Universitatea Babeş-Bolyai Cluj-Napoca
Facultatea de Matematică şi Informatică
Ciclul de studii: Masterat
Domeniul: Matematica
Programul de studii: Analiză reală şi complexă (în limba engleză)
Limba de predare: Engleză

SYLLABUS

I. General data
Code	Subject	
MO260	Numerical Methods in Optimization	

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

II. Full status faculty members
Name and	Scientific	Didactic	Chair	Type of activity	
surname	title	title		C	S	L	
LUPSA Liana	Ph.D.	Prof.	Mathematical Analysis and Optimization	*			

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

III. Course objectives
Getting to know some significant numerical methods for solving the optimization problems.	

IV. Course contents
1. Numerical methods to minimize the unimodal functions. 
2. Numerical methods to minimize the unconstrained function: decreasing methods, conjugated directions methods, relaxation methods, methods whithout the hypothesis of differentiability.
3. Numerical methods with feasible directions,
4 .Numerical methods based on reducing constrained problems to unconstrained ones
5. Cutting methods, 
6. Inner point methods 
7. Branch and bound methods.
8. Specific methods to solve fractional, hyperbolic and quadratic programming problems are studied, too. 

V. Bibliography
1) Andrei N.: Programarea matematică avansată. Teorie, metode computaţionale, aplicaţii. Bucureşti, Editura Tehnică, 1999.
2) Blaga L., Lupşa L.: Cercetare operaţională. Cluj-Napoca, Editura Argonaut, 2006.
3) Breckner W.W.: Cercetare operaţională. Cluj-Napoca, Universitatea Babeş-Bolzai, 1981
4) Ehrgot M.: Multicriteria Optimization. Berlin Heidelberg New York: Springer, 2005. 
5) Forgo F.: Nonconvex Programming, Budapest: Akademiai Kiado, 1988.
6), Jongen H. Th., Meer K., Triesch E.: Optimization Theory. Kluwer Academic Publishers, Boston, 2004.
7) Lupşa L.: Numerical optimization methods. Special issues in discrete optimization. Cluj-Napoca: Risoprint, 2005.
9) Snyman J. A.: Practical Mathematical Optimization. An Introduction to Basic Optimization Theory and Classical and New Gradient-Based Algorithms. Springer, 2005.
10) Wenzu Sun, Za-Xiang Zuan: Optimization Theory and Methods. Nonlinear Programming. Springer, 2006

VI. Thematic of didactic activities per weeks
Week 1.  Introduction: some about history of operation researche and of numerical methods for solving optimization problems.[2]
Week 2. Numerical methods to minimize the unimodal functions: the method of successive section, Fibonacci’s algorithm, algorithm of gold ratio, bisection algorithm, the method of equidistant cuts.[7]
Week 3. Numerical methods to minimize the unimodal functions: Newton’s method, Powell’s method.[1]
Week 4. Heuristic algorithms for finding a minimum point without computing the derivatives: Rosenbrock’s algorithm.[1]
Week 4. Descending methods and their convergence.[3]
Week 5. Methods of conjugate directions: Fletcher-Reeves’s methods,  Polak-Ribière’s methods, Daniel’s method.[3] and [9]
Week 6. Numerical methods based on reduction of constrained problems to unconstrained ones (penalties methods): exterior penalty methods, barrier methods.[3] and [10]
Week 7. Cutting methods: direct cut methods and indirect cut methods. The cutting plane method.[2], [5]
Week 8. Application of cutting methods:to solve integer programming problem. (Gomory's Discreet and Continuous Algorithms)[7]
Week 9. Branch and Bound Methods: the principles of branch and bound methods; application to solve integer programming problems.[7]
Week 10 Little’s algorithm for solving travel salesman problems;
Week 11. Dynamic programming [3]
Week 12. Specific methods for solving fractional and hyperbolic programming problems.[10]
Week 13. Specific methods for solving quadratic programming problems.[10]
Week 14. Some about Numerical Methods of Global Optimization. [9]

Schedule of seminars

Week 1: Administration of seminar activities
Week 2. Applications of Fibonacci’s algorithm and of the bisection algorithm.
Week 3. Applications of Newton’s method and of Powel’s method.
Week 4. Applications of some heuristic algorithms for finding a minimum point without computing the derivatives.
Week 4. Applications of gradient methods. .
Week 5. Applications of Hestenes-Stiefel’s algorithm.
Week 6. Applications of exterior penalty methods.
Week 7. Applications of cutting plane method.
Week 8. Applications of Gomory's continuous algorithms.
Week 9. Some applications of branch and bound methods to solve an integer programming problems.
Week 10 Applications of to Little’s.
Week 11. Applications of Belmann’s theorem.
Week 12. Applications of cuts methods to solve fractional programming problems.
Week 13. Applications of Beale’s algorithm to solve quadratic programming problems.
Week 14. Applications of  trust region method.

VII. Didactic methods used
Expositions: description, explanation, class lectures, dialog-based lectures, lectures with demonstrations, introductive lectures, current lectures, synthesis lectures. 
Conversations: debate, dialog, introductive conversations, conversations for knowledge consolidation, conversations to systematize and synthesize knowledge 
Use of problems: use of problem questions, problems and problem situations. 
Discovery: directed and independent rediscovery, creative discovery, inductive discovery, deductive discovery, discovery by documenting, experimental discovery. 
Other methods: case study; cooperation, simulation, brainstorming, individual study, exercise, homework study	

VIII. Assessment
The activity ends with a final exam (grade E). During the semester, the students will have to prepare a theoretical report (R1) and a program to solve a specific task (P) or two theoretical reports (grades R1, R2) The students’ activity during the semester will be also considered (grade A).  The final grade is the weighted mean of the five grades mentioned above. The final grade = 30%E + 20%R1 + 20%R2 or P + 20%P + 10%A. Successful passing of the exam is conditioned by the final grade that has to be at least 5.

IX. Additional references

	Date,							Course responsible,
     September	1, 2008				      Prof. LUPSA Liana, Ph.D.

   Dean,					 	   Chair head,
  Prof. TAMBULEA Leon, Ph.D.			Prof. COBZAS Stefan, Ph.D.