Introduction to the Scenario Approach - CNR - introduction to... · thanks to: Algo Care’...
Transcript of Introduction to the Scenario Approach - CNR - introduction to... · thanks to: Algo Care’...
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Introduction to the Scenario Approach
Marco C. Campi University of Brescia
Italy
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thanks to:
Algo
Care’
Giuseppe
Calafiore
Maria Prandini
Bernardo
Pagnoncelli
Federico
Ramponi
Simone
Garatti
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PART I: Principles
PART II: Algorithms
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PART I: Principles
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system design
controller synthesis
portfolio selection
optimization
program
Optimization
management
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Uncertain environment
exercise caution
system design
controller synthesis
portfolio selection
optimization
program
Optimization
management
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U-OP:
Uncertain Optimization Program
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U-OP:
not well-defined
Uncertain Optimization Program
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Uncertainty
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Uncertainty
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[G. Zames, 1981]
Uncertainty
optimization [A. Ben-Tal & A. Nemirovski, 2002]
control theory
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Probabilistic uncertainty
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Probabilistic uncertainty
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Probabilistic uncertainty
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Probabilistic uncertainty
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Probabilistic uncertainty
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Probabilistic uncertainty
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Probabilistic uncertainty
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[A. Charnes, W.W. Cooper, and G.H. Symonds, 1958]
Probabilistic uncertainty
chance-constrained approach:
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[A. Charnes, W.W. Cooper, and G.H. Symonds, 1958]
Probabilistic uncertainty
chance-constrained approach:
very difficult to solve, … with exceptions
[A. Prékopa, 1995]
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[A. Charnes, W.W. Cooper, and G.H. Symonds, 1958]
Probabilistic uncertainty
chance-constrained approach:
very difficult to solve, … with exceptions
[A. Prékopa, 1995]
the scenario approach provides algorithmic tools
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a look at optimization in the space
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performance cloud
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worst-case
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average
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chance-constrained approach
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chance-constrained approach
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performance - violation plot
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PART II: Algorithms
(convex case)
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The “scenario” paradigm
[G. Calafiore & M. Campi, Math. Programming, 2005]
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SPN = scenario program
The “scenario” paradigm
SPN is a standard finite convex optimization problem
[G. Calafiore & M. Campi, Math. Programming, 2005]
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Fundamental
question: what’s the risk of ?
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Example: feedforward noise compensation
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Example: feedforward noise compensation
Compensator ARMAX
System
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Example: feedforward noise compensation
Compensator ARMAX
System
Objective: reduce the effect of noise on y
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Example: feedforward noise compensation
Compensator ARMAX
System
ARMAX System:
Compensator:
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Example: feedforward noise compensation
Compensator ARMAX
System
Goal:
ARMAX System:
Compensator:
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Example: feedforward noise compensation
Compensator ARMAX
System
ARMAX System:
Compensator:
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Example: feedforward noise compensation
system parameters unknown:
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Example: feedforward noise compensation
system parameters unknown:
sample:
solve:
scenario approach:
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more examples: minimax prediction
[M. Campi, G. Calafiore & S. Garatti, Automatica, 2009]
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more examples: machine learning
[M. Campi, Machine Learning, 2010]
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more examples: portfolio optimization
= return of asset , = instance in the record
[M. Campi, B. Pagnoncelli & D. Reich, 2012]
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Fundamental
question: what’s the risk of ?
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Fundamental
question:
that is: how guaranteed is against other
what’s the risk of ?
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Fundamental
question:
from the “visible” to the “invisible”
what’s the risk of ?
that is: how guaranteed is against other
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[M. Campi & S. Garatti, SIAM J. on Optimization, 2008;
T. Alamo, R. Tempo and A. Luque, Springer-Verlag, 2010]
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[M. Campi & S. Garatti, SIAM J. on Optimization, 2008;
T. Alamo, R. Tempo and A. Luque, Springer-Verlag, 2010]
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[M. Campi & S. Garatti, SIAM J. on Optimization, 2008;
T. Alamo, R. Tempo and A. Luque, Springer-Verlag, 2010]
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[M. Campi & S. Garatti, SIAM J. on Optimization, 2008;
T. Alamo, R. Tempo and A. Luque, Springer-Verlag, 2010]
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Comments
generalization need for structure
good news: the structure we need
is only convexity
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… more comments
N easy to compute
N depends on the problem through only
N independent of
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Example: feedforward noise compensation
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Example: feedforward noise compensation
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Example: feedforward noise compensation
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Example: feedforward noise compensation
sample:
solve:
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Example: feedforward noise compensation
sample:
solve:
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Example: feedforward noise compensation
Output variance below 5.8 for all plants but a
small fraction ( = 0.5%)
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Example: feedforward noise compensation
performance profile
Output variance below 5.8 for all plants but a
small fraction ( = 0.5%)
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Risk-Return Tradeoff
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Risk-Return Tradeoff
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Risk-Return Tradeoff
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Risk-Return Tradeoff
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Risk-Return Tradeoff
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Risk-Return Tradeoff
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Risk-Return Tradeoff
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[M. Campi & S. Garatti, JOTA, 2011]
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[M. Campi & S. Garatti, JOTA, 2011]
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Comments
the result does not depend on the
algorithm for eliminating k scenarios
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Comments
… do it greedy
the result does not depend on the
algorithm for eliminating k scenarios
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Comments
the result does not depend on the
algorithm for eliminating k scenarios
… do it greedy
value can be inspected
the risk is guaranteed by the
theorem
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performance - violation plot
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Example: feedforward noise compensation
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Example: feedforward noise compensation
sample:
solve:
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Example: feedforward noise compensation
sample:
solve:
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Example: feedforward noise compensation
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performance - violation plot
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performance profile
Example: feedforward noise compensation
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performance profile
Example: feedforward noise compensation
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performance profile
Example: feedforward noise compensation
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performance profile
Example: feedforward noise compensation
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performance profile
Example: feedforward noise compensation
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performance profile
Example: feedforward noise compensation
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performance profile
Example: feedforward noise compensation
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performance profile
Example: feedforward noise compensation
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performance profile
Example: feedforward noise compensation
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REFERENCES
M.C. Campi and S. Garatti.
The Exact Feasibility of Randomized Solutions of Uncertain Convex Programs.
SIAM J. on Optimization, 19, no.3: 1211-1230, 2008.
M.C. Campi and S. Garatti.
A Sampling-and-Discarding Approach to Chance-Constrained Optimization: Feasibility and Optimality.
J. of Optimization Theory and Application, 148: 257-280, 2011.
G. Calafiore and M.C. Campi.
Uncertain Convex Programs: randomized Solutions and Confidence Levels.
Mathematical Programming, 102: 25-46, 2005.
G. Calafiore and M.C. Campi.
The Scenario Approach to Robust Control Design.
IEEE Trans. on Automatic Control, AC-51: 742-753, 2006.
M.C. Campi, G. Calafiore and S. Garatti.
Interval Predictor Models: Identification and Reliability.
Automatica, 45: 382-392, 2009.
M.C. Campi.
Classification with guaranteed probability of error.
Machine Learning, 80: 63-84, 2010.
T. Alamo, R. Tempo and E.F. Camacho
A randomized strategy for probabilistic solutions of uncertain feasibility and optimization problems.
IEEE Trans. on Automatic Control, AC-54: 2545–2559, 2009.