# Hybrid Particle Swarm Optimization for High-Dimensional Latin Hypercube Design Problem
###### Abstract
In addition to the above optimization algorithms, iterative heuristic algorithms are designed to optimize different types of problems in various fields because of their easy implementation and ability to optimize complex issues effectively. Particle Swarm Optimization (PSO) [26; 27; 28; 29] is one famous algorithm proposed by [PERSON] and [PERSON] in 1995. The idea of this algorithm comes from a simplified model of bird flocks' food-seeking behavior [26]. The respective positions are updated through information exchange between individual birds, combining the current best position with the individual's history to find the optimal value in the range. Due to its simplicity and high efficiency, it is widely used in various fields in the real world [30; 31; 32; 33; 34]. PSO can solve high-dimensional optimization problems with multiple optimal solutions, optimized and applied to the HLHD problem. [PERSON] et al. [35] proposed a LaPSO to optimize LHD by reducing the Hamming distance to obtain the optimal LHD. [PERSON] et al. [36] proposed the Adaptive Memetic Particle Swarm (AMPSO) algorithm, which excels in solving large-scale LHD problems. The efficiency of finding the optimal LHD combined with the PSO can still be improved. The above works are novel and can serve as a valuable reference. However, there are still some problems that need to be addressed in applying PSO to LHD optimization:  
(1) Since the PSO algorithm's evolution method is not suitable for discrete issues in LHD, subsequent discretization operations are required when using the solution obtained by PSO. (2) PSO is highly dependent on the dynamic equation of its evolution. The value of the control parameters in the equation determines the convergence speed and the pros and cons of evolution. There may be insufficient space exploration, and the optimal solution cannot be obtained [31]. (3) Each particle in the PSO learns its personal best particle and global best particle simultaneously. The inappropriate selection of parameters in the dynamic equation will make the population change less frequently during the evolution process, which may lead to the phenomenon of \"oscillation\" [37; 38] and the premature convergence of the population [39]. It will reduce the search efficiency of the algorithm, be unable to fully explore the research space, and fall into the local optimal solution [31; 40]. (4) Although PSO can quickly find the optimal solution when dealing with low-dimensional problems, optimizing high-dimensional issues makes it more prone to premature convergence [41]. In practical applications, as the dimension increases, the PSO algorithm is more prone to boundary violation problems during the search process, which affects the evolution effect of particles. In this work, we proposed a high-dimensional LHD design framework based on the improved hybrid PSO (IHPSO). First, the diversity of the population is ensured by using comprehensive learning. Second, the oscillation problem of the PSO is solved by the minimum point distance (MPD), accelerating the calculation process. Third, the ranked ordered value (ROV) rule is adopted to realize the discretization of the PSO. Finally, local and global searches are executed to search for the near-optimal HLHD. The remainder of this paper is organized as follows: Section 2 introduces the technical background, including the LHD definition, the PSO algorithm, and the space-filling criteria. In Section 3, the composition of the proposed PSO algorithm for the fast construction of optimal LHD is presented. Numerical results and discussion are shown in Section 4. Finally, Section 5 concludes this paper and proposes an outlook for improving the performance of the proposed algorithm.