introduction to the scenario approach - cnr - introduction to... · thanks to: algo care’...

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.