python - Scipy optimize.minimize function

Question

I try to solve nonlinear programming task using scipy.optimize.minimize

max r
x1**2 + y1**2 <= (1-r)**2
(x1-x2)**2 + (y1-y2)**2 >= 4*r**2
0 <= r <= 1

So I've wrote next code:

r = np.linspace(0, 1, 100)
x1 = np.linspace(0, 1, 100)
y1 = np.linspace(0, 1, 100)
x2 = np.linspace(0, 1, 100)
y2 = np.linspace(0, 1, 100)


fun = lambda r: -r
cons = ({'type': 'ineq',
     'fun': lambda x1, r: [x1[0] ** 2 + x1[1] ** 2 - (1 - r) ** 2],
     'args': (r,)},
    {'type': 'ineq',
     'fun': lambda x2, r: [x2[0] ** 2 + x2[1] ** 2 - (1 - r) ** 2],
     'args': (r,)},
    {'type': 'ineq',
     'fun': lambda x1, x2, r: [(x1[0] - x2[0]) ** 2 + (x1[1] - x2[1]) ** 2 - 4 * r ** 2],
     'args': (x2, r,)})
bnds = ((0, 1), (-1, 1), (-1, 1), (-1, 1), (-1, 1))
x0 = [0, 0, 0, 0, 0]
minimize(fun, x0, bounds=bnds, constraints=cons)

But I've got next error

File "C:Anaconda2libsite-packagesscipyoptimizeslsqp.py", line 377, in _minimize_slsqp
c = concatenate((c_eq, c_ieq))
ValueError: all the input arrays must have same number of dimensions

Please, help me to find out my mistakes and write correct code

UPD: Thx to @unutbu i've understand how to build it correctly.

fun = lambda x: -x[0]
cons = ({'type': 'ineq',
     'fun': lambda x: -x[1] ** 2 - x[2] ** 2 + (1 - x[0]) ** 2},
    {'type': 'ineq',
     'fun': lambda x: -x[3] ** 2 - x[4] ** 2 + (1 - x[0]) ** 2},
    {'type': 'ineq',
     'fun': lambda x: (x[1] - x[3]) ** 2 + (x[1] - x[4]) ** 2 - 4 * x[0] ** 2})
bnds = ((0, 1), (-1, 1), (-1, 1), (-1, 1), (-1, 1))
x0 = [0.5, 0.3, 0.5, 0.3, 0.5]
answer = minimize(fun, x0, bounds=bnds, constraints=cons)

In task of minimization we have to lead constraints to such form:

g(x) >= 0

that's why constraints look like in that way.

Answer 1

Your parameter space appears to be 5-dimensional. A point in your parameter space would be z = (r, x1, y1, x2, y2). Therefore the function to be minimized -- and also the constraint functions -- should accept a point z and return a scalar value.

Thus instead of

fun = lambda r: -r

use

def func(z):
    r, x1, y1, x2, y2 = z
    return -r

and instead of

lambda x1, r: [x1[0] ** 2 + x1[1] ** 2 - (1 - r) ** 2]

use

def con1(z):
    r, x1, y1, x2, y2 = z
    return x1**2 + y1**2 - (1-r)**2

and so on.

---

Note that simple constraints such as 0 <= r <= 1 can be handled by setting the bounds parameter instead of defining a constraint. And if the bounds for x1, y1, x2, y2 are from -1 to 1, then you might also want change

x1 = np.linspace(0, 1, 100)
...

to

x1 = np.linspace(-1, 1, 100)
...

However, the arrays r, x1, y1, x2, y2 are not needed to minimize func, so you could just as well eliminate them from the script entirely.

---

import numpy as np
import scipy.optimize as optimize

"""
max r
x1**2 + y1**2 <= (1-r)**2
(x1-x2)**2 + (y1-y2)**2 >= 4*r**2
0 <= r <= 1
"""

def func(z):
    r, x1, y1, x2, y2 = z
    return -r

def con1(z):
    r, x1, y1, x2, y2 = z
    return x1**2 + y1**2 - (1-r)**2

def con2(z):
    r, x1, y1, x2, y2 = z
    return 4*r**2 - (x1-x2)**2 - (y1-y2)**2 

cons = ({'type': 'ineq', 'fun': con1}, {'type': 'ineq', 'fun': con2},)
bnds = ((0, 1), (-1, 1), (-1, 1), (-1, 1), (-1, 1))
guess = [0, 0, 0, 0, 0]
result = optimize.minimize(func, guess, bounds=bnds, constraints=cons)
print(result)

yields

     fun: -1.0
     jac: array([-1.,  0.,  0.,  0.,  0.,  0.])
 message: 'Optimization terminated successfully.'
    nfev: 14
     nit: 2
    njev: 2
  status: 0
 success: True
       x: array([ 1.,  0.,  0.,  0.,  0.])