cvxpy contrained normalization equations (abs)

Question

I am working in an optimization problem (A*v = b) where I would like to rank a set of alternatives X = {x1,x2,x3,x4}. However, I have the following normalization constraint: |v[i] - v[j]| <= 1, which can be in the form -1 <= v[i] - v[j] <= 1. My code is as follows:

import cvxpy as cp

n = len(X) #set of alternatives
v = cp.Variable(n)
objective = cp.Minimize(cp.sum_squares(A*v - b))
constraints = [0 <= v]
#Normalization condition -1 <= v[i] - v[j] <= 1
for i in range(n):
    for j in range(n):
        constraints = [-1 <= v[i]-v[j], 1 >= v[i]-v[j]]
prob = cp.Problem(objective, constraints)
# The optimal objective value is returned by `prob.solve()`.
result = prob.solve()
# The optimal value for v is stored in `v.value`.
va2 = v.value

Which outputs:

[-0.15  0.45 -0.35  0.05]

Result, which is not close to what should be and even have negative values. I think, my code for the normalization contraint most probably is wrong.

Answer 1

You are not appending your constraints, instead you are overwriting them each time. Instead of this line

constraints = [-1 <= v[i]-v[j], 1 >= v[i]-v[j]]

You should have

constraints += [-1 <= v[i]-v[j], 1 >= v[i]-v[j]]

For cleanliness you may want to change this

for i in range(n):
    for j in range(n):

To only consider each pair once:

for i in range(n):
    for j in range(i+1, n):