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

numpy, python, scipy, sympy

Solution

If I understand your question correctly, I think this is what you're after:

from scipy.optimize import minimize
import numpy as np

def f(coord, x, y, r):
    return np.sum(((coord[0] - x) ** 2) + ((coord[1] - y) ** 2) - (r ** 2))

x = np.array([0, 2, 0])
y = np.array([0, 0, 2])
r = np.array([.88, 1, .75])

# initial (bad) guess at (x,y) values
initial_guess = np.array([100, 100])

res = minimize(f, initial_guess, args=(x, y, r))

Which yields:

>>> print res.x
[0.66666665 0.66666665]

You might also try the least squares method which expects an objective function that returns a vector. It wants to minimize the sum of the squares of this vector. Using least squares, your objective function would look like this:

def f2(coord, x, y, r):
    # notice that we're returning a vector of dimension 3
    return ((coord[0] - x) ** 2) + ((coord[1] - y) ** 2) - (r ** 2)

And you'd minimize it like so:

from scipy.optimize import leastsq
res = leastsq(f2, initial_guess, args=(x, y, r))

Which yields:

>>> print res[0]
>>> [0.77958134 0.8580946 ]

This is basically the same as using `minimize` and re-writing the original objective function as:

def f(coord, x, y, r):
    vec = ((coord[0] - x) ** 2) + ((coord[1] - y) ** 2) - (r ** 2)
    # return the sum of the squares of the vector
    return np.sum(vec ** 2)

This yields:

>>> print res.x
>>> [0.77958326 0.85809648]

Note that `args` are handled a bit differently with `leastsq`, and that the data structures returned by the two functions are also different. See the documentation for `scipy.optimize.minimize` and `scipy.optimize.leastsq` for more details.

See the `scipy.optimize` documentation for more optimization options.

Problem

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-y3)^2 - r3^2 = 0 ``` Given the values a point(x,y) and a radius (r): ``` x1, y1, r1 = (0, 0, 0.88) x2, y2, r2 = (2, 0, 1) x3, y3, r3 = (0, 2, 0.75) ``` What is the best way to find the optimal solution for the point (x,y) Using the above example it would be: ~ (1, 1)

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