How to Find the Global Maximum of (1‑x³)·sin(3x) Using Python and SciPy

1 Find the Global Optimum

1.1 Problem

Find the maximum value of the function f(x) = (1 - x³)·sin(3x).

1.2 Solution

To aid understanding, first plot the function.

<code>import numpy as np
from matplotlib.pyplot import rc, plot, show, savefig
from scipy.optimize import fminbound, fmin
rc('font', size=16)
fx = lambda x: (1 - x**3) * np.sin(3 * x)
x0 = np.linspace(-2 * np.pi, 2 * np.pi, 100)
y0 = fx(x0)
plot(x0, y0)
show()
</code>

The plot shows the function reaches its maximum near x = -6 and x = 6, with a value close to 195. Using SciPy’s fminbound to solve yields:

<code>from scipy.optimize import fminbound
xm2 = fmin(lambda x: -fx(x), -2 * np.pi)
ym2 = fx(xm2)
print(xm2, ym2, '\n--------------')
</code>

Result:

-3.7505026680508093 52.00462222535155

This only finds a local maximum at x ≈ -3.75 with value ≈ 52, which is incorrect because fminbound can become trapped in local extrema, a common difficulty for many optimization algorithms.

Using a random sampling approach—generating 100 random points in the interval—produces a maximum around 194, much closer to the true global maximum.

<code>x = np.random.uniform(-2 * np.pi, 2 * np.pi, 100)
y = fx(x)
ym = y.max()
xm = x[y == ym]
print(xm, ym)
</code>

Result:

[-5.81863236] 194.9001725139532