Conjugate Gradient Method

Copyright (C) 2020 Andreas Kloeckner

MIT License

Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.

import numpy as np
import numpy.linalg as la
import scipy.optimize as sopt
import matplotlib.pyplot as pt

Let's make up a random linear system with an SPD $A$:

np.random.seed(25)
n = 2
Q = la.qr(np.random.randn(n, n))[0]
A = Q @ (np.diag(np.random.rand(n)) @ Q.T)

b = np.random.randn(n)

Here's the objective function for CG:

def phi(xvec):
    x, y = xvec
    return 0.5*(A[0,0]*x*x + 2*A[1,0]*x*y +  A[1,1]*y*y) - x*b[0] - y*b[1]

def dphi(xvec):
    x, y = xvec
    return np.array([
            A[0,0]*x + A[0,1]*y - b[0],
            A[1,0]*x + A[1,1]*y - b[1]
        ])

Here's the function $\phi$ as a "contour plot":

xmesh, ymesh = np.mgrid[-10:10:50j,-10:10:50j]
phimesh = phi(np.array([xmesh, ymesh]))
pt.axis("equal")
pt.contour(xmesh, ymesh, phimesh, 50)

<matplotlib.contour.QuadContourSet at 0x7fc6d591b780>

Running Conjugate Gradients ("CG")

Initialize the method:

x0 = np.array([2, 2./5])
#x0 = np.array([2, 1])

iterates = [x0]
gradients = [dphi(x0)]
directions = [-dphi(x0)]

Evaluate this cell many times in-place:

x = iterates[-1]
s = directions[-1]

def f1d(alpha):
    return phi(x + alpha*s)

alpha_opt = sopt.golden(f1d)
next_x = x + alpha_opt*s

g = dphi(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(phi(next_x))

iterates.append(next_x)

# plot function and iterates
pt.axis("equal")
pt.contour(xmesh, ymesh, phimesh, 50)
it_array = np.array(iterates)
pt.plot(it_array.T[0], it_array.T[1], "x-")

-4.61671051783

[<matplotlib.lines.Line2D at 0x7fc6d5583dd8>]