Geometric Facility Location Problems

Given a set of data points (x 1
, y 1
), (x 2
, y 2
), …., (x n
, y n
) we have to find the
equation of the straight line y = ax + b that minimizes the L 2
error, i.e.,
Minimize
Here the variables are in fact a and b. So we have two unknowns and we
require at least two equations to solve. They are obtained from the minima
criterion after taking partial derivatives w.r.t. a and b respectively:
The result can be extended for any dimensional data. For example in d­
dimensions we have to find the Regression Hyper­plane a 1
x 1
+ a 2
x 2
+ … +
a d
x d
= b. Thus we have to
Minimize
n
∑
i=
1
2
[ yi − ( axi
+ b)]
=
n
∑
i=
1
d
[ b − ∑
j=
1
a
j
∂E
∂a
x
i
j
]
2
E
∂E
= 0 , = 0
∂b
=
E