Optimization Applications - EECS at UC Berkeleyelghaoui/Talks/talkIMA2003b.pdf · Optimization...
Transcript of Optimization Applications - EECS at UC Berkeleyelghaoui/Talks/talkIMA2003b.pdf · Optimization...
![Page 2: Optimization Applications - EECS at UC Berkeleyelghaoui/Talks/talkIMA2003b.pdf · Optimization Applications Laurent El Ghaoui u 03. Thanks. models. ... (IMA) Mike Todd, Scot Adams](https://reader036.fdocuments.us/reader036/viewer/2022062601/5cf699c088c993d5258c211d/html5/thumbnails/2.jpg)
Thanks
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Optimization models
![Page 4: Optimization Applications - EECS at UC Berkeleyelghaoui/Talks/talkIMA2003b.pdf · Optimization Applications Laurent El Ghaoui u 03. Thanks. models. ... (IMA) Mike Todd, Scot Adams](https://reader036.fdocuments.us/reader036/viewer/2022062601/5cf699c088c993d5258c211d/html5/thumbnails/4.jpg)
Pitfalls
![Page 5: Optimization Applications - EECS at UC Berkeleyelghaoui/Talks/talkIMA2003b.pdf · Optimization Applications Laurent El Ghaoui u 03. Thanks. models. ... (IMA) Mike Todd, Scot Adams](https://reader036.fdocuments.us/reader036/viewer/2022062601/5cf699c088c993d5258c211d/html5/thumbnails/5.jpg)
Robust Optimization Paradigm
![Page 6: Optimization Applications - EECS at UC Berkeleyelghaoui/Talks/talkIMA2003b.pdf · Optimization Applications Laurent El Ghaoui u 03. Thanks. models. ... (IMA) Mike Todd, Scot Adams](https://reader036.fdocuments.us/reader036/viewer/2022062601/5cf699c088c993d5258c211d/html5/thumbnails/6.jpg)
Approximating a robust solution
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Agenda
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LP as a conic problem
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Second-order cone programming
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Semidefinite programming
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Dual form of conic program
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Robust conic programming
![Page 13: Optimization Applications - EECS at UC Berkeleyelghaoui/Talks/talkIMA2003b.pdf · Optimization Applications Laurent El Ghaoui u 03. Thanks. models. ... (IMA) Mike Todd, Scot Adams](https://reader036.fdocuments.us/reader036/viewer/2022062601/5cf699c088c993d5258c211d/html5/thumbnails/13.jpg)
Polytopic uncertainty
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Robust LP
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Robust LP with ellipsoidal uncertainty
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Robust LP as SOCP
![Page 17: Optimization Applications - EECS at UC Berkeleyelghaoui/Talks/talkIMA2003b.pdf · Optimization Applications Laurent El Ghaoui u 03. Thanks. models. ... (IMA) Mike Todd, Scot Adams](https://reader036.fdocuments.us/reader036/viewer/2022062601/5cf699c088c993d5258c211d/html5/thumbnails/17.jpg)
Example: robust portfolio design
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Solution of robust portfolio problem
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Robust SOCP
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Example: robust least-squares
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Robust SDP
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Example: robust control
![Page 23: Optimization Applications - EECS at UC Berkeleyelghaoui/Talks/talkIMA2003b.pdf · Optimization Applications Laurent El Ghaoui u 03. Thanks. models. ... (IMA) Mike Todd, Scot Adams](https://reader036.fdocuments.us/reader036/viewer/2022062601/5cf699c088c993d5258c211d/html5/thumbnails/23.jpg)
Analysis of robust conic problems
![Page 24: Optimization Applications - EECS at UC Berkeleyelghaoui/Talks/talkIMA2003b.pdf · Optimization Applications Laurent El Ghaoui u 03. Thanks. models. ... (IMA) Mike Todd, Scot Adams](https://reader036.fdocuments.us/reader036/viewer/2022062601/5cf699c088c993d5258c211d/html5/thumbnails/24.jpg)
Relaxations
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Quality estimates
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Quality estimates: some results
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restriction
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Sampling
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Variations on Robust Conic Programming
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A Boolean problem
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Max-quad as a robust LP
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Rank relaxation
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Boolean optimization: geometric approach
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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
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Robust boolean optimization
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SDP relaxation of robust problem
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Chance-constrained programming
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Problems with adjustable parameters
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Adjustable parameters: some results
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Link with feedback control
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Challenges
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Set estimation
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Part I: summary
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Part II: Contextual Applications
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Robust path planning
![Page 47: Optimization Applications - EECS at UC Berkeleyelghaoui/Talks/talkIMA2003b.pdf · Optimization Applications Laurent El Ghaoui u 03. Thanks. models. ... (IMA) Mike Todd, Scot Adams](https://reader036.fdocuments.us/reader036/viewer/2022062601/5cf699c088c993d5258c211d/html5/thumbnails/47.jpg)
Uncertainty in Markov Decision Process
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Agenda
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Markov decision problem
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Previous Work
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Robust dynamic programming
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Inner problem
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Worst-case performance of a policy
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Describing uncertainty
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Joint estimation and optimization
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Estimating a transition matrix
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Likelihood regions
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likelihood regions
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Reduction to a 1-D problem
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Complexity results
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Application to aircraft routing
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Markov chain model for the storms
0 1
p q
1-p
1-q
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information update and recourse
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Dynamic programming model
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Nominal algorithm
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Sample path planning
![Page 67: Optimization Applications - EECS at UC Berkeleyelghaoui/Talks/talkIMA2003b.pdf · Optimization Applications Laurent El Ghaoui u 03. Thanks. models. ... (IMA) Mike Todd, Scot Adams](https://reader036.fdocuments.us/reader036/viewer/2022062601/5cf699c088c993d5258c211d/html5/thumbnails/67.jpg)
Improvements over obvious strategies
49.81%54.78%Scenario 2
42.76%66.42%Scenario 1
Over-optimistic Strategy (ignore storm and apply recourse at the last moment, if needed)
Conservative Strategy (avoid storm)
Improvement
Scenario
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Robustness
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Optimality vs. uncertainty level
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Errors in uncertainty level
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Extensions
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Summary of results
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Some references
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Robust Classification
![Page 75: Optimization Applications - EECS at UC Berkeleyelghaoui/Talks/talkIMA2003b.pdf · Optimization Applications Laurent El Ghaoui u 03. Thanks. models. ... (IMA) Mike Todd, Scot Adams](https://reader036.fdocuments.us/reader036/viewer/2022062601/5cf699c088c993d5258c211d/html5/thumbnails/75.jpg)
Linear Classification
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What is a classifier?
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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
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minimax probability machine
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Problem statement
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SOCP formulation
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Dual problem
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Geometric interpretation
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Robust classification: summary of results
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Wrap-up