PREPRINT 2009:3
Approximating the Pareto Optimal
Set using a Reduced Set of Objective
Functions
PETER LINDROTH
MICHAEL PATRIKSSON
ANN-BRITH STRÖMBERG
Department of Mathematical Sciences
Division of Mathematics
CHALMERS UNIVERSITY OF TECHNOLOGY
UNIVERSITY OF GOTHENBURG
Göteborg Sweden 2009
Preprint 2009:3
Approximating the Pareto Optimal Set
using a Reduced Set of Objective Functions
Peter Lindroth, Michael Patriksson,
Ann-Brith Strömberg
Department of Mathematical Sciences
Division of Mathematics
Chalmers University of Technology and University of Gothenburg
SE-412 96 Göteborg, Sweden
Göteborg, January 2009
Preprint 2009:3
ISSN 1652-9715
Matematiska vetenskaper
Göteborg 2009
∗†
†
†
! "# $ ! % & ' % (
min {f1 (x), . . . , fk (x)} ,
x∈X
x ∈ Rn X ⊆ Rn fi : X → R ! f = {f1, . . . , fk } " K = {1, . . . , k} ! Z " Z = f (X) = {z = f (x) | x ∈ X} # " $ %
$ # & '
∗ x∗ ∈ X ∗
x ∈ X fi (x) ≤ fi (x ), i ∈ K fj (x) < fj (x ) j ∈ K z∗ = f (x∗ ) '
x∗ x∗ ∈ X P ⊆ X x ∈ X y ∈ X fi (x) ≤ fi (y), i ∈
K
fj (x) < fj (y) j ∈ K
! '
" (
z
z ∗ ! " ! # "
!$! # %& ' ('& )
*# %& ' ((&(
† +
, - . +
, - '( /& ! # 0"! "
1$
!
z
z
∈
R k
z =
min f1 (x), . . . , min fk (x) ,
x∈P
z =
x∈P
∈ Rk
max f1 (x), . . . , max fk (x) .
x∈P
x∈P
x∗ ∈ X ) '
x ∈ X fi (x) < fi (x∗ ), i ∈ K z∗ = f (x∗ ) ) '
x∗ Pw * X = {xj | j ∈ N }
N = {1, . . . , N } +
& '
* $ ,"
∗ ! N ≥ 1 X = {x1, . . . , xN } M 1
x
∈X
fi (x∗ ) < fi (xj ) + M (1 − uij ),
∗
j
i∈K fi (x ) ≤
i∈K fi (x ) + M (1 − u0j ),
i∈{0}∪K uij
≥ 1,
j ∈ N , i ∈ K,
j ∈ N,
j ∈ N,
uij ∈ {0, 1},
j ∈ N , i ∈ {0} ∪ K.
-
-
-
-
! ,"
'
(
x∗ ∈ X
'
∗
& x ∈ X fi (x) ≤ fi (x ), i ∈ K i∈K fi (x) <
∗
f
(x
) & x ∈ X k k + 1 i
i∈K
" x∗ ∈ X '
$ j ∈ N k + 1 fi (xj ) > fi (x∗ ), i ∈ K i∈K fi (xj ) ≥ i∈K fi (x∗ ) "
! & - . "
- #$ "
"
- / " P ⊆ X 0 k ) " $
P 1 2345 264 +
k !
* $ ! 7 8 2-4 9 '
# fi X +
24 9
" ) # 2:4 " ;
< 5 ;
< x ∈ X & 0 24 9 "
& &
1 =
"
2>4 = " 2.4 & & , $ 234 -
' ?
0 '?0 &5 ) Z ! ) '
%
* ; < / " k $ '
# 2>4 ! x, y ∈ X x y x y ! $ '
! %
X 9
'
! @
) 2>4 # & 2:4 ( %
# '
P β P P * '
! ) # 5 & '
' " '
" * " A A
'
B / $ '
9 '
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/ '
)
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! "$ " # 6 '
. " ! $ !
> $
1
# . C $ $ "
)
# '
* & '
6
-6
'
D 9 '
# 2C4
& $ '
P̂ !
! $ + P̂
" P 0 $
P̂ P P̂ P̂ P B 2:4 2-E4 " %& (P̂) = max min c(x, y) ,
x∈P
6
y∈P̂
c : Rn × Rn → R+ ! %& P P̂ * " P β '
$ P ( P
$ P β !
$ %& )
%
9 * d(·, ·) %
9 E F
" dH (E, F ) = max max min d(u, v); max min d(v, u) .
>
u∈E v∈F
v∈F u∈E
7 &
& P 2F4 +
&
$ n = 2 1 %
$ / P,
1, x ∈
ξ(x) =
0, x ∈ P.
A
A # 234 " " $ '
" +
" $ $ $ # g : X → Rr r < k # " g $ '
* ! +
! 2>4( 8 K 2K = {K1, . . . , K2 } B " 2K r {s1, . . . , sr } ⊂ K̂ = {1, . . . , 2k } "
# {f1 , . . . , fk } {gs , . . . , gs } r < k, gs = |K1 | i∈K fi Ks gs * & fi gs ∪rj=1 Ks = K * k
1
j
sj
j
sj
j
j
>
j
r
βp =
1,
0,
p .
p ∈ K̂.
8 A ∈ Bk×2 $ k
aip =
1,
0,
fi p
i ∈ K, p ∈ K̂.
! $ r " β "
Aβ ≥ 1k ,
β Ì 12 ≤ r,
C
k
k
β ∈ {0, 1}2 ,
n
# " β P β 1n
P̂
'
* $ '
,
)
'
$ 5 & ?
2.4 # ! Kβ = {j ∈ K | βj = 1} ! Pw ⊆ X (Pwβ ⊆ X) K (Kβ ) Pwβ ⊆ Pw y∗ ∈ Pwβ ! y ∈ X fi(y) < fi(y∗ ),
β
K ⊆ K, y ∈ X
fi (y) < fi (y
∗
), i ∈ K
y ∈ Pw ∗
i ∈ Kβ ' - k = n = 2, {f1 (x), f2 (x)} X f1 (x) = x1 f2 (x) = x2 X = {(1, 2); (2, 1); (3, 1)} P = {(1, 2); (2, 1)} ( f1 β = (0, 1)
P β = {(2, 1); (3, 1)} ⊆ P ! 2>4 '
) * P β ⊆ P ! Kβ {w1 f1 + w2 f2 , f3 , . . . , fk }, w1 , w2 > 0
! P ⊆ X (P β ⊆ X) K (Kβ ) P β ⊆ P D $
'
! " ) '
# P β 9 P ) " ρ ∈ [0, 1] " x ∈ E
E X P P β ρ
ρ
.
ρ ) ρ
E ρ ⊆ E E ⊆ X E ρ = x ∈ E fi (x) ≤ (1 − ρ)zi + ρzi, i ∈ K .
:
* E ⊆ X E ρ = E ∩ X ρ
# ρ '
P ρ = P ∩X ρ '
P ρ 0 E ⊆ X E 0 = {x ∈ E f (x) ≤ z }
E 1 = {x ∈ E f (x) ≤ z } = ∅ + f2
z
f(X)
f(P \P )
ρ
f(E \E ρ )
f(E ρ )
f(E \E ρ )
f(P ρ )
f(P \P ρ )
z
f1
+ ( 0 ρ %
E X ρ ≈ 0.2
! " * $ '
!
τ ≥ 0 '
9 '
* " τ Pτ τ * τ ≥ 0 x∗ ∈ X τ '
x ∈ X fi (x) + τ ≤ fi (x∗ ), i ∈ K fj (x) + τ < fj (x∗ ) j ∈ K z∗ = f (x∗ ) τ '
x∗ τ τ Pτ ⊆ X / P ⊆ Pτ̃ ⊆ Pτ
τ ≥ τ̃ ≥ 0 =
Pτβ β
P τ
# $%& &
* ( 7 r < k ρ ∈ [0, 1] " {s1 , . . . , sr } ⊂ K̂ τ ≥ 0 %
9 dH (f (P ρ ), f (Pτβ,ρ )) + - ! ½
2 Pτβ,ρ - (Pτβ )ρ !
C
,τ
β
δ(β, τ ) := dH (f (P ρ ), f (Pτβ,ρ )),
Aβ ≥ 1k ,
β Ì 12 ≤ r,
3
k
k
β ∈ {0, 1}2 ,
τ ≥ 0.
X
E
F
dH
+ -( 0 %
9 E ⊆ X F ⊆ X '
'
. ;< 3 ' ( ? Pτβ,ρ β τ ! & $ '
'
. # $ τ '
τ '
* 6- G - & X '
'
?
- - - &
w , ∈ N - " x * xj x " X " uij =
u0j =
vj =
w =
1,
0,
1,
0,
1,
0,
1,
0,
fi (x ) < fi (xj ),
,
i∈K fi (x ) ≤ i∈K fi (xj ),
,
xj x xj x ,
x ∈ P vj = 1 ∀j),
x ∈/ P,
:
j, ∈ N , i ∈ K,
j, ∈ N ,
i∈{0}∪K uij
≥ 1),
j, ∈ N ,
∈ N.
G ∈ N &(
−M uij ≤ fi (x ) − fi (xj ) < M (1 − uij ),
−M u0j < i∈K fi (x ) − i∈K fi (xj ) ≤ M (1 − u0j ),
vj ≤ i∈{0}∪K uij ≤ (k + 1)vj ,
N w ≤ j∈N vj ≤ w + N − 1,
j ∈ N , i ∈ K,
j ∈ N,
j ∈ N,
uij , vj , w ∈ {0, 1},
j ∈ N , i ∈ {0} ∪ K.
F
F
F
F
F
* (
* F "
x ∈ X, ∈
N w , ∈ N "
F #$ "
ξ = min |fi (xj ) − fi (x )| : i ∈ K, j, ∈ N , |fi (xj ) − fi (x )| > 0 > 0,
"
"
F E . F
+ P
$
⊆ X
"
F ∈N w ,
− fi (xj )
fi (x )
< M (1 − uij ),
j
i∈K fi (x ) −
i∈K fi (x ) ≤ M (1 − u0j ),
vj ≤ i∈{0}∪K uij ,
N w ≤ j∈N vj ,
uij , vj , w ∈ {0, 1},
j, ∈ N , i ∈ K,
j, ∈ N ,
j, ∈ N ,
∈ N,
j, ∈ N , i ∈ {0} ∪ K.
E
E
E
E
E
' ( ! 2k 5 βp , p ∈ K̂ = {1, . . . , 2k } %
τ '
{f1, . . . , fk } F {β1 g1 , . . . , β2 g2 } ! hi x ∈ X x∗ ∈ X τ > 0 $
x ∈ X hi (x) + τ ≤ hi (x∗ )5 x ∈ X τ '
! gp βp = 0 βp gp ≡ 0 !
F
" & gp βp = 0 * " u, v w k
upj =
u0j =
vj =
w =
1,
0,
1,
0,
1,
0,
1,
0,
gp(x ) < gp (xj ) + τ p ,
,
p∈K̂ βp gp (x ) ≤ p∈K̂ (βp gp (xj ) + τ ),
,
xj τ x p∈{0}∪K̂ upj ≥ 1,
xj τ x ,
x ∈ Pτβ vj = 1 ∀j),
x ∈/ Pτβ ,
3
k
j, ∈ N , p ∈ K̂,
j, ∈ N ,
j, ∈ N ,
∈ N.
8 βp , p ∈ K̂ " ∈ N w = 1 x ∈ Pτβ (
−M upj ≤ M (1 − βp ) + βp gp (x ) − βp gp (xj ) + τ < 2M (1 − upj ),
−M u0j < p∈K̂ βp gp (x ) − p∈K̂ βp gp (xj ) + τ ≤ M (1 − u0j ),
vj ≤ p∈{0}∪K̂ upj ≤ (r + 1)vj ,
N w ≤ j∈N vj ≤ w + N − 1,
upj , vj , w ∈ {0, 1},
j ∈ N , p ∈ K̂,
j ∈ N,
j ∈ N,
j ∈ N , p ∈ {0} ∪ K̂.
! 9 F τ "
τ '
H
9 & F ) upj = 0 βp = 0 0
k + 1 F r + 1 +
! {g1, . . . , g2 } β ∈ {0, 1}2 # r$ x ∈ X, ∈ N τ τ k
k
' ( ! dH (f (P ρ ), f (Pτβ,ρ )) β τ ρ !
,"
-> $ # & X ρ ρ '
dim(P) = k fi (x) − zi
fi (x) :=
, i ∈ K,
-
zi
− zi
f (X) ⊆ Rk+ f (x) ∈ [0, 1]k , x ∈ P '
6. $ & ρ
+ fi : X → R+ , i ∈ K X = {x1, . . . , xN } ρ ∈ [0, 1] M 1 ! P ⊆ X
ρ
w = 1 x ∈ P x ∈ X i ∈ K m ∈ N j ∈ N fi (x ) ≤ (1 − ρ)wj fi (xj ) + ρfi (xm ) + M (1 − wm ).
6
+ "
: ( +
x ∈ X
ρ 1 i ∈ K &
fi (x ) ≤ (1 − ρ) max fi (xj ) + ρ
j: wj =1
min
m: wm =1
fi (xm ),
>
8
ĵ ∈ arg maxj∈N {fi (xj ) | wj = 1} m̂ ∈ arg minm∈N {fi (xm ) | wm = 1} H
{j | wj = 1} X '
H
& 6 > $ xj xm ⇐ # > ĵ 6 j ∈ N # > m̂ 6 m ∈ N m wm = 1 fi (xm ) ≥ fi (xm̂ ) wm = 0
⇒ # 6 j ∈ N ĵ j fi (xĵ ) ≥ wj fi (xj ) ! 6 m = m̂ M 1 wm = 1 $ m ∈ N # & m ∈ N m̂ > 0 '
6. fi 5 - F
* bijm =
cij =
ei =
a =
1,
0,
1,
0,
1,
0,
1,
0,
fi (x ) ≤ (1 − ρ)wj fi (xj ) + ρfi (xm ) + M (1 − wm ),
,
m∈N bijm ≥ N,
,
j∈N cij ≥ 1,
,
x ∈ X ρ i∈K ei ≥ k),
x ∈/ X ρ ,
j, m, ∈ N , i ∈ K,
j, ∈ N , i ∈ K,
∈ N , i ∈ K,
∈ N,
∈ N & a = 1 x ∈ X ρ x ∈ X ρ(
−2M bijm < fi (x ) − (1 − ρ)wj fi (xj ) + ρfi (xm ) + M (1 − wm ) ,
M (1 − bijm ) ≥ fi (x ) − (1 − ρ)wj fi (xj ) + ρfi (xm ) + M (1 − wm ) ,
N cij ≤ m∈N bijm ≤ cij + N − 1,
ei ≤ j∈N cij ≤ N ei ,
ka ≤ i∈K ei ≤ a + k − 1,
bijm , cij , ei , a ∈ {0, 1},
j, m ∈ N , i ∈ K,
j, m ∈ N , i ∈ K,
j ∈ N , i ∈ K,
i ∈ K,
j, m ∈ N , i ∈ K.
.
.
.
.
.
.
* fi : X → R+ i ∈ K x ∈
ρ
ρ
X, ∈ N ρ
X ρ
X \ X . "
?
F . η =
1,
0,
x ∈ P ρ ,
x ∈/ P ρ ,
∈ N,
w + a − 1 ≤ 2η ≤ w + a ,
∈ N,
η ∈ {0, 1},
∈ N,
C
C
" " η = 1 ∈ N x ∈ P ρ x '
ρ η = 0 ∈ N x ∈ X \ P ρ ! ( * $ ) B x ∈ X ρ )
{gs , . . . , gs } 8 X β,ρ ⊆ X ! ,"
-> & )
{β1 g1 , . . . , β2 g2 } p βp = 0 &
: " %
"
. # & 6 (
1
k
∃j ∈ N :
r
k
fi (x ) ≤ (1 − ρ)fi (xj ) − M (1 − wj ) + ρfi (xm ) + M (1 − wm ),
i ∈ K, m ∈ N .
:
p ∈ K̂, m ∈ N .
3
D fi βp gp $ j ∈ N βp gp (x ) ≤ (1 − ρ)βp gp (xj ) − M (1 − wj ) + ρβp gp (xm ) + M (1 − wm ),
E
0
66 p ∈ K̂ j, m, ∈ N " "
" (
bpjm =
cpj =
ep =
a =
1,
0,
1,
0,
1,
0,
1,
0,
βp gp (x ) ≤ (1−ρ)βpgp (xj ) − M (1−wj ) + ρβp gp (xm ) + M (1−wm),
,
m∈N bpjm ≥ N,
,
j∈N cpj ≥ 1,
,
x ∈ X β,ρ p∈K̂ ep ≥ 2k ),
x ∈/ X β,ρ
+
{β1 g1 , . . . , β2
(
k
g2k } ρ . $
−2M bpjm < βp gp (x ) − (1−ρ)βp gp (xj )−M (1−wj )+ρβp gp (xm )+M (1−wm) ,
j, m, ∈ N , p ∈ K̂,
j
m
M (1−bpjm ) ≥ βp gp (x ) − (1−ρ)βp gp (x )−M (1−wj )+ρβp gp (x )+M (1−wm) ,
F
j, m, ∈ N , p ∈ K̂,
F
F
F
F
F
N cpj ≤
ep ≤
k
2 a ≤
m∈N bpjm
j∈N cpj
p∈K̂ ep
≤ cpj + N − 1,
j, ∈ N , p ∈ K̂,
≤ N ep ,
∈ N , p ∈ K̂,
k
≤ a + 2 − 1,
∈ N,
bpjm , cpj , ep , a ∈ {0, 1},
j, m, ∈ N , p ∈ K̂.
0 F η =
1,
0,
x ∈ Pτβ,ρ,
x ∈/ Pτβ,ρ,
∈ N,
w + a − 1 ≤ 2η ≤ w + a ,
∈ N,
η ∈ {0, 1},
∈ N,
-E
-E
η = 1 x ∈ X ρ τ '
x ∈ Pτβ,ρ ∈ N # )& G $ '
ρ 3 0 ρ '
Q " Q = {1, . . . , Q} ! P ρ = x1 , . . . , xQ ! xq ∈ P ρ
x ∈ X dql = ||f (xq ) − f (x )||, q ∈ Q, ∈ N #
$ θ ∈ R+ 3 θ,
θ ≥ min dq , q ∈ Q,
∈N :
η =1
θ ≥ min dq , ∈ N : η = 1,
q∈Q
-
η β τ ! - $ θ,
(1 − η )M + dq , q ∈ Q,
θ ≥ min
--
∈N
θ ≥ min dq − (1 − η )M , ∈ N ,
q∈Q
M ≥ max dq | q ∈ Q, ∈ N ! min
1, θ ≥ (1 − η )M + dq ,
yq =
0, ,
1, θ ≥ dq − (1 − η )M,
zq =
0, ,
q ∈ Q, ∈ N ,
q ∈ Q, ∈ N ,
" 3 θ,
(1 − η )M + dql − θ ≤ 2M (1 − yq ), q ∈ Q,
−(1 − η )M + dql − θ ≤ M (1 − zq ),
yq ≥ 1,
∈N
q∈Q zq ≥ 1,
yq , zq ∈ {0, 1},
τ ≥ 0,
β
(u, v, w, β, τ )
(b, c, e, a)
(w, a, η)
" (6),
" (11),
" (19),
" (20).
∈ N,
q ∈ Q, ∈ N ,
q ∈ Q,
∈ N,
q ∈ Q, ∈ N ,
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
! -6 N 3 2k $ ! N k
* $ $
'
H
)
) ) $ 1 $
$ !
$
$ β τ & ! -6 $ β τ !
τ '
ρ 6 ! $
# $ β τ ! ) C
! A FA # " -C > β ) τ ! " E'
ρ X -: > " τ β
B
$ -6 )
β )
$ τ 9
-
'
0
$ %
β $ ! $ !
! , {x1 , . . . , xN } " fi
sij
ρ̂(fi , fj ) = √
∈ [−1, 1],
sii sjj
1 sij =
fi (x ) −
N
∈N
1
N
fi (x ) fj (x ) −
m
m∈N
fj
->
i, j ∈ K,
1
N
X =
fj (x ) ,
m
i, j ∈ K.
m∈N
! 1 X = {x1 , . . . , xN } # ρ̂(fi , fj ) =
1 '
P 5
G * >- " $
! β $ 1
fi ∈ {f1, . . . , fk } {gs , . . . , gs } 0 + 6 {f1 , . . . , fk } I I fi J ! " r Ks j = 1, . . . , r K fi i ∈ Ks $
1
r
j
j
Δρ
+ 6( 0 ) r = 2 ;
< Δρ ! )& # 2K K !
$ α α
+
α ∈ [−1, 1] " Kα := ∈ K̂ | ρ̂(fi , fj ) ≥ α, ∀i, j ∈ K ,
K ∈ 2K
{gs
-.
∈ K̂ 0 α
{f1 , . . . , fk } 1
,
.
.
.
,
g
}
g
=
s
s
1
r
j
i∈Ks fi sj ∈ K 0
- |Ks |
j
j
¾ 3 ρ̂(f , f ) = 1 - P 4 !! - f (x) = x i j
i
5 "6
" fj (x) = x2 ρ̂(fi , fk ) = ρ̂(fj , fk ) - k ∈ {i, j}!
6
β ∈ {0, 1}|K | gs $ A ∈ Bk×|K | "
fi gs 8 ψp := mini∈K ρ̂(gp , fi ) $ z ∈ R M ≥ 2 !
r α
α
j
j
p
$
z,β
z,
z ≤ ψp + (1 − βp )M, p ∈ Kα ,
-C
Aβ ≥ 1k ,
α
βÌ 1|K | ≤ r,
α
β ∈ {0, 1}|K | ,
r < k +
α = −1 2F4 2k %
α $ -C 0 '
> & + 6
! -C B (z ∗ , β∗ ) ∈ B r p∈Kα
βp∗ = r
8 s = 0 (z s, βs ) -C p∈K
α
βps < r
* r < k ≤ |K | α ∈ [0, 1] i ∈ K i ∈ K fi *
k " Kα K = {} ∈ K ! ψ = 1 = 1, . . . , k $ q ∈ K fq βqs = 0 8
s+1
s+1
s
s+1
β
∈ Kα \ {q} βqs+1 = 1 (z s+1
= z s p p
p = βs+1
, β )s∈ B z
s
= p∈Kα βp + 1 8 s = s + 1 p∈Kα βp = r ! p∈Kα βp
α
α
! )& -C (z ∗ , β∗ ) " B ;
< $ -C (z ∗ , β∗ ) -6 / X '
P β $ + ) ρ ,"
-> x ∈ P β ρ P β ,ρ ⊆ P β $ ! A P0β ,ρ A
τ " τ ∗ %
9 f (P ρ ) f (Pτβ ,ρ ) ! ∗
∗
∗
∗
∗
∗
τ
-:
∗
dH (f (P ρ ), f (Pτβ ,ρ )),
τ ≥ 0.
! $ + >
⎤
X
⎦ =⇒
⎣
ρ
{f1 , . . . , fk }
⎡
-C
=⇒ β
∗
=⇒
Pρ
∗
P β ,ρ
=⇒
-:
=⇒ τ ∗ (β ∗ )
+ >( ! $ -C -:
? τ '
ρ ,"
-> !
1 τ )
-: ! ρ
-: " 0
>
τ ) >
$
( +$ ρ $ Pτβ P ρ
,"
->
∗
∗
Pτβ ,ρ :≈
∗
x ∈ Pτβ
gp (x) ≤ (1 − ρ) max gp (y) + ρ min gp (y), ∀p ∈ Kα .
y∈P
y∈P
-3
+
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-> P ρ ⊆ P * P β ,ρ ⊆ P ρ P β ,ρ = P β ∩ X ρ ⊆ P ∩ X ρ = P ρ " & $
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6
τ Pτβ ,ρ φ1 (τ ) 0
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6-
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66
τ ≥0
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