Geoscience Reference
In-Depth Information
v
3
D
.
1;0/, v
4
D
.
2;
1/ and v
5
D
.1;
2
/. In the dual interpretation these
points are transferred to the five lines in the right part of the figure.
L
1
D
H
0;1;
2
Df
.x
1
;x
2
/
W
x
2
D
1
2
g
L
2
D
H
0;1;1
Df
.x
1
;x
2
/
W
x
2
D
1
g
L
3
D
H
1;1;0
Df
.x
1
;x
2
/
W
x
2
D
x
1
g
L
4
D
H
2;1;1
Df
.x
1
;x
2
/
W
x
2
D
2x
1
C
1
g
L
5
D
H
1;1;
2
Df
.x
1
;x
2
/
W
x
2
D
x
1
C
1
2
g
It can also be seen that the line H
through the two points v
1
and v
3
is
transformed to the point v
D
.
2
;
2
/ in dual space which lies on the intersection
of L
1
and L
3
. Furthermore, note that in the point .
1;
1/ in dual space three of
the lines meet, namely, L
2
;L
3
; and L
4
. Hence, this point corresponds to the line
H
1;1;1
Df
.x
1
;x
2
/
W
x
2
D
x
1
C
1
g
which passes through the three points v
2
;v
3
;
and v
4
.
1
2
;1;
1
2
7.3.3
The Minsum Hyperplane Location Problem
Let us now start with the
minsum hyperplane location problem
defined as follows:
Given a set of existing points V
Df
v
1
;:::;v
n
g
D
with positive weights
w
j
>
0;j
D
1;:::;n, find a hyperplane H
a;b
which minimizes
R
X
n
f
1
.H
a;b
/
D
w
j
d.H
a;b
;v
j
/:
jD1
A hyperplane H minimizing f
1
.H/ is called
minsum hyperplane w.r.t the distance
d. Let us assume throughout this section that there are n>Daffinely independent
points, otherwise an optimal solution is the hyperplane containing all of them.
7.3.3.1
Minsum Hyperplane Location with Vertical Distance
We first look at the problem with vertical distance d
ver
. As explained after
Lemma
7.1
we may without loss of generality assume that a
D
D
1. This simplifies
the problem formulation to the question of finding a
1
;:::;a
D1
;b
2
R
such that
n
X
w
j
j
v
t
j
a
C
b
j
f
1
.a;b/
D
(7.4)
jD1
