Geoscience Reference
In-Depth Information
the piecewise polynomials that appear in
OMPF
as polynomials in the new set of
variables.
X
m
X
m
D
minimize
j
f
i
.
u
/
w
ij
WD
p
.x;
u
;v;
w
/
(10.13)
jD1
iD1
m
X
subject to
w
ij
D
1; for i
D
1;:::;m;
(10.14)
jD1
X
m
w
ij
D
1; for j
D
1;:::;m;
(10.15)
iD1
X
m
X
m
w
ij
f
i
.
u
/
w
ij
C1
f
i
.
u
/;j
D
1;:::;m
1; (10.16)
iD1
iD1
w
ij
w
ij
D
0; for i;j
D
1;:::;m;
(10.17)
v
2
k`
D
.x
`
a
k`
/
2r
;k
D
1;:::;n; `
D
1;:::;d; (10.18)
X
d
u
k
D
.
v
k`
/
s
;k
D
1;:::;n;
(10.19)
`D1
X
m
w
ij
1; i
D
1;:::;m;
(10.20)
jD1
d
X
v
ij
M
2
;i
D
1;:::;n;
(10.21)
jD1
w
ij
2
R
;
8
i;j
D
1;:::;m;
(10.22)
v
k`
0;
u
k
0;k
D
1;:::;n;`
D
1;:::;d;
(10.23)
x
2
K
:
(10.24)
By means of the
w
variables, the objective function (
10.13
) is the ordered
weighted sum of the f
i
polynomials which can be written as the polynomial p
.
The first set of constraints (
10.14
) ensures that for each x, f
i
.x/ is sorted in a unique
position. The second set (
10.15
) ensures that the jth position is only assigned to one
polynomial function. The next constraints (
10.16
) state that f
.1/
.
u
/
f
.m/
.
u
/.
Constraints (
10.17
) are added to assure that
w
ij
2f
0;1
g
. Next, the two families of
constraints (
10.18
)and(
10.19
)set
u
k
as the correct value of
k
a
k
x
k
(recall that
D
r=s). The last set of constraints (
10.2
0
)
and(
10.21
) ensure that Archimedean
property holds for the new feasible region
K
of the above auxiliary problem. (Note
that this last set of constraints are redundant but it is convenient to add them for a
better description of the feasible set.)
