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# Semi-Infinite and Robust Optimization

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

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### Semi-Infinite and Robust Optimization

1. 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. 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. 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. 4. Generalized Semi-Infinite Optimization 2 C I, K, L finite
5. 5. Semi-Infinite Optimization Hubertus Th. Jongen Semi-Infinite Optimization, EURO XXIII 2009, July 5-9, 2009, Bonn, Germany
6. 6. Semi-Infinite Optimization
7. 7. Semi-Infinite Optimization
8. 8. Generalized Semi-Infinite Optimization
9. 9. Generalized Semi-Infinite Optimization ψ (τ ) τ ψ ϕ (⋅,τ ) homeom. structurally stable asymptotic effect ε (⋅) IR n global local global
10. 10. Generalized Semi-Infinite Optimization Thm. (W. 1999/2003, 2006): ⇔ ξ
11. 11. Generalized Semi-Infinite Optimization constructions max-type, nonsmooth functions Morse theory, topology perturbed given given feasible set perturbed nonsmooth GSIP
12. 12. Generalized Semi-Infinite Optimization B time-minimal cooling (or heating) of r R ∀T > 0 ∃! GSIP
13. 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. 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. 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. 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. 17. Generalized Semi-Infinite Optimization numerical methods by discretization 123 4 4 {
18. 18. Generalized Semi-Infinite Optimization numerical methods parametrically, by approximation
19. 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. 20. Laurent El Ghaoui Robust Optimization Robust Optimization and Applications, IMA Tutorial, March 11, 2003 .
21. 21. Robust Optimization LP as a conic problem . . .
22. 22. Robust Optimization CQP . . . .
23. 23. Robust Optimization semidefinite programming (SDP) . . CQP . . .
24. 24. Robust Optimization dual of conic program . . , .
25. 25. Robust Optimization robust conic programming . . . .
26. 26. Robust Optimization robust conic programming . . semi-infinite . .
27. 27. Robust Optimization polytopic uncertainty . . .
28. 28. Robust Optimization robust portfolio optimization . . , \$T r .
29. 29. Robust Optimization solution of robust portfolio optimization problem CQP , . .
30. 30. Robust Optimization robust CQP . , CQP , .
31. 31. Robust Optimization Ex.: robust least-squares . .
32. 32. References