Copyright (C) 2020 Andreas Kloeckner
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 are two functions. The first one is an oblong "bowl-shaped" one made of quadratic functions.
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]])
def ddf(x):
return np.array([
[1,0],
[0,5]
])
The second one is a challenge problem for optimization algorithms known as Rosenbrock's banana function.
def f(X):
x = X[0]
y = X[1]
val = 100.0 * (y - x**2)**2 + (1.0 - x)**2
return val
def df(X):
x = X[0]
y = X[1]
val1 = 400.0 * (y - x**2) * x - 2 * x
val2 = 200.0 * (y - x**2)
return np.array([val1, val2])
def ddf(X):
x = X[0]
y = X[1]
val11 = 400.0 * (y - x**2) - 800.0 * x**2 - 2
val12 = 400.0
val21 = -400.0 * x
val22 = 200.0
return np.array([[val11, val12], [val21, val22]])
Let's take a look at these functions. First in 3D:
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,
alpha=0.3, cmap=pt.cm.coolwarm, rstride=3, cstride=3)
<mpl_toolkits.mplot3d.art3d.Poly3DCollection at 0x7f04fcb90160>
Then as a "contour plot":
pt.axis("equal")
pt.contour(xmesh, ymesh, fmesh, 50)
<matplotlib.contour.QuadContourSet at 0x7f04fdd7c780>
First, initialize:
guesses = [np.array([2, 2./5])]
Then evaluate this cell lots of times:
x = guesses[-1]
s = la.solve(ddf(x), df(x))
next_guess = x - s
print(f(next_guess), next_guess)
guesses.append(next_guess)
0.0 [ 0. 0.]
Here's some plotting code to see what's going on:
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-")
[<matplotlib.lines.Line2D at 0x7f04fc7b6dd8>]
Initialize the method:
x0 = np.array([2, 2./5])
#x0 = np.array([2, 1])
iterates = [x0]
gradients = [df(x0)]
directions = [-df(x0)]
Evaluate this cell many times in-place:
# Evaluate this cell many times in-place
x = iterates[-1]
s = directions[-1]
def f1d(alpha):
return f(x + alpha*s)
alpha_opt = sopt.golden(f1d)
next_x = x + alpha_opt*s
g = df(next_x)
last_g = gradients[-1]
gradients.append(g)
beta = np.dot(g, g)/np.dot(last_g, last_g)
directions.append(-g + beta*directions[-1])
print(f(next_x))
iterates.append(next_x)
# plot function and iterates
pt.axis("equal")
pt.contour(xmesh, ymesh, fmesh, 50)
it_array = np.array(iterates)
pt.plot(it_array.T[0], it_array.T[1], "x-")
7.81186907775e-19
[<matplotlib.lines.Line2D at 0x7f04fc66e6d8>]