The course should provide an overview on a wide range of methods for various optimization problems, such as unconstrained and constrained optimization problems as well as global optimization problems. The students should learn about the essential theoretical results and { equally important { also about numerical algorithms for solving optimization problems. Solving exercise problems will be an integral part of the course. It is recommended that the course will be modi¯ed in the future in order to focus more on topics which are represented as research topics at the partner universities.
The syllabi of the course is created by
Franz Kappel
Institute for Mathematics and Scientific Computing
University of Graz
Petar Kenderov
Institute of Mathematics and Informatics
Bulgarian Academy of Sciences
Lecturers of the course
Franz Kappel
Institute for Mathematics and Scientific Computing
University of Graz
Mikhail Ivanov Krastanov
Bulgarian Academy of Sciences
Gregory D. von Winckel
University of Graz
Prerequisites on the students side
Participating students need to have programming skills and a sound knowledge of numerical mathematics (in particular numerical linear algebra). Fundamentals of functional analysis are also required. Of course, English language proficiency is an absolute necessity.
Since the group of students who attend the course is large, and the number of computers in the computing lab BIM is limited, it is desirable that students bring their own laptops. It is also desirable to have MatLab software installed.
Modules of the course
PART I: Constraint and Unconstraint Optimization (20 units)
| Module | No. of units | Contents |
| I: Unconstrained optimiza-tion | 10 | FundamentalsLine search methods Trust-region methods Conjugate gradient methods Quasi-Newton methods BFGS-methods Calculus of variations and optimal control |
| II: Constraint optimization | 10 | Optimality conditions Linear programming, interior point methods Quadratic programming SQP methods PDE-constraint optimization |
PART II: Optimal Control and Stochastic Optimization (40 units)
| Module | No. of units | Contents |
| III: Dynamic programming | 9 | The Hamilton-Jacobi-Bellman equation Linear-quadratic control problems |
| IV: Stochastic optimizatio | 11 | Implicit ltering Direct search algorithm |
| V: Global optimizatio | 20 | Branch and bound methods Cutting plane methodsInterval methods Simulated annealing Clustering methods Genetic agorithms |
| Total no. of units: | 60 |
Literature
[1] J. E. Dennis and R. B. Schnabel, Numerical Methods for Unconstrained Optimization,SIAM, Philadelphia 1996.
[2] P. Dorato, C. Abdallah, and V. Cerone, Linear-Quadratic Control, Prentics Hall, EnglewoodCli®s, N. J., 1995.
[3] C. Geiger and C. Kanzow, Numerische Verfahren zur LÄosung unrestringierter Optimierungs-aufgaben, Springer-Verlag, Berlin 1999.
[4] C. Geiger and C. Kanzow, Theorie und Numerik restringierter Optimierungsaufgaben,Springer-Verlag, Berlin 2002.
[5] D. E. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning,Addison-Wesley, New York 1989.
[6] E. Hansen, Global Optimization Using Interval Analysis, Pure and Applied MathematicsVol. 165, Marcel Dekker, New York 1992.
[7] C. Hillermeier, Nonlinear Multiobjective Optimization: A Generalized Homotopy Approach,BirkhÄauser Verlag, Basel 2001.
[8] R. Horst and H. Tuy, Global Optimization: Deterministic Approaches, Springer-Verlag,Berlin 1990.
[9] F. Jarre and J. Stoer, Optimierung, Springer-Verlag, Berlin 2004.
[10] C. T. Kelley, Iterative Methods for Optimization, Frontiers in Applied Mathematics Vol. 18,SIAM, Philadelphia 1999.
[11] D. G. Luenberger, Linear and Nonlinear Programming, Addison-Wesley, Reading (USA)1984.
[12] K. Mietinen, Nonlinear Multiobjective Optimization, Kluwer, Dordrecht 1999.
[13] J. Nocedal and S. Wright, Numerical Optimization, 2nd ed., Springer-Verlag, NewYork2006.
[14] R. E. Steuer, Multiple Criteria Optimization: Theory, Computations and Applications,John Wiley & Sons, New York 1986.
[15] B. Rustem, Algorithms for Nonlinear Programming and Multiobjective Design, John Wiley& Sons, Chichester 1998.
[16] A. A. TÄorn and A. Zilinskas, Global Optimization, Lecture Notes in Computer Science Vol.·350, Springer-Verlag, Berlin 1989.
[17] A. A. Zhigljavsky, Theory of Global Random Search, Mathematics and Its ApplicationsVol. 65, Kluwer, Dordrecht 1991.
Teaching
The course should be accompanied by homework exercises which should require at most 2 of theafternoon sessions as indicated below. The major part of the afternoon session should be spentby working independently in teams on little projects on practical or pseudo-practical problems.The results also should be presented in the afternoon sessions. During the afternoon session theteacher should be available for questions respectively be present in order to get an impression onperformance of the students. Homework exercises and projects for teamwork should also involveprogramming of algorithms respectively use of available software.
The course is planned for 4 weeks, each week from Monday till Friday. This implies that therewill be 3 teaching units ( 45 minutes) per day.
The following schedule is proposed for each day:
- 8:00 till 11:00: three units with breaks in between;
- 11:30 till 12:30: discussion with the teacher;
- 15:00 till 17:30: work on homework exercises,
work in teams on problems posed by the lecturer,
presentation of results, respectively
Grading
The basis for grading is provided by the performance of students in the following items:
a) Exercises for homework involving also numerical computations will be regularly given inorder to provide possibilities for a better understanding of the material presented in the course.
b) Team projects and presentation of results.
c) An oral examination concerning the course.
The oral examination could consist of several parts taken at different times and should give thelecturer an impression on how well the student has understood the material of the course.
In order to obtain the grade for the course the following weights will be used for the items a),b) and c) from above:
- Homework exercises 20%
- Team projects 50%
- Oral examination 30%
Course Brochure
