighted-sum scalarization method	Direct instruction, mathematical proof, exemplification		
4.	Bicriteria linear optimization; the parametric scalarization method	Direct instruction, mathematical proof, exemplification		
5.	Transportation problems; statement of the problem and existence of solutions	Direct instruction, mathematical proof, exemplification		
6.	Graphs associated to a transportation problem	Direct instruction, mathematical proof, exemplification		
7.	Numerical solution of transportation problems	Direct instruction, mathematical proof, exemplification		
8.	Nonlinear optimization problems; optimality conditions	Direct instruction, mathematical proof, exemplification		
9.	Gradient methods for solving unconstrained optimization problems 	Direct instruction, mathematical proof, exemplification		
10.	The penalty function method for solving constrained optimization problems	Direct instruction, mathematical proof, exemplification		
11.	The barrier function method for solving constrained optimization problems	Direct instruction, mathematical proof, exemplification		
12.	Dynamic optimization models; the Bellman’s principle of dynamic optimization and applications to economics and network-type problems.	Direct instruction, mathematical proof, exemplification		
13.	Optimization models via calculus of variations: preliminary results concerning integrals depending on parameters;the fundamental Lemma in variational calculus	Direct instruction, mathematical proof, exemplification		
14.	The fundamental problem of the calculus of variations; the Euler equation and some of its special cases; applications.	Direct instruction, mathematical proof, exemplification		
Bibliography
1. BOYD, S., VANDENBERGHE, L.: Convex Optimization, Cambridge University Press, 2004.
2. BRECKNER, W.W.: Cercetare operaţională, Universitatea “Babeş-Bolyai”, Facultatea de Matematică, Cluj-Napoca, 1981.
3. CHIANG, A.C.: Elements of Dynamic Optimization. McGraw-Hill, New York, 1992.
4. EHRGOT, M.: Multicriteria Optimization. Springer, Berlin Heidelberg New York, 2005.
5. MARUŞCIAC, I.: Programare matematică, Universitatea Babeş-Bolyai, Cluj-Napoca, 1975.
6. POPOVICI, N.: Optimizare vectorială, Casa Carţii de Ştiinţă, Cluj-Napoca, 2005.
7. VANDERBEI, R.: Linear Programming. Foundations and Extensions, Springer, New York, 2008.	
8.2 Seminar	Teaching methods	Remarks	
1.	Mathematical modeling of different real-life optimization problems	Direct instruction, case studies, debate		
2.	Integer optimization problems solved by the Gomory algorithm	Problem-based instruction, debate		
3.	Multicriteria optimization problems solved by the weighted-sum scalarization method	Problem-based instruction, debate		
4.	Bicriteria linear optimization problems solved by the parametric scalarization method	Problem-based instruction, debate		
5.	Theoretical exercises concerning transportation problems	Direct instruction, exemplification, mathematical proofs		
6.	Numerical solution of transportation problems	Problem-based instruction, debate, mathematical proofs		
7.	Different practical problems which can be modelled as transportation problems	Problem-based instruction, case studies, debate		
8.	Nonlinear optimization problems solved by using the optimality conditions	Problem-based instruction, debate		
9.	Nonlinear optimization problems solved by the gradient method	Problem-based instruction, debate		
10.	Constrained optimization problems solved by the penalty function method 	Problem-based instruction, debate		
11.	Constrained optimization problems solved by the barrier function method	Problem-based instruction, debate		
12.	Economic problems solved by dynamic optimization, based on Bellman’s principle in continuous case	Problem-based instruction, debate		
13.	Economic problems solved by dynamic optimization, based on Bellman’s principle in discrete case	Problem-based instruction, debate		
14.	Network-type problems solved by dynamic optimization	Problem-based instruction, debate		
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., DUCA, D.: Culegere de probleme de cercetare operationala, Universitatea Babes-Bolyai, Facultatea de Matematica, Cluj-Napoca, 1983.	

9. Corroborating the content of the discipline with the expectations of the epistemic community, professional associations and representative employers within the field of the program

The course ensures a solid theoretical background, according to national and international standards 	

10. Evaluation
Type of activity	10.1 Evaluation criteria	10.2 Evaluation methods	10.3 Share in the grade (%)	
10.4 Course	- Knowledge of theoretical concepts and capacity to rigorously prove the main theorems;
- Ability to solve practical exercises and theoretical problems	Written exam	80%	
10.5 Seminar/lab activities	- Attendance and active class participation	Continuous evaluation 	20%
	
10.6 Minimum performance standards	
The grade [as weighted average (80 * Written exam +20 * Continuous evaluation)/100] should be greater than or equal to 5.	

Date		Signature of course coordinator 		Signature of seminar coordinator
30.04.13	Assoc. Prof. Nicolae Popovici, Ph.D.	Assoc. Prof. Nicolae Popovici, Ph.D.

Date of approval  						Signature of the head of department
...........................................					Prof. Bazil Pârv, Ph.D.