1. Robust Optimization and 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

51. Robust dynamic programming

52. Inner problem

53. Worst-case performance of a policy

54. Describing uncertainty

55. Joint estimation and optimization

56. Estimating a transition matrix

57. Likelihood regions

58. likelihood regions

59. Reduction to a 1-D problem

60. Complexity results

61. Application to aircraft routing

62. Markov chain model for the storms 0 1 p q 1-p 1-q

63. information update and recourse

64. Dynamic programming model

65. Nominal algorithm

66. Sample path planning

67. Improvements over obvious strategies Improvement Conservative Strategy (avoid storm) Over-optimistic Strategy (ignore storm and apply recourse at the last moment, if needed) Scenario 166.42%42.76% Scenario 254.78%49.81% Scenario

68. Robustness

69. Optimality vs. uncertainty level

70. Errors in uncertainty level

71. Extensions

72. Summary of results

73. Some references

74. Robust Classification

75. Linear Classification

76. What is a classifier?

77. Classification constraints

78. robust classification: support vector machine

79. box uncertainty model

80. formulations

81. extensions

82. minimax probability machine

83. Problem statement

84. SOCP formulation

85. Dual problem

86. Geometric interpretation

87. Robust classification: summary of results

88. Wrap-up