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Robust Optimizationand Applications
Laurent El [email protected]
IMA Tutorial, March 11, 2003
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Optimization models
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Robust Optimization Paradigm
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Approximating a robust solution
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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
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Polytopic uncertainty
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Robust LP with ellipsoidal uncertainty
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Robust LP as SOCP
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Example: robust portfolio design
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Solution of robust portfolio problem
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Example: robust least-squares
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Example: robust control
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Analysis of robust conic problems
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Quality estimates
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Quality estimates: some results
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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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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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Part II: Contextual Applications
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Robust path planning
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Uncertainty in Markov Decision Process
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Markov decision problem
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Robust dynamic programming
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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
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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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Optimality vs. uncertainty level
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Errors in uncertainty level
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Summary of results
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Robust Classification
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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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minimax probability machine
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Problem statement
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Geometric interpretation
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Robust classification: summary of results