AACIMP 2009 Summer School lecture by Gerhard Wilhelm Weber. "Modern Operational Research and Its Mathematical Methods" course.

1. 4th International Summer School
Achievements and Applications of Contemporary
Informatics, Mathematics and Physics
National University of Technology of the Ukraine
Kiev, Ukraine, August 5-16, 2009
Further Advanced Methods from
Mathematical Optimization
Gerhard-Wilhelm Weber * and Başak Akteke-Öztürk
Gerhard- Akteke-
Institute of Applied Mathematics
Middle East Technical University, Ankara, Turkey
* Faculty of Economics, Management and Law, University of Siegen, Germany
Center for Research on Optimization and Control, University of Aveiro, Portugal
EURO CBBM
EURO EURO ORD
EURO CE*OC

2. 4th International Summer School
Achievements and Applications of Contemporary
Informatics, Mathematics and Physics
National University of Technology of the Ukraine
Kiev, Ukraine, August 5-16, 2009
Motivatio On Foundations of Continuous Optimization
Gerhard-
Gerhard-Wilhelm Weber *, Başak Akteke-Öztürk
Akteke-
Institute of Applied Mathematics
n Programs of Financial Mathematics, Actuarial Sciences and Scientific Computing
Department of Biomedical Engineering
Middle East Technical University, Ankara, Turkey
* Faculty of Economics, Management and Law, University of Siegen, Germany
Center for Research on Optimization and Control, University of Aveiro, Portugal

3. Mathematics
• differential topology, singularity theory, Morse theory, algebraic topology
• discrete math., discrete optimization, coding theory and cryptography,
complexity theory, theoretical informatics, logic
• probability theory, statistics, stochastic calculus,
financial math., actuarial sciences
• data mining and computational statistics
• calculus, nonsmooth analysis, theory of functions
• numerical mathematics
• dynamical systems, hybrid systems

4. Mathematics
• differential topology, singularity theory, Morse theory, algebraic topology
optimization
• discrete math., discrete optimization, coding theory and cryptography,
complexity theory, theoretical informatics, logic opt.
• probability theory, statistics, stochastic calculus,
financial math., actuarial sciences opt.
• data mining and computational statistics opt.
• calculus, nonsmooth analysis, theory of functions opt.
• numerical mathematics opt.
• dynamical systems, hybrid systems opt.

5. Optimization
• linear optimization, conic opt., robust (& stochastic) opt.
• convex opt.
• nonsmooth opt., global opt.
• discontinuous opt. (in preparation)
• nonlinear opt., semi-infinite opt., infinite programming
• calculus of variations, optimal control, opt. of PDEs
• variational inequalities
• discrete optimization, hybrid opt.

6. Networks and Optimization

7. Networks and Optimization
GSIP relaxation
2
l ∗ −1 )
min ∑
α =0
∗ &
M Eκα + C E κα + D∗ − Eκα
∗
(mij ∗ ), (cil∗ ), (d i ∗ ) ∞
subject to
n
∑
i =1
p ij ( m ij ∗ , y ) ≤ α j ( y ) ( j = 1, ..., n )
n
∑ q il ( c il ∗ , y ) ≤ β l ( y ) ( l = 1, ..., m ) ( y ∈ Y (C ∗ , D∗ ))
i =1
n
∑
i =1
ζ i ( d i∗ , y ) ≤ γ ( y ) set of combined environmental effects
m ii ≥ δ i , m in ( i = 1, . . . , n ) Y (C ∗ , D∗ ) :=
& o v e r a ll b o x c o n s t r a in t s ( ∏
i =1,..., n
0, ci∗l  ) × (
  ∏
i =1,..., n
0, d i∗  )
 
l =1,..., m

8. Optimization
semi-infinite
I, K, L finite

9. Optimization
generalized semi-infinite
I, K, L finite

10. Generalized Semi-Infinite Optimization
ψ (τ )
τ
ψ
ϕ (⋅,τ )
homeom.
structurally stable asymptotic
effect
ε (⋅)
IR n
global local global

11. Clustering
clustering problem (for a finite set A)
nonsmooth problem:
.
2

12. Clustering and Classification
2
max-min separability incremental approach

13. Incremental Approach to Classification
2
Adil Bagirov

14. Separable Functions
desirability functions (quality opt.)
max-, min-type
continuous selections
normal forms smoothening
stability
instability
bilevel
problems

15. Separable Functions
desirability functions (quality opt.)
max-, min-type
continuous selections
normal forms
stability
instability modified
subgradient
algorithm

16. Examples: International
• EURO
EURO WG on Complex Societal Problems
EURO WG “OR in Comput. Biology, Bioinformatics & Medicine”
EURO WG “OR for Development” etc. 29 EURO WGs
• EUROPT 9 Years
EUROPT
EUROPT
Rhodes Ankara
Reykjavik Prague
Neringa Bonn
• SIAM Kiev Lisbon
SIAG/OPT The Pacific Optimization
Optimization Society
Research Activity GrouP

17. Examples: Pacific Region
• Infrastructure planning is very important for many developing
countries such as China, Korea and Vietnam in Asian Pacific area.
Such planning and design were usually done by city designers,
industrial people, or politicians. Optimization problems arise naturally
in this kind of planning and may play an important role in saving costs
and improving performance.
• Traffic control and network optimization. Traffic has long been
a real problem for big cities in China and other Asian countries.
How to apply the network optimization and traffic control theory
to improve the traffic is a challenge for optimization community.
• High quality optimization journals of POP.
Xiaoling Sun (Fudan University, Shanghai)

18. Pacific Region
• Journal of Industrial and Management Optimization
• Pacific Journal of Optimization
• Dynamics of Continuous, Discrete and Impulsive Systems (Series B)
• Global Journal of Technology and Optimization
(in preparation)
and , , etc.

19. 4th International Summer School
Achievements and Applications of Contemporary
Informatics, Mathematics and Physics
National University of Technology of the Ukraine
Kiev, Ukraine, August 5-16, 2009
Motivatio On Foundations of Continuous Optimization
Parametric Unconstrained Optimization
and Related Topics
n
Gerhard-
Gerhard-Wilhelm Weber *, Başak Akteke-Öztürk
Akteke-
Institute of Applied Mathematics
Programs of Financial Mathematics, Actuarial Sciences and Scientific Computing
Department of Biomedical Engineering
Middle East Technical University, Ankara, Turkey
* Faculty of Economics, Management and Law, University of Siegen, Germany
Center for Research on Optimization and Control, University of Aveiro, Portugal