# Optimization with scipy.optimize¶

In [27]:
%pylab inline
import numpy as np
from scipy import optimize

Populating the interactive namespace from numpy and matplotlib

/home/fabian/anaconda3/lib/python3.6/site-packages/IPython/core/magics/pylab.py:160: UserWarning: pylab import has clobbered these variables: ['f']
%matplotlib prevents importing * from pylab and numpy
"\n%matplotlib prevents importing * from pylab and numpy"


# A simple example¶

In [28]:
def f(x):   # The rosenbrock function
return .5*(1 - x[0])**2 + (x[1] - x[0]**2)**2

def fprime(x):
return np.array((-2*.5*(1 - x[0]) - 4*x[0]*(x[1] - x[0]**2), 2*(x[1] - x[0]**2)))

In [46]:
# plot the function f(x)
x = np.linspace(-1, 2, 100)
y = np.linspace(-2, 10, 100)

X, Y = np.meshgrid(x, y)
Z = np.zeros((x.size, y.size))
for i in range(x.size):
for j in range(y.size):
Z[i, j] = f((x[i], y[j]))

plt.contour(X, Y, Z.T, 100);
plt.colorbar()
plt.show()

In [47]:
# using scipy.optimize is easy when we have access to f
optimize.minimize(f, [2, 2], method='BFGS')

Out[47]:
      fun: 1.78380307372662e-11
hess_inv: array([[0.95489061, 1.90006631],
[1.90006631, 4.27872379]])
jac: array([9.88094725e-07, 2.41748897e-06])
message: 'Optimization terminated successfully.'
nfev: 36
nit: 8
njev: 9
status: 0
success: True
x: array([1.00000573, 1.00001265])
In [48]:
# in this case scipy.optimize will estimate the gradient using finite differences
# to speed up convergence, we can provide him with a gradient using the keyword 'jac':
optimize.minimize(f, [2, 2], jac=fprime, method='BFGS')

Out[48]:
      fun: 1.8414093407262628e-11
hess_inv: array([[0.95489113, 1.90006768],
[1.90006768, 4.27872719]])
jac: array([9.88085521e-07, 2.41739812e-06])
message: 'Optimization terminated successfully.'
nfev: 9
nit: 8
njev: 9
status: 0
success: True
x: array([1.00000582, 1.00001285])

# Optimization with constraints¶

scipy can deal efficiently with some constraints. In particular, the L-BFGS implementation can deal with box constraints, hence the name L-BFGS-B (B for Box constraints). The L stands for Limited memory: instead of using the full Hessian, we will keep an low rank representation as estimated by the last (typically) 20 gradients.

In [49]:
import numpy as np
import pylab as pl
from scipy import optimize

x, y = np.mgrid[-2.9:5.8:.05, -2.5:5:.05]
x = x.T
y = y.T

# Create 2 figure: only the second one will have the optimization
# path
pl.clf()
pl.axes([0, 0, 1, 1])

contours = pl.contour(np.sqrt((x - 3)**2 + (y - 2)**2),
extent=[-3, 6, -2.5, 5],
cmap=pl.cm.gnuplot)
pl.clabel(contours,
inline=1,
fmt='%1.1f',
fontsize=14)
pl.plot([-1.5, -1.5,  1.5,  1.5, -1.5],
[-1.5,  1.5,  1.5, -1.5, -1.5], 'k', linewidth=2)
pl.fill_between([ -1.5,  1.5],
[ -1.5, -1.5],
[  1.5,  1.5],
color='.8')
pl.axvline(0, color='k')
pl.axhline(0, color='k')

pl.text(-.9, 4.4, '$x_2$', size=20)
pl.text(5.6, -.6, '$x_1$', size=20)
pl.axis('equal')
pl.axis('off')

Out[49]:
(-3.0, 6.0, -2.5, 5.0)
In [50]:
def f(x):
# Store the list of function calls
return np.sqrt((x[0] - 3)**2 + (x[1] - 2)**2)

# We don't use the gradient, as with the gradient, L-BFGS is too fast,
# and finds the optimum without showing us a pretty path
def f_prime(x):
r = np.sqrt((x[0] - 3)**2 + (x[0] - 2)**2)
return np.array(((x[0] - 3)/r, (x[0] - 2)/r))

In [51]:
optimize.minimize(f, np.array([0, 0]), method='L-BFGS-B',
bounds=((-1.5, 1.5), (-1.5, 1.5)))

Out[51]:
      fun: 1.5811388300841898
hess_inv: <2x2 LbfgsInvHessProduct with dtype=float64>
jac: array([-0.94868331, -0.31622778])
x: array([1.5, 1.5])