Semi-Infinite and Robust Optimization
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The document discusses the achievements and applications presented during the 4th International Summer School on contemporary informatics, mathematics, and physics held at the National University of Technology in Ukraine. It focuses on topics related to semi-infinite and robust optimization, including their mathematical frameworks and various applications such as in biomedical engineering and environmental controls. Additionally, it outlines numerical methods and case studies within the field of optimization.
![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.](https://image.slidesharecdn.com/kievsummerschool2009-semi-infiniteandrobustoptimization-willi-090901160150-phpapp01/85/Semi-Infinite-and-Robust-Optimization-14-320.jpg)
![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.](https://image.slidesharecdn.com/kievsummerschool2009-semi-infiniteandrobustoptimization-willi-090901160150-phpapp01/85/Semi-Infinite-and-Robust-Optimization-15-320.jpg)
![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.](https://image.slidesharecdn.com/kievsummerschool2009-semi-infiniteandrobustoptimization-willi-090901160150-phpapp01/85/Semi-Infinite-and-Robust-Optimization-16-320.jpg)
Semi-Infinite and Robust Optimization
- 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
- 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