paref.pareto_reflections.minimize_weighted_norm_to_utopia#

Classes

MinimizeWeightedNormToUtopia(utopia_point, ...)

Find the Pareto point closest to some utopia point

class paref.pareto_reflections.minimize_weighted_norm_to_utopia.MinimizeWeightedNormToUtopia(utopia_point: ndarray, potency: ndarray | float, scalar: ndarray)[source]#

Bases: ParetoReflection

Find the Pareto point closest to some utopia point

When to use#

This Pareto reflection should be used if a Pareto point is desired which is closest to some utopia point. This, in particular, implies the knowledge of some utopia point.

What it does#

The Pareto points of this map are the ones which minimize the (weighted) distance to the utopia point.

Mathematical formula#

\[p(x) = \sum_{i=1,...,n}(a_{i}x_{i}-a_{i}u_{i})^p\]

where n denotes the input dimension, u denotes some utopia point, a denotes some vector of dimension n with strictly positive entries and p is the potency.

u,a and p are fixed when the WeightedNormToUtopia is initialized. Calling returns the weighted p-norm to the given utopia point.

Examples

Define the utopia point, potency (p) and the scalar

>>> import numpy as np
>>> utopia_point, potency, scalar = np.zeros(2), np.array([2]), np.ones(2)

Initialize the WeightedNormToUtopia

>>> pareto_reflection = MinimizeWeightedNormToUtopia(utopia_point=utopia_point, potency=potency, scalar=scalar)

Calling it to (1,1), i.e. calculating

\[p(1,1)=(1-0)^2+(1-0)^2=2\]

yields:

>>> pareto_reflection(np.ones(2))
2

Specify the utopia point, the potency and the scalar used in the weighted p-norm

param utopia_point:
param np.ndarray:: utopia point stored in 1 dimensional array of length n
param potency:
param np.ndarray:: potency (i.e. value of p for the used p-norm)
param scalar:
param np.ndarray:: scalar vector stored in 1 dimensional array of length n

__call__(x: ndarray) → ndarray[source]#

property dimension_codomain: int#

property dimension_domain: int#