1. Robust Optimizationand Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003

2. Thanks

3. Optimization models

4. Pitfalls

5. Robust Optimization Paradigm

6. Approximating a robust solution

7. Agenda

8. LP as a conic problem

9. Second-order cone programming

10. Semidefinite programming

11. Dual form of conic program

12. Robust conic programming

13. Polytopic uncertainty

14. Robust LP

15. Robust LP with ellipsoidal uncertainty

16. Robust LP as SOCP

17. Example: robust portfolio design

18. Solution of robust portfolio problem

19. Robust SOCP

20. Example: robust least-squares

21. Robust SDP

22. Example: robust control

23. Analysis of robust conic problems

24. Relaxations

25. Quality estimates

26. Quality estimates: some results

27. restriction

28. Sampling

29. Variations on Robust Conic Programming

30. A Boolean problem

31. Max-quad as a robust LP

32. Rank relaxation

33. Boolean optimization: geometric approach

34. SDP for boolean / nonconvex optimization • geometric and algebraic approaches are dual (see later), yield the same upper bound • SDP provides upper bound • may recover primal variable by sampling • approach extends to many problems • eg, problems with (nonconvex) quadratic constraints & objective • in some cases, quality of relaxation is provably good

35. Robust boolean optimization

36. SDP relaxation of robust problem

37. Chance-constrained programming

38. Problems with adjustable parameters

39. Adjustable parameters: some results

40. Link with feedback control

41. Challenges

42. Set estimation

43. Part I: summary

45. Part II: Contextual Applications

46. Robust path planning

47. Uncertainty in Markov Decision Process

48. Agenda

49. Markov decision problem

50. Previous Work