Newton's method in $n$ dimensions

Copyright (C) 2020 Andreas Kloeckner

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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:

ax = pt.figure().add_subplot(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>

• You may need to add contours to seee more detail.
• The function is not symmetric about the y axis!

Newton

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>]

For comparison: Conjugate Gradients ("CG") -- later in the class

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>]