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

Code	Subject	
MO057	Optimization	

Section	Semester	Hours: C+S+L	Category	Type	
Applied Mathematics	7	2+2+0		compulsory	

Teaching Staff in Charge	Prof. BRECKNER Wolfgang, Ph.D., breckner@math.ubbcluj.ro	
Aims	The main goal is to improve the students&#8217; knowledge of finding the optimal points of real-valued functions, a knowledge acquired within the courses in Mathematical Analysis (1 and 2). For this end there are described diverse numerical methods for solving certain important types of optimization problems in the Euclidean space R^n. In addition the role played by the optimization theory in solving many real-world problems is revealed.	
Content	1. Preliminaries (4 hours course + 2 hours seminar). The Euclidean space R^n. Classes of special functions defined on the space R^n. Optimal points and locally optimal points. Constrained optimization problems.

2. Methods for solving linear constrained optimization problems (8 hours course + 12  hours seminar). Primally feasible bases and dualy feasible bases. The primal simplex method. The dual simplex method. Solving the dual of a linear constrained optimization problem in the standard form. Solving a linear constrained optimization problem in the standard form obtained by adding an equation containing an additional nonnegative variable. Linear optimization problems in integers.

3. Elements of game theory (6 hours course + 4 hours seminar). The mathematical concept of game. Matrix games. Optimum strategies for the players. Pure strategies. The domination principle. Solving a strictly determined matrix game. Solving a matrix game by means of linear constrained optimization problems.

4. Methods for solving nonlinear constrained optimization problems (6 hours course + 6 hours seminar). The cutting plane method. The penalty function method. The barrier function method.

5. Methods for finding the optimal points of a differentiable real-valued function on R^n (4 hours course + 4 hours seminar). Descent methods. Methods of conjugate directions. 	
Bibliography	1. ALT W.: Nichtlineare Optimierung. Vieweg, Braunschweig &#8211; Wiesbaden, 2002
2. BRECKNER B. E., POPOVICI N.: Probleme de cercetare operaţională. Editura Fundaţiei pentru Studii Europene, Cluj-Napoca, 2006
3. BRECKNER W. W.: Cercetare operaţională. Universitatea Babeş-Bolyai, Facultatea de Matematică, Cluj-Napoca, 1981
4. BRECKNER W. W., DUCA D.: Culegere de probleme de cercetare operaţională. Universitatea Babeş-Bolyai, Facultatea de Matematică, Cluj-Napoca, 1983
5. DOMSCHKE W., DREXL A.: Einführung in Operations Research. 6. Aufl. Springer, Berlin, 2005
6. DOMSCHKE W., DREXL A., KLEIN R., SCHOLL A., VOSS S.: Übungen und Fallbeispiele zum Operations Research. 5. Aufl. Springer, Berlin, 2005
7. MARTI K., GRÖGER D.: Einführung in die lineare und nichtlineare Optimierung. Physica-Verlag, Heidelberg, 2000	
Assessment	A test paper during the semester (20%) + Exam (80%).