# 2.7.4.8. Constraint optimization: visualizing the geometry¶

A small figure explaining optimization with constraints

```import numpy as np
import matplotlib.pyplot as plt
import scipy as sp

x, y = np.mgrid[-2.9:5.8:0.05, -2.5:5:0.05]
x = x.T
y = y.T

for i in (1, 2):
# Create 2 figure: only the second one will have the optimization
# path
plt.figure(i, figsize=(3, 2.5))
plt.clf()
plt.axes([0, 0, 1, 1])

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

plt.text(-0.9, 4.4, "\$x_2\$", size=20)
plt.text(5.6, -0.6, "\$x_1\$", size=20)
plt.axis("equal")
plt.axis("off")

# And now plot the optimization path
accumulator = []

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))

sp.optimize.minimize(
f, np.array([0, 0]), method="L-BFGS-B", bounds=((-1.5, 1.5), (-1.5, 1.5))
)

accumulated = np.array(accumulator)
plt.plot(accumulated[:, 0], accumulated[:, 1])

plt.show()
```

Total running time of the script: (0 minutes 0.079 seconds)

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