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.

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

>>> from DIRECT import solve

Then we need to define the objective of the function:

>>> def obj(x, user_data):
  ...     """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, 0

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

>>> l = [-3, -2]
>>> u = [3, 2]

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

>>> x, fmin, ierror = solve(
  ...                 obj,
  ...                 l,
  ...                 u
  ...                 )

In the above we use the default settings of the DIRECT algorithm. It us possible to costumize the algorithm using the paramters of the solve function (see DIRECT.solve()).

The solve function returns the optimal point, x, the optimal objective, fmin, and a status message ierror.

We can visualize the problem using matplotlib:

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

>>> 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]], None)[0]

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

This results in

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