cma.fitness_transformations.ScaleCoordinates

Domain interface (lower and upper variable values)

Let u and l be vectors (or a scalar) of (approximate) lower and upper variable values, respectively. Assigning fun2 from fun like:

fun2 = cma.ScaleCoordinates(fun, upper=u, lower=l)

gets fun2(x) == fun(l + (u - l) * x). Now, passing 0 as x[i] to fun2 evaluates fun at l[i] while passing 1 evaluates fun at u[i].

To match the size of x, the sizes of u and l are shortened or their last entry is recycled if necessary.

The default value for lower is zero in which case upper just becomes a scaling multiplier.

Bounding the search domain of fun2 to [0, 1] now bounds fun to the domain [l, u]. The 'bounds' option of CMAOptions allows to set these bounds.

More general, the affine transformation is defined such that x[i]==from_lower_upper[0] evaluates fun at l[i] and x[i]==from_lower_upper[1] evaluates fun at u[i] where from_lower_upper == [0, 1] by default.

Scaling interface

After fun2 = cma.ScaleCoordinates(fun, multipliers, zero), we have fun2(x) == fun(multipliers * (x - zero)), where the vector size of multipliers and zero is adapated to the size of x, in case by recycling their last entry. This awkwardly asks to pass the zero argument of the preimage space where it has little meaning. Hence more conveniently, fun2 = cma.ScaleCoordinates(fun, multipliers, lower=lower) gets fun2(x) == fun(multipliers * x + lower)).

Examples and Doctest

>>> import numpy as np
>>> import cma
>>> fun = cma.ScaleCoordinates(cma.ff.sphere, upper=[30, 1])
>>> fun([1, 1]) == 30**2 + 1**2
True
>>> list(fun.transform([1, 1])), list(fun.transform([0.2, 0.2]))
([30.0, 1.0], [6.0, 0.2])
>>> list(fun.inverse(fun.transform([0.1, 0.3])))
[0.1, 0.3]
>>> fun = cma.ScaleCoordinates(cma.ff.sphere, upper=[31, 3], lower=[1, 2])
>>> -1e-9 < fun([1, -1]) - (31**2 + 1**2) < 1e-9
True
>>> f = cma.ScaleCoordinates(cma.ff.sphere, [100, 1])
>>> assert f[0] == cma.ff.sphere  # first element of f-composition
>>> assert f(range(1, 6)) == 100**2 + sum([x**2 for x in range(2, 6)])
>>> assert f([2.1]) == 210**2 == f(2.1)
>>> assert f(20 * [1]) == 100**2 + 19
>>> assert np.all(f.inverse(f.scale_and_offset([1, 2, 3, 4])) ==
...               np.asarray([1, 2, 3, 4]))
>>> f = cma.ScaleCoordinates(f, [-2, 7], [2, 3, 4]) # last is recycled
>>> f([5, 6]) == sum(x**2 for x in [100 * -2 * (5 - 2), 7 * (6 - 3)])
True

See also these [Practical Hints](https://cma-es.github.io/cmaes_sourcecode_page.html#practical) for encoding variables.