Constraint optimization: visualizing the geometryΒΆ

A small figure explaining optimization with constraints

Python source code: plot_constraints.py

```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

for i in (1, 2):
# Create 2 figure: only the second one will have the optimization
# path
pl.figure(i, figsize=(3, 2.5))
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')

# And now plot the optimization path
accumulator = list()

def f(x):
# Store the list of function calls
accumulator.append(x)
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))