# coding: utf-8
# # Steepest Descent
# In[1]:
import numpy as np
import numpy.linalg as la
import scipy.optimize as sopt
import matplotlib.pyplot as pt
from mpl_toolkits.mplot3d import axes3d
# Here's a function. It's an oblong bowl made of two quadratic functions.
#
# This is pretty much the easiest 2D optimization job out there.
# In[2]:
def f(x):
return 0.5*x[0]**2 + 2.5*x[1]**2
def df(x):
return np.array([x[0], 5*x[1]])
# Let's take a look at the function. First in 3D:
# In[3]:
fig = pt.figure()
ax = fig.gca(projection="3d")
xmesh, ymesh = np.mgrid[-2:2:50j,-2:2:50j]
fmesh = f(np.array([xmesh, ymesh]))
ax.plot_surface(xmesh, ymesh, fmesh)
# And then as a "contour plot":
# In[4]:
pt.axis("equal")
pt.contour(xmesh, ymesh, fmesh)
# Next, initialize steepest descent with a starting guess:
# In[15]:
guesses = [np.array([2, 2./5])]
# Next, run Steepest Descent:
# In[27]:
x = guesses[-1]
s = -df(x)
def f1d(alpha):
return f(x + alpha*s)
alpha_opt = sopt.golden(f1d)
next_guess = x + alpha_opt * s
guesses.append(next_guess)
print(next_guess)
# Here's some plotting code to illustrate what just happened:
# In[28]:
pt.axis("equal")
pt.contour(xmesh, ymesh, fmesh, 50)
it_array = np.array(guesses)
pt.plot(it_array.T[0], it_array.T[1], "x-")
# In[ ]: