2.7.4.11. Alternating optimization

The challenge here is that Hessian of the problem is a very ill-conditioned matrix. This can easily be seen, as the Hessian of the first term in simply 2*np.dot(K.T, K). Thus the conditioning of the problem can be judged from looking at the conditioning of K.

import time

import numpy as np
from scipy import optimize
import pylab as pl

np.random.seed(0)

K = np.random.normal(size=(100, 100))

def f(x):
    return np.sum((np.dot(K, x - 1))**2) + np.sum(x**2)**2


def f_prime(x):
    return 2*np.dot(np.dot(K.T, K), x - 1) + 4*np.sum(x**2)*x


def hessian(x):
    H = 2*np.dot(K.T, K) + 4*2*x*x[:, np.newaxis]
    return H + 4*np.eye(H.shape[0])*np.sum(x**2)

Some pretty plotting

pl.figure(1)
pl.clf()
Z = X, Y = np.mgrid[-1.5:1.5:100j, -1.1:1.1:100j]
# Complete in the additional dimensions with zeros
Z = np.reshape(Z, (2, -1)).copy()
Z.resize((100, Z.shape[-1]))
Z = np.apply_along_axis(f, 0, Z)
Z = np.reshape(Z, X.shape)
pl.imshow(Z.T, cmap=pl.cm.gray_r, extent=[-1.5, 1.5, -1.1, 1.1],
          origin='lower')
pl.contour(X, Y, Z, cmap=pl.cm.gnuplot)

# A reference but slow solution:
t0 = time.time()
x_ref = optimize.minimize(f, K[0], method="Powell").x
print('     Powell: time %.2fs' % (time.time() - t0))
f_ref = f(x_ref)

# Compare different approaches
t0 = time.time()
x_bfgs = optimize.minimize(f, K[0], method="BFGS").x
print('       BFGS: time %.2fs, x error %.2f, f error %.2f' % (time.time() - t0,
    np.sqrt(np.sum((x_bfgs - x_ref)**2)), f(x_bfgs) - f_ref))

t0 = time.time()
x_l_bfgs = optimize.minimize(f, K[0], method="L-BFGS-B").x
print('     L-BFGS: time %.2fs, x error %.2f, f error %.2f' % (time.time() - t0,
    np.sqrt(np.sum((x_l_bfgs - x_ref)**2)), f(x_l_bfgs) - f_ref))


t0 = time.time()
x_bfgs = optimize.minimize(f, K[0], jac=f_prime, method="BFGS").x
print("  BFGS w f': time %.2fs, x error %.2f, f error %.2f" % (
    time.time() - t0, np.sqrt(np.sum((x_bfgs - x_ref)**2)),
    f(x_bfgs) - f_ref))

t0 = time.time()
x_l_bfgs = optimize.minimize(f, K[0], jac=f_prime, method="L-BFGS-B").x
print("L-BFGS w f': time %.2fs, x error %.2f, f error %.2f" % (
    time.time() - t0, np.sqrt(np.sum((x_l_bfgs - x_ref)**2)),
    f(x_l_bfgs) - f_ref))

t0 = time.time()
x_newton = optimize.minimize(f, K[0], jac=f_prime, hess=hessian, method="Newton-CG").x
print("     Newton: time %.2fs, x error %.2f, f error %.2f" % (
    time.time() - t0, np.sqrt(np.sum((x_newton - x_ref)**2)),
    f(x_newton) - f_ref))

pl.show()

Out:

Powell: time 0.26s
       BFGS: time 5.86s, x error 10.39, f error 31006.85
     L-BFGS: time 0.05s, x error 0.02, f error -0.02
  BFGS w f': time 5.87s, x error 10.39, f error 31006.85
L-BFGS w f': time 0.00s, x error 0.02, f error -0.02
     Newton: time 0.00s, x error 6.24, f error 8444.13

Total running time of the script: ( 0 minutes 12.396 seconds)