tutorial.rst

Tutorial - Solving the six-hump camelback function

Filename: test/C6.py

The following tutorial shows how to find the global minimum of a Six-hump camelback function using the DIRECT algorithm.

f(x₁, x₂) = (4 − 2.1x₁² + x₁⁴ + x₁⁴/3)x₁² + x₁x₂ + ( − 4 + 4x₂²)x₂²,

Ω = [ − 3, 3] × [ − 2, 2].

First we need to import the solve function from the DIRECT package:

>>> from scipydirect import minimize

Then we need to define the objective of the function:

>>> def obj(x):
  ...     """Six-hump camelback function"""
  ...     x1 = x[0]
  ...     x2 = x[1]
  ...     f = (4 - 2.1*(x1*x1) + (x1*x1*x1*x1)/3.0)*(x1*x1) + x1*x2 + (-4 + 4*(x2*x2))*(x2*x2)
  ...     return f

We need to define the domain of the problem using block constraints:

>>> bounds = [(-3, 3), (-2, 2)]

We use the DIRECT algorithm to solve the optimization problem. The algoritm is called using the minimize function. The solve functions accepts the problem objective obj and block constraints:

>>> res = minimize(obj, bounds)

In the above we use the default settings of the DIRECT algorithm. It us possible to costumize the algorithm using the parameters of the minimize function (see :pyscipydirect.minimize).

The minimize function returns a result object res making accessible the optimal point, res.x, the value of the objective at the optimum, res.fun, and a status message res.ierror.

We can visualize the problem using `matplotlib`:

>>> fig = plt.figure()
>>> ax = fig.add_subplot(111, projection='3d')

>>> x = res.x
>>> X, Y = np.mgrid[x[0]-1:x[0]+1:50j, x[1]-1:x[1]+1:50j]
>>> Z = np.zeros_like(X)

>>> for i in range(X.size):
  ...     Z.ravel()[i] = obj([X.flatten()[i], Y.flatten()[i]])

>>> ax.plot_wireframe(X, Y, Z, rstride=1, cstride=1, cmap=cm.jet)
>>> ax.scatter(x[0], x[1], res.fun, c='r', marker='o')
>>> ax.set_title('Six-hump Camelback Function')
>>> ax.view_init(30, 45)
>>> plt.show()

This results in

tutorialfig.py

More examples can be found in the source distribution under the test/ folder.