Benchmark Results - Problem U4
I. Problem Description
A. Overall Approach
Samples are drawn in a cone-shaped region around the line spanning \((0,0,\ldots,0) \in \mathbb{R}^d\) to \((1,1,\ldots,1) \in \mathbb{R}^d\), where each sample is constructed as follows:
- determine radius \(r = \frac{1}{10}\sqrt{d}\) (ensuring a cone with constant angular width as \(d\) increases)
- choose a point randomly at a distance \(r\) from \((1,1,\ldots,1) \in \mathbb{R}^d\) in a random direction
- scale point (component-wise) with scaling factor \(c\) drawn from a uniform distribution over \([0, 1]\)
B. Visualization
This image shows problem U4 with size parameter \(s=2\) (thus \(d=2\), \(n=200\), \(k=20\), \(m=0\)):
The image below shows an example solution, obtained by using the DEFAULT solver preset over 10.000 iterations
using the L2 distance metric and the geomean_separation diversity metric:
C. Separation statistics
The image below shows distribution of vector separations (distances to nearest neighbor for all vectors in the population), for different problem sizes:
