# 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[ ]: