Numpy: Linear system with specific conditions. No negative solutions

Question

I\'m writing a Python code using numpy. In my code I use \"linalg.solve\" to solve a linear system of n equations in n variables. Of course the solutions could be either positiv

Answer 1

You have a minimization problem, i.e.

min ||Ax - b||
s.t. x_i >= 0 for all i  in [0, n-1]

You can use the Optimize module from Scipy

import numpy as np
from scipy.optimize import minimize

A = np.array([[1., 2., 3.],[4., 5., 6.],[7., 8., 10.]], order='C')
b = np.array([6., 12., 21.])
n = len(b)

# Ax = b --> x = [1., -2., 3.]

fun = lambda x: np.linalg.norm(np.dot(A,x)-b)
# xo = np.linalg.solve(A,b)
# sol = minimize(fun, xo, method='SLSQP', constraints={'type': 'ineq', 'fun': lambda x:  x})
sol = minimize(fun, np.zeros(n), method='L-BFGS-B', bounds=[(0.,None) for x in xrange(n)])

x = sol['x'] # [2.79149722e-01, 1.02818379e-15, 1.88222298e+00]

With your method I get x = [ 0.27272727, 0., 1.90909091].

In the case you still want to use your algorithm, it is below

n = len(b)
x = np.linalg.solve(A,b)
pos = np.where(x>=0.)[0]

while len(pos) < n:
    Ap = A[pos][:,pos]
    bp = b[pos]
    xp = np.linalg.solve(Ap, bp)
    x = np.zeros(len(b))
    x[pos] = xp
    pos = np.where(x>=0.)[0]

But I don't recommend you to use it, you should use the minimize option.