Non-negative least squares for underdetermined system

Question

Consider the following problem:

Find: x_1, x_2, x_3 > 0 such that

67.5 = 60*x_1 +  90*x_2 + 120*x_3  
60   = 30*x_1 + 120*x_2 +  90*x_3

Is there a way to solve this equation in Python? Perhaps with scipy.nnls()?

Answer 1

Using sympy to solve the equation set symbolically

from sympy import * 

x_1, x_2, x_3 = symbols('x_1 x_2 x_3')

res = solve([Eq(60*x_1+90*x_2+120*x_3, 67.5),
             Eq(30*x_1+120*x_2+90*x_3, 60)],
             [x_1, x_2, x_3])
print res
#{x_1: -1.4*x_3 + 0.6, x_2: -0.4*x_3 + 0.35}

using scipy.optimize.nnls

import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import nnls 

A = np.array([[60, 90, 120], 
              [30, 120, 90]])

b = np.array([67.5, 60])

x, rnorm = nnls(A,b)

print x
#[ 0.          0.17857143  0.42857143]
print rnorm
#0.0

Altough this only promises a solution where the parameters are x>=0 so you can get zeros, as you did for this example.