RosenbrockFunctional

class odl.functionals.example_funcs.RosenbrockFunctional(*args, **kwargs)[source]

Bases: Functional

The well-known Rosenbrock function on R^n.

The Rosenbrock function is often used as a test problem in smooth optimization.

Notes

The functional is defined for $x \\in \\mathbb{R}^n$ , $n \\geq 2$ , as

$\sum_{i=1}^{n - 1} c (x_{i+1} - x_i^2)^2 + (1 - x_i)^2,$

where $c$ is a constant, usually set to 100, which determines how “ill-behaved” the function should be. The global minimum lies at $x = (1, \\dots, 1)$ , independent of $c$ .

There are two definitions of the n-dimensional Rosenbrock function found in the literature. One is the product of 2-dimensional Rosenbrock functions, which is not the one used here. This one extends the pattern of the 2d Rosenbrock function so all dimensions depend on each other in sequence.

References

__init__(space, scale=100.0)[source]

Initialize a new instance.

Parameters

spaceTensorSpace: Domain of the functional.
scalepositive float, optional: The scale c in the functional determining how “ill-behaved” the functional should be. Larger value means worse behavior.

Examples

Initialize and call the functional:

>>> r2 = odl.rn(2)
>>> functional = RosenbrockFunctional(r2)
>>> functional([1, 1])  # optimum is 0 at [1, 1]
0.0
>>> functional([0, 1])
101.0

The functional can also be used in higher dimensions:

>>> r5 = odl.rn(5)
>>> functional = RosenbrockFunctional(r5)
>>> functional([1, 1, 1, 1, 1])
0.0

We can change how much the function is ill-behaved via scale:

>>> r2 = odl.rn(2)
>>> functional = RosenbrockFunctional(r2, scale=2)
>>> functional([1, 1])  # optimum is still 0 at [1, 1]
0.0
>>> functional([0, 1])  # much lower variation
3.0