# Hybrid Particle Swarm Optimization for High-Dimensional Latin Hypercube Design Problem
###### Abstract  
Latin Hypercube Design (LHD) is widely used in computer simulation to solve large-scale, complex, nonlinear problems. The high-dimensional LHD (HLHD) problem is one of the crucial issues and has been a large concern in the long run. This paper proposes an improved Hybrid Particle Swarm Optimization (IHPSO) algorithm to find the near-optimal HLHD by increasing the particle evolution speed and strengthening the local search. In the proposed algorithm, firstly, the diversity of the population is ensured through comprehensive learning. Secondly, the Minimum Point Distance (MPD) method is adopted to solve the oscillation problem of the PSO algorithm. Thirdly, the Ranked Ordered Value (ROV) rule is used to realize the discretization of the PSO algorithm. Finally, local and global searches are executed to find the near-optimal HLHD. The comparisons show the superiority of the proposed method compared with the existing algorithms in obtaining the near-optimal HLHD. adaptive particle swarm optimization; high-dimensional Latin hypercube design; space-filling quality; design points +
Footnote †: journal: Applied Sciences  
**Citation:** [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] Hybrid Particle Swarm Optimization for High-Dimensional Latin Hypercube Design Problem. **Citation:** [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] Hybrid Particle Swarm Optimization for High-Dimensional Latin Hypercube Design. **Optimization for High-Dimensional Latin Hypercube Design Problem. **  
**Latin Hypercube Design Problem. **  
**Appl. Sci. 2023, 13, 7066. [[https://doi.org/10.3390/app13127066](https://doi.org/10.3390/app13127066)]([https://doi.org/10.3390/app13127066](https://doi.org/10.3390/app13127066))**  
**Academic Editor: [PERSON]**  
Received: 27 April 2023  
Received: 2 June 2023  
Accepted: 7 June 2023  
Published: 12 June 2023  
**Copyright:** (c) 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license ([[https://creativecommons.org/licenses/by/](https://creativecommons.org/licenses/by/)]([https://creativecommons.org/licenses/by/](https://creativecommons.org/licenses/by/))). **This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license ([[https://creativecommons.org/licenses/by/](https://creativecommons.org/licenses/by/)]([https://creativecommons.org/licenses/by/](https://creativecommons.org/licenses/by/))). **AJin [16], is named the Enhanced Stochastic Evolution (ESE) algorithm. The contribution of this work is to obtain better solutions through the cooperation of the inner and outer loops. The optimization problem also has limitations, and it is usually challenging to obtain the optimal solution. [PERSON] [19] and [PERSON] [20] used a genetic algorithm (GA) to generate low-discrepancy sequences with a high degree of concordance, which was limited in local search ability and easily trapped in locally optimal solutions. [PERSON] et al. [21] proposed a population algorithm based on a GA, using a permutation genetic algorithm (PermGA) combined with local search and global search to construct the optimal LHD. The algorithm can achieve better results in the optimization of medium-scale problems. However, due to the limitations of the local search algorithm, it is not easy to find and achieve the optimal solution for the large-scale problem. In addition, some other optimized LHD algorithms include Iterative Local Search (ILS) [6], Sequential Local Enumeration (SLE) [22], Sliced Latin Hypercube Design (SLHD) [23], Simulation-based Sequencing Optimization for Annealing (SOBSA) [24], and Inflation, Expansion, and Stacking (IES) algorithms [25].