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Robust Optimization and Applications

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Robust Optimization and Applications. Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003. Thanks. Optimization models. Pitfalls. Robust Optimization Paradigm. Approximating a robust solution. Agenda. LP as a conic problem. Second-order cone programming.

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Robust Optimization and Applications

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  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

  44. Part II: Contextual Applications

  45. Robust path planning

  46. Uncertainty in Markov Decision Process

  47. Agenda

  48. Markov decision problem

  49. Previous Work

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