Geometric Facility Location Problems

Thus we have d ­ unknowns a 1
, a 2
, …, a d
. They are solved from the following
system of equations:
∂E
∂a
1
∂E
= 0,
∂a
2
∂E
= 0,...,
∂
a d
= 0
For L 1
approximation of the problem we have to
Minimize
n
∑
i=
1
| b
d
−∑
j=
1
a
j
x
i
j
| =
E
This absolute value function is not a linear function. But we can easily
convert it to a standard LP problem as follows [Chvatal]:
Minimize
Subject to:
n
∑
i=
1
b
e i
d
−∑
j=1
−b
+
d
j=1
a
∑
j
a
j
x
x
i
j
i
j
≤ e
i
≤ e
i
i =1,2 ,...,
i =1,2 ,...,
n
n