Auteur: Alexandre Gramfort, Joseph Salmon joseph.salmon@umontpellier.fr
On cherche à minimiser la fonctiont suivante:
$f(x_1, x_2) = (x_1^2 + x_2 - 11)^2 + (x_1 + x_2^2 - 7)^2$
Question : est-ce une fonction convexe?
import numpy as np
def f(x):
x1, x2 = x
return (x1**2 + x2 - 11)**2 + (x1 + x2**2 - 7)**2
%matplotlib notebook
import matplotlib.pyplot as plt
import matplotlib
cmap_reversed = matplotlib.cm.get_cmap('RdBu_r')
from mpl_toolkits import mplot3d
X1, X2 = np.meshgrid(np.linspace(-5.5, 5.5, 50),
np.linspace(-5.5, 5.5, 50))
Z = f([X1, X2]) # Altitude
fig = plt.figure(figsize=(6, 6))
ax = plt.axes(projection='3d')
ax.plot_surface(X1, X2, Z, rstride=1, cstride=1,
cmap=cmap_reversed, edgecolor='none')
ax.set_xlim(-5, 5)
ax.set_ylim(-5, 5)
ax.set_zlim(0, 500)
plt.show()
def my_plot(xs=None, cmap=cmap_reversed):
levels = list(1.7 ** np.linspace(0, 10, 30) - 1.) + [300]
plt.figure(figsize=(5, 5))
plt.contourf(X1, X2, np.sqrt(Z), levels=np.sqrt(
levels), cmap=cmap)
plt.colorbar(extend='both')
if xs is not None:
x1, x2 = np.array(xs).T
plt.plot(x1, x2, 'k')
plt.plot(x1, x2, 'o', color='purple')
plt.show()
plot()
plot(cmap=matplotlib.cm.get_cmap('RdBu'))
def f_grad(x):
x1, x2 = x
df_x1 = 2 * (-7 + x1 + x2**2 + 2 * x1 * (-11 + x1**2 + x2))
df_x2 = 2 * (-11 + x1**2 + x2 + 2 * x2 * (-7 + x1 + x2**2))
return np.array([df_x1, df_x2])
x0 = [0.5, -4]
def grad_descent(x_init=x0, step_size=0.01, max_iter=20):
"""Descente de gradient avec un pas constant"""
x = x_init
xs = [x]
for k in range(max_iter):
x = ### XXXX
xs.append(x)
my_plot(xs)
plt.show()
grad_descent(x_init=x0, step_size=0.01, max_iter=20)
from ipywidgets import interact, fixed
interact(grad_descent, x_init=fixed(x0), step_size=(0., .05, 0.005), max_iter=(0, 50, 1))