Semi-Infinite and Robust Optimization
description
Transcript of Semi-Infinite and Robust Optimization
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Elements of Semi-Infinite and Robust Optimization
Motivatio
4th International Summer SchoolAchievements and Applications of Contemporary Informatics, Mathematics and PhysicsNational University of Technology of the UkraineKiev, Ukraine, August 5-16, 2009
GerhardGerhardGerhardGerhardGerhardGerhardGerhardGerhard--------Wilhelm Weber Wilhelm Weber Wilhelm Weber Wilhelm Weber Wilhelm Weber Wilhelm Weber Wilhelm Weber Wilhelm Weber *,*, Başak AktekeBaşak AktekeBaşak AktekeBaşak AktekeBaşak AktekeBaşak AktekeBaşak AktekeBaşak Akteke--------Öztürk Öztürk Öztürk Öztürk Öztürk Öztürk Öztürk Öztürk
Institute of Applied Mathematics Institute of Applied Mathematics Programs of Financial Mathematics, Actuarial Sciences and Scientific ComputingPrograms of Financial Mathematics, Actuarial Sciences and Scientific Computing
Department of Biomedical EngineeringDepartment of Biomedical EngineeringMiddle East Technical University, Ankara, TurkeyMiddle East Technical University, Ankara, Turkey
* * * * * * * * Faculty of Economics, Management and Law, University of Siegen, GermanyFaculty of Economics, Management and Law, University of Siegen, GermanyCenter for Research on Optimization and Control, University of Aveiro, Portugal
n
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GENE time 0 9.5 11.5 13.5 15.5 18.5 20.5
'YHR007C' 0.224 0.367 0.312 0.014 -0.003 -1.357 -0.811
'YAL051W' 0.002 0.634 0.31 0.441 0.458 -0.136 0.275
'YAL054C' -1.07 -0.51 -0.22 -0.012 -0.215 1.741 4.239
'YAL056W' 0.09 0.884 0.165 0.199 0.034 0.148 0.935
'PRS316' -0.046 0.635 0.194 0.291 0.271 0.488 0.533
'KAN-MX' 0.162 0.159 0.609 0.481 0.447 1.541 1.449
'E. COLI #10' -0.013 0.88 -0.009 0.144 -0.001 0.14 0.192
Networks and Optimization
'E. COLI #10' -0.013 0.88 -0.009 0.144 -0.001 0.14 0.192
'E. COLI #33' -0.405 0.853 -0.259 -0.124 -1.181 0.095 0.027
ex.: yeast data
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min( ), ( ), ( )ij i im c d∗ ∗ ∗
l
( 1, . . . , )( , ) ( )n
j np m y yα∗ =≤∑
subject to
)2
1
0
l
M E C E D Eαα ακκ κ
α
∗ −∗ ∗ ∗
= ∞
+ + −∑ &
GSIP relaxation
Networks and Optimization
set of combined environmental effects combined environmental effects combined environmental effects combined environmental effects
( , ) :Y C D∗ ∗ =
1,..., 1,...,1,...,
( 0, ) ( 0, )i ii n i n
m
c d∗ ∗
= ==
× ∏ ∏l
l
1
1
1
, m in
( 1, . . . , )
( 1, . . . , )
( , ) ( )
( , ) ( )
( , ) ( )
&
i j i j ji
n
i ii
n
i ii
i i i
j n
m
p m y y
q c y y
d y y
m
α
β
ζ γ
δ
∗
=
∗
=
∗
=
=
=
≤
≤
≤
≥
∑
∑
∑
l l l l
o v e ra ll b o x c o n s t ra in ts
( ( , ))y Y C D∗ ∗∈
( 1, . . . , )i n=
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Generalized Semi-Infinite Optimization
2C
I, K, L finite
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Semi-Infinite Optimization
Hubertus Th. JongenSemi-Infinite Optimization,EURO XXIII 2009, July 5-9, 2009, Bonn, Germany
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Semi-Infinite Optimization
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Semi-Infinite Optimization
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Generalized Semi-Infinite Optimization
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)(τψτ
ψ
Generalized Semi-Infinite Optimization
structurally stable
global local global
)(⋅ε
nIR
asymptotic
effect
homeom.
),( τϕ ⋅
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Thm. (W. 1999/2003, 2006):
⇔ξ
Generalized Semi-Infinite Optimization
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constructions
Morse theory, topologymax-type, nonsmooth functions
Generalized Semi-Infinite Optimization
given
feasible set
perturbed
perturbed
given
nonsmooth GSIP
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r R
B
time-minimal cooling (or heating) of
Generalized Semi-Infinite Optimization
!0 ∃>∀T
GSIP
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further ex. : thermo-regulation of premature infants
control of global warming
•
•
Generalized Semi-Infinite Optimization
anticipation
maximization of time-horizon longest term description•
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Hoffmann, Reinhard
Ex.: approx. of a thermo-couple characteristic
thermo-couple f (y) : spline of polynomials with deg. 3 – 13, on [a,b]
to be approx. by :
bounds on error•
Generalized Semi-Infinite Optimization
Bernhard
some interpol.
(= y)•
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Ex.: approx. of a thermo-couple characteristic
thermo-couple f (y) : spline of polynomials with deg. 3 – 13, on [a,b]
to be approx. by :
bounds on error•
Generalized Semi-Infinite Optimization
some interpol.
•
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Ex.: approx. of a thermo-couple characteristic
thermo-couple f (y) : spline of polynomials with deg. 3 – 13, on [a,b]
to be approx. by :
bounds on error•
Generalized Semi-Infinite Optimization
some interpol.
•time
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by discretization
43421 {
Generalized Semi-Infinite Optimization
numerical methods
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parametrically,by approximation
Generalized Semi-Infinite Optimization
numerical methods
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by local linearization & transversal intersection
1
3
Generalized Semi-Infinite Optimization
numerical methods
reduction ansatzexchange methodsemismooth Newton’s method
O. Stein, G. Still W.A. Tezel
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Robust Optimization Laurent El Ghaoui
.
Robust Optimizationand Applications,IMA Tutorial, March 11, 2003
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LP as a conic problem
Robust Optimization
.
.
.
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CQP
.
.
Robust Optimization
.
.
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semidefinite programming (SDP)
.
.
Robust Optimization
CQP .
.
.
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dual of conic program
.
Robust Optimization
,
.
.
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robust conic programming
Robust Optimization
.
.
.
.
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robust conic programming
Robust Optimization
.
semi-infinite
.
.
.
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polytopic uncertainty
Robust Optimization
.
.
.
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robust portfolio optimization
Robust Optimization
.
$T
r
.
.
,
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solution of robust portfolio optimization problem
Robust Optimization
CQP
,,
.
.
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robust CQP
Robust Optimization
.
CQP
,
.
,
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Ex.: robust least-squares
Robust Optimization
.
..
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References