How to perform non-linear optimization with scipy/numpy or sympy?

Question

I am trying to find the optimal solution to the follow system of equations in Python:

(x-x1)^2 + (y-y1)^2 - r1^2 = 0
(x-x2)^2 + (y-y2)^2 - r2^2 = 0
(x-x3)^2 + (y         


        

Answer 1

I made an example script by the following. Note that the last line will find an optimal solution (a,b):

import numpy as np
import scipy as scp
import sympy as smp
from scipy.optimize import minimize

a,b = smp.symbols('a b')
x_ar, y_ar = np.random.random(3), np.random.random(3)
x = np.array(smp.symbols('x0:%d'%np.shape(x_ar)[0]))
y = np.array(smp.symbols('y0:%d'%np.shape(x_ar)[0]))
func = np.sum(a**2+b**2-x*(a+b)+2*y)
print func
my_func = smp.lambdify((x,y), func)
print 1.0/3*my_func(x_ar,y_ar)
ab = smp.lambdify((a,b),my_func(x_ar,x_ar))
print ab(1,2)

def ab_v(x):
   return ab(*tuple(x))

print ab_v((1,2))

minimize(ab_v,(0.1,0.1))

The outputs are :

3*a**2 + 3*b**2 - x0*(a + b) - x1*(a + b) - x2*(a + b) + 2*y0 + 2*y1 + 2*y2
1.0*a**2 - 0.739792011558683*a + 1.0*b**2 - 0.739792011558683*b    +0.67394435712335


12.7806239653
12.7806239653
Out[33]:
  status: 0
 success: True
 njev: 3
 nfev: 12
 hess_inv: array([[1, 0],
   [0, 1]])
 fun: 3.6178137388030356
 x: array([ 0.36989601,  0.36989601])
 message: 'Optimization terminated successfully.'
 jac: array([  5.96046448e-08,   5.96046448e-08])