generate_ndset: Generate a random set of mutually nondominated points

In moocore: Core Mathematical Functions for Multi-Objective Optimization

Generate a random set of mutually nondominated points

Description

Generate a random set of n mutually nondominated points of dimension d with the shape defined by method.

When integer = FALSE (the default), the points are generated within the hypercube (0,1)^d and can be scaled to another range using normalise(). Otherwise, points are scaled to a non-negative integer range that keeps the points mutually nondominated.

See the visualisations in the vignette \HTMLVignettegenerateSampling Random Nondominated Sets.

Usage

generate_ndset(n, d, method, seed = NULL, integer = FALSE)

Arguments

n	integer(1) Number of rows in the output.
d	integer(1) Number of columns in the output.
method	character(1) Method used to generate the random nondominated set. See Details below for more information.
seed	integer(1) Integer seed for random number generation. If NULL, a random seed is generated.
integer	logical(1) If TRUE, return integer-valued points.

Details

The available methods are:

'simplex', 'linear', or 'L'

Uniformly samples points on the standard simplex. This shape of nondominated set is also called 'linear' in the literature \citepLacKlaFon2017box.

The standard (d-1)-simplex is defined by \{x \in \mathbb{R}_+^d : \sum_i x_i = 1\}. Each point \vec{z} \in (0,1)^d \subset \mathbb{R}^d is generated by sampling d independent and identically distributed values (x_1,x_2, \dots, x_d) from the exponential distribution, then dividing each value by the L1-norm of the vector, z_i = x_i / \sum_{i=1}^d x_i \citepRubMel1998simulation. Values sampled from the exponential distribution are guaranteed to be positive. Sampling from either the standard normal distribution \citepGueFonPaq2021hv or the uniform distribution \citepLacKlaFon2017box does not produce a uniform distribution when projected onto the simplex.

'concave-sphere', 'sphere', or 'C'

Uniformly samples points on the positive orthant of the hyper-sphere, which is concave when all objectives are minimised.

Each point \vec{z} \in (0,1)^d \subset \mathbb{R}^d is generated by sampling d independent and identically distributed values \vec{x}=(x_1,x_2, \dots, x_d) from the standard normal distribution, then dividing each value by the l2-norm of the vector, z_i = \frac{|x_i|}{\|\vec{x}\|_2} \citepMuller1959sphere. The absolute value in the numerator ensures that points are sampled on the positive orthant of the hyper-sphere. Sampling from the uniform distribution \citepLacKlaFon2017box would not result in a uniform sampling when projected onto the surface of the hyper-sphere.

'convex-sphere' or 'X'

Equivalent to 1 - generate_ndset(..., method='concave-sphere'), which is convex for minimisation. This shape has also been called inverted convex \citepIshHeSha2019regular. This sampling is uniform.

It corresponds to translating points from the negative orthant of the hyper-sphere to the positive orthant. Thus, the sampling remains uniform.

'convex-simplex'

Equivalent to generate_ndset(..., method='simplex')^2, which is convex for minimisation. Such a set cannot be obtained by any affine transformation of a subset of the hyper-sphere. This sampling is not uniform.

The corresponding surface is equivalent to a simplex curved towards the origin. The generated points \vec{z} \in (0,1)^d \subset \mathbb{R}^d satisfy \sum_{i=1}^d \sqrt{z_i} = 1. Although the sampling on the simplex is uniform, the transformed points are not.

'concave-simplex'

Equivalent to 1 - generate_ndset(..., method='convex-simplex'), which is concave for minimisation. This shape has also been called inverted concave \citepIshHeSha2019regular. This sampling is not uniform because method='convex-simplex' is also not uniform.

'inverted-simplex' or 'inverted-linear'

Equivalent to 1 - generate_ndset(..., method='simplex'). This sampling is uniform.

Methods 'inverted-simplex', 'concave-simplex' and 'convex-sphere' are translations of 'simplex', 'convex-simplex' and 'concave', respectively. These translations have been called inverted shapes in the literature and have different properties than their regular counterparts \citepIshHeSha2019regular.

Value

A numeric matrix of size ⁠n × d⁠ containing nondominated points.

References

\insertAllCited

Examples

generate_ndset(5, 3, "simplex", seed = 42)
generate_ndset(5, 3, "simplex", seed = 42, integer = TRUE)
generate_ndset(4, 2, "sphere", seed = 123)
generate_ndset(3, 5, "convex-sphere", seed = 123)
generate_ndset(4, 4, "convex-simplex", seed = 123)

moocore documentation built on July 12, 2026, 5:06 p.m.