Hyperellipsoid Density Sampling: Exploitative Sequences to Accelerate High-Dimensional Optimization

Abstract

The curse of dimensionality remains a persistent challenge in modern optimization problems. Expanding the search space into higher dimensions exponentiates the difficulty of finding optimal solutions, rendering traditional algorithms inefficient. An efficient sampling strategy is presented to accelerate high-dimensional optimization as an alternative to uniform quasi-Monte Carlo (QMC) methods. This method, referred to as Hyperellipsoid Density Sampling (HDS), generates its sequences by defining multiple hyperellipsoids throughout the search space. HDS utilizes three types of unsupervised learning algorithms to bypass high-dimensional geometric calculations, producing a non-uniform sample sequence that exploits statistically promising regions of the parameter space. The ability to influence its distribution towards regions of interest makes HDS versatile for applications beyond global optimization, where models benefit from samples focused in specific regions. HDS was evaluated against Sobol, a highly uniform QMC sampling method, using differential evolution (DE) on the challenging set of 29 CEC2017 benchmark test functions. The results show statistically significant improvements in final solution geometric mean error (p<0.05), with average performance gains ranging from 37% in 10-D to 11% in 100-D. This paper demonstrates the efficacy of HDS as a robust alternative to uniform QMC sampling in high-dimensional optimization.