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
Motivatio Elements of
Semi-Infinite and Robust Optimization
Gerhard-
Gerhard-Wilhelm Weber *, Başak Akteke-Öztürk
Akteke-
n
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

2. Networks and Optimization
GENE time 0 9.5 11.5 13.5 15.5 18.5 20.5
'YHR007C' 0.224 0.367 0.312 0.014 -0.003 -1.357 -0.811
'YAL051W' 0.002 0.634 0.31 0.441 0.458 -0.136 0.275
'YAL054C' -1.07 -0.51 -0.22 -0.012 -0.215 1.741 4.239
'YAL056W' 0.09 0.884 0.165 0.199 0.034 0.148 0.935
'PRS316' -0.046 0.635 0.194 0.291 0.271 0.488 0.533
'KAN-MX' 0.162 0.159 0.609 0.481 0.447 1.541 1.449
'E. COLI #10' -0.013 0.88 -0.009 0.144 -0.001 0.14 0.192
'E. COLI #33' -0.405 0.853 -0.259 -0.124 -1.181 0.095 0.027
ex.: yeast data

3. 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

4. Generalized Semi-Infinite Optimization
2
C
I, K, L finite

5. Semi-Infinite Optimization
Hubertus Th. Jongen
Semi-Infinite Optimization,
EURO XXIII 2009,
July 5-9, 2009, Bonn, Germany

6. Semi-Infinite Optimization

7. Semi-Infinite Optimization

8. Generalized Semi-Infinite Optimization

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

10. Generalized Semi-Infinite Optimization
Thm. (W. 1999/2003, 2006):
⇔
ξ

11. Generalized Semi-Infinite Optimization
constructions
max-type, nonsmooth functions Morse theory, topology
perturbed
given
given
feasible set
perturbed
nonsmooth GSIP

12. Generalized Semi-Infinite Optimization
B
time-minimal cooling (or heating) of
r R
∀T > 0 ∃!
GSIP

13. Generalized Semi-Infinite Optimization
further ex. : • thermo-regulation of premature infants
• control of global warming
• maximization of time-horizon longest term description
anticipation

14. Generalized Semi-Infinite Optimization
Ex.: approx. of a thermo-couple characteristic
Hoffmann, Reinhard
thermo-couple f (y) : spline of polynomials with deg. 3 – 13, on [a,b]
to be approx. by :
• bounds on error
Bernhard
(= y)
•
some interpol.

15. Generalized Semi-Infinite Optimization
Ex.: approx. of a thermo-couple characteristic
thermo-couple f (y) : spline of polynomials with deg. 3 – 13, on [a,b]
to be approx. by :
• bounds on error
•
some interpol.

16. Generalized Semi-Infinite Optimization
Ex.: approx. of a thermo-couple characteristic
thermo-couple f (y) : spline of polynomials with deg. 3 – 13, on [a,b]
to be approx. by :
• bounds on error
time •
some interpol.

17. Generalized Semi-Infinite Optimization
numerical methods
by discretization
123
4 4 {

18. Generalized Semi-Infinite Optimization
numerical methods
parametrically,
by approximation

19. Generalized Semi-Infinite Optimization
numerical methods
by local linearization &
transversal intersection
3
1
reduction ansatz
exchange method O. Stein, G. Still W.
A. Tezel
semismooth Newton’s method

20. Laurent El Ghaoui
Robust Optimization Robust Optimization
and Applications,
IMA Tutorial, March 11, 2003
.

21. Robust Optimization
LP as a conic problem
.
.
.

22. Robust Optimization
CQP
.
.
.
.

23. Robust Optimization
semidefinite programming (SDP)
.
.
CQP .
.
.

24. Robust Optimization
dual of conic program
.
.
,
.

25. Robust Optimization
robust conic programming
.
.
.
.

26. Robust Optimization
robust conic programming
.
.
semi-infinite
.
.

27. Robust Optimization
polytopic uncertainty
.
.
.

28. Robust Optimization
robust portfolio optimization
.
.
,
$T
r .

29. Robust Optimization
solution of robust portfolio optimization problem
CQP
,
.
.

30. Robust Optimization
robust CQP
.
,
CQP ,
.

31. Robust Optimization
Ex.: robust least-squares
.
.

32. References