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TRANSCRIPT
Introduction to the Scenario Approach
Marco C. Campi University of Brescia
Italy
thanks to:
Algo
Care’
Giuseppe
Calafiore
Maria Prandini
Bernardo
Pagnoncelli
Federico
Ramponi
Simone
Garatti
PART I: Principles
PART II: Algorithms
PART I: Principles
system design
controller synthesis
portfolio selection
optimization
program
Optimization
management
Uncertain environment
exercise caution
system design
controller synthesis
portfolio selection
optimization
program
Optimization
management
U-OP:
Uncertain Optimization Program
U-OP:
not well-defined
Uncertain Optimization Program
Uncertainty
Uncertainty
[G. Zames, 1981]
Uncertainty
optimization [A. Ben-Tal & A. Nemirovski, 2002]
control theory
Probabilistic uncertainty
Probabilistic uncertainty
Probabilistic uncertainty
Probabilistic uncertainty
Probabilistic uncertainty
Probabilistic uncertainty
Probabilistic uncertainty
[A. Charnes, W.W. Cooper, and G.H. Symonds, 1958]
Probabilistic uncertainty
chance-constrained approach:
[A. Charnes, W.W. Cooper, and G.H. Symonds, 1958]
Probabilistic uncertainty
chance-constrained approach:
very difficult to solve, … with exceptions
[A. Prékopa, 1995]
[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
a look at optimization in the space
performance cloud
worst-case
average
chance-constrained approach
chance-constrained approach
performance - violation plot
PART II: Algorithms
(convex case)
The “scenario” paradigm
[G. Calafiore & M. Campi, Math. Programming, 2005]
SPN = scenario program
The “scenario” paradigm
SPN is a standard finite convex optimization problem
[G. Calafiore & M. Campi, Math. Programming, 2005]
Fundamental
question: what’s the risk of ?
Example: feedforward noise compensation
Example: feedforward noise compensation
Compensator ARMAX
System
Example: feedforward noise compensation
Compensator ARMAX
System
Objective: reduce the effect of noise on y
Example: feedforward noise compensation
Compensator ARMAX
System
ARMAX System:
Compensator:
Example: feedforward noise compensation
Compensator ARMAX
System
Goal:
ARMAX System:
Compensator:
Example: feedforward noise compensation
Compensator ARMAX
System
ARMAX System:
Compensator:
Example: feedforward noise compensation
system parameters unknown:
Example: feedforward noise compensation
system parameters unknown:
sample:
solve:
scenario approach:
more examples: minimax prediction
[M. Campi, G. Calafiore & S. Garatti, Automatica, 2009]
more examples: machine learning
[M. Campi, Machine Learning, 2010]
more examples: portfolio optimization
= return of asset , = instance in the record
[M. Campi, B. Pagnoncelli & D. Reich, 2012]
Fundamental
question: what’s the risk of ?
Fundamental
question:
that is: how guaranteed is against other
what’s the risk of ?
Fundamental
question:
from the “visible” to the “invisible”
what’s the risk of ?
that is: how guaranteed is against other
[M. Campi & S. Garatti, SIAM J. on Optimization, 2008;
T. Alamo, R. Tempo and A. Luque, Springer-Verlag, 2010]
[M. Campi & S. Garatti, SIAM J. on Optimization, 2008;
T. Alamo, R. Tempo and A. Luque, Springer-Verlag, 2010]
[M. Campi & S. Garatti, SIAM J. on Optimization, 2008;
T. Alamo, R. Tempo and A. Luque, Springer-Verlag, 2010]
[M. Campi & S. Garatti, SIAM J. on Optimization, 2008;
T. Alamo, R. Tempo and A. Luque, Springer-Verlag, 2010]
Comments
generalization need for structure
good news: the structure we need
is only convexity
… more comments
N easy to compute
N depends on the problem through only
N independent of
Example: feedforward noise compensation
Example: feedforward noise compensation
Example: feedforward noise compensation
Example: feedforward noise compensation
sample:
solve:
Example: feedforward noise compensation
sample:
solve:
Example: feedforward noise compensation
Output variance below 5.8 for all plants but a
small fraction ( = 0.5%)
Example: feedforward noise compensation
performance profile
Output variance below 5.8 for all plants but a
small fraction ( = 0.5%)
Risk-Return Tradeoff
Risk-Return Tradeoff
Risk-Return Tradeoff
Risk-Return Tradeoff
Risk-Return Tradeoff
Risk-Return Tradeoff
Risk-Return Tradeoff
[M. Campi & S. Garatti, JOTA, 2011]
[M. Campi & S. Garatti, JOTA, 2011]
Comments
the result does not depend on the
algorithm for eliminating k scenarios
Comments
… do it greedy
the result does not depend on the
algorithm for eliminating k scenarios
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
performance - violation plot
Example: feedforward noise compensation
Example: feedforward noise compensation
sample:
solve:
Example: feedforward noise compensation
sample:
solve:
Example: feedforward noise compensation
performance - violation plot
performance profile
Example: feedforward noise compensation
performance profile
Example: feedforward noise compensation
performance profile
Example: feedforward noise compensation
performance profile
Example: feedforward noise compensation
performance profile
Example: feedforward noise compensation
performance profile
Example: feedforward noise compensation
performance profile
Example: feedforward noise compensation
performance profile
Example: feedforward noise compensation
performance profile
Example: feedforward noise compensation
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.