Question 2
a
We have seen in class that a 
norm penalization has the property to provide results with a lot of entries equal to zero. We exploit this property of the 
norm to study spatial traffic patterns. For this we study the following optimization problem:
b
The matrix 
represents information on the traffic patterns. For example, we can expect some neighboring links to have the same traffic conditions. In particular, we can encourage neighboring links to have the same speed. We realize that two links 
and 
with travel times 
and 
and length 
and 
have the same speed if and only if 
c
In this section, we will experiment our traffic pattern with different K matrices. K matrix is produced as follows: For every intersection u, there are incoming links 
and outgoing links 
. For every (
,
) pair, there is a row at index i with value 1/
and at index o with value -1/
.