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Transcript of Robust Optimization and Applications Laurent El Ghaoui [email protected] IMA Tutorial,...
![Page 2: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/2.jpg)
Thanks
![Page 3: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/3.jpg)
Optimization models
![Page 4: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/4.jpg)
Pitfalls
![Page 5: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/5.jpg)
Robust Optimization Paradigm
![Page 6: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/6.jpg)
Approximating a robust solution
![Page 7: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/7.jpg)
Agenda
![Page 8: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/8.jpg)
LP as a conic problem
![Page 9: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/9.jpg)
Second-order cone programming
![Page 10: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/10.jpg)
Semidefinite programming
![Page 11: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/11.jpg)
Dual form of conic program
![Page 12: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/12.jpg)
Robust conic programming
![Page 13: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/13.jpg)
Polytopic uncertainty
![Page 14: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/14.jpg)
Robust LP
![Page 15: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/15.jpg)
Robust LP with ellipsoidal uncertainty
![Page 16: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/16.jpg)
Robust LP as SOCP
![Page 17: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/17.jpg)
Example: robust portfolio design
![Page 18: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/18.jpg)
Solution of robust portfolio problem
![Page 19: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/19.jpg)
Robust SOCP
![Page 20: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/20.jpg)
Example: robust least-squares
![Page 21: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/21.jpg)
Robust SDP
![Page 22: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/22.jpg)
Example: robust control
![Page 23: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/23.jpg)
Analysis of robust conic problems
![Page 24: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/24.jpg)
Relaxations
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Quality estimates
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Quality estimates: some results
![Page 27: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/27.jpg)
restriction
![Page 28: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/28.jpg)
Sampling
![Page 29: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/29.jpg)
Variations on Robust Conic Programming
![Page 30: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/30.jpg)
A Boolean problem
![Page 31: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/31.jpg)
Max-quad as a robust LP
![Page 32: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/32.jpg)
Rank relaxation
![Page 33: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/33.jpg)
Boolean optimization: geometric approach
![Page 34: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/34.jpg)
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
![Page 35: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/35.jpg)
Robust boolean optimization
![Page 36: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/36.jpg)
SDP relaxation of robust problem
![Page 37: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/37.jpg)
Chance-constrained programming
![Page 38: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/38.jpg)
Problems with adjustable parameters
![Page 39: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/39.jpg)
Adjustable parameters: some results
![Page 40: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/40.jpg)
Link with feedback control
![Page 41: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/41.jpg)
Challenges
![Page 42: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/42.jpg)
Set estimation
![Page 43: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/43.jpg)
Part I: summary
![Page 44: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/44.jpg)
![Page 45: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/45.jpg)
Part II: Contextual Applications
![Page 46: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/46.jpg)
Robust path planning
![Page 47: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/47.jpg)
Uncertainty in Markov Decision Process
![Page 48: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/48.jpg)
Agenda
![Page 49: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/49.jpg)
Markov decision problem
![Page 50: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/50.jpg)
Previous Work
![Page 51: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/51.jpg)
Robust dynamic programming
![Page 52: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/52.jpg)
Inner problem
![Page 53: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/53.jpg)
Worst-case performance of a policy
![Page 54: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/54.jpg)
Describing uncertainty
![Page 55: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/55.jpg)
Joint estimation and optimization
![Page 56: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/56.jpg)
Estimating a transition matrix
![Page 57: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/57.jpg)
Likelihood regions
![Page 58: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/58.jpg)
likelihood regions
![Page 59: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/59.jpg)
Reduction to a 1-D problem
![Page 60: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/60.jpg)
Complexity results
![Page 61: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/61.jpg)
Application to aircraft routing
![Page 62: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/62.jpg)
Markov chain model for the storms
0 1
p q
1-p
1-q
![Page 63: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/63.jpg)
information update and recourse
![Page 64: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/64.jpg)
Dynamic programming model
![Page 65: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/65.jpg)
Nominal algorithm
![Page 66: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/66.jpg)
Sample path planning
![Page 67: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/67.jpg)
Improvements over obvious strategies
Improvement
Conservative Strategy (avoid storm)
Over-optimistic Strategy (ignore storm and apply recourse at the last moment, if needed)
Scenario 1 66.42% 42.76%
Scenario 2 54.78% 49.81%
Scenario
![Page 68: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/68.jpg)
Robustness
![Page 69: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/69.jpg)
Optimality vs. uncertainty level
![Page 70: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/70.jpg)
Errors in uncertainty level
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Extensions
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Summary of results
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Some references
![Page 74: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/74.jpg)
Robust Classification
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Linear Classification
![Page 76: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/76.jpg)
What is a classifier?
![Page 77: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/77.jpg)
Classification constraints
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robust classification: support vector machine
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box uncertainty model
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formulations
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extensions
![Page 82: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/82.jpg)
minimax probability machine
![Page 83: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/83.jpg)
Problem statement
![Page 84: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/84.jpg)
SOCP formulation
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Dual problem
![Page 86: Robust Optimization and Applications Laurent El Ghaoui elghaoui@eecs.berkeley.edu IMA Tutorial, March 11, 2003.](https://reader037.fdocuments.in/reader037/viewer/2022110213/5697bf8e1a28abf838c8c93e/html5/thumbnails/86.jpg)
Geometric interpretation
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Robust classification: summary of results
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Wrap-up