Branin function

\[f(x) = (x_2 - \frac{5.1}{4\pi^2}x_1^{2} + \frac{5}{\pi}x_1 - 6)^2 + 10(1-\frac{1}{8\pi})\cos(x_1) + 10,\]

where \(x = (x_1, x_2)\) with \(-5\leq x_1 \leq 10, 0 \leq x_2 \leq 15\)

The Branin function has three global minima:

\(f(x^*) = 0.397887\), at \(x^* = (-\pi, 12.275), (\pi, 2.275)\) and \((9.42478, 2.475)\)

Usage

import numpy as np
import matplotlib.pyplot as plt

from smt.problems import Branin

ndim = 2
problem = Branin(ndim=ndim)

num = 100
x = np.ones((num, ndim))
x[:, 0] = np.linspace(-5.0, 10.0, num)
x[:, 1] = np.linspace(0.0, 15.0, num)
y = problem(x)

yd = np.empty((num, ndim))
for i in range(ndim):
    yd[:, i] = problem(x, kx=i).flatten()

print(y.shape)
print(yd.shape)

plt.plot(x[:, 0], y[:, 0])
plt.xlabel("x")
plt.ylabel("y")
plt.show()

(100, 1)
(100, 2)

Options

List of options

Option	Default	Acceptable values	Acceptable types	Description
ndim	2	[2]	[‘int’]
return_complex	False	None	[‘bool’]
name	Branin	None	[‘str’]