[Paper Review] Musings about Constructions of Efficient Latin Hypercube Designs with Flexible Run-sizes

Hongzhi Wang, Qian Xiao+1 more|arXiv (Cornell University)|Oct 19, 2020

Advanced Multi-Objective Optimization Algorithms46 references4 citations

TL;DR

This paper provides a comprehensive review and comparative analysis of algebraic constructions and search algorithms for generating optimal Latin hypercube designs (LHDs) with flexible run sizes. It evaluates methods including simulated annealing, genetic algorithms (GA), particle swarm optimization (PSO), and algebraic techniques, concluding that GA offers the best balance of performance and computational efficiency for near-optimal LHDs across moderate to large designs.

ABSTRACT

Efficient Latin hypercube designs (LHDs), including maximin distance LHDs, maximum projection LHDs and orthogonal LHDs, are widely used in computer experiments. It is challenging to construct such designs with flexible sizes, especially for large ones. In the current literature, various algebraic methods and search algorithms have been proposed for identifying efficient LHDs, each having its own pros and cons. In this paper, we review, summarize and compare some currently popular methods aiming to provide guidance for experimenters on what method should be used in practice. Using the R package we developed which integrates and improves various algebraic and searching methods, many of the designs found in this paper are better than the existing ones. They are easy to use for practitioners and can serve as benchmarks for the future developments on LHDs.

Motivation & Objective

• To evaluate and compare existing algebraic constructions and search algorithms for optimal Latin hypercube designs (LHDs) with flexible run sizes.
• To identify the strengths and limitations of each method under various optimality criteria, including maximin distance, maximum projection, and orthogonality.
• To provide practical guidance for researchers and practitioners in selecting appropriate design strategies based on design size, computational budget, and desired optimality criteria.
• To highlight open challenges and serve as a benchmark for future development in optimal LHD construction.

Proposed method

• Systematic review and comparison of widely used search algorithms: simulated annealing (SA), SA2008 (Joseph and Hung, 2008), sliced LHD (SLHD), LaPSO (particle swarm optimization), and genetic algorithm (GA).
• Evaluation of algebraic constructions such as Costas arrays (Xiao and Xu, 2017), Williams transformation (Wang et al., 2018), and others (Ye, 1998; Sun et al., 2010) for provable optimality in specific run sizes.
• Empirical benchmarking using multiple criteria: average and maximum |q| values (for MaxPro), CPU time, and optimality under L2 distance and maximin criteria.
• Implementation and testing of algorithms across diverse design sizes (n × k), including both rule-of-thumb sizes and non-standard sizes.
• Use of R packages LHD and MaxPro for GA and MaxPro-based LHD generation, with performance measured via average and maximum |q|, and computational time.
• Statistical comparison of algorithm performance across 14 different design size configurations, with results summarized in Tables 11–14.

Experimental results

Research questions

• RQ1 Which search algorithm consistently produces the highest-quality Latin hypercube designs across a wide range of run sizes and factor counts?
• RQ2 How do algebraic constructions compare to metaheuristic search algorithms in terms of optimality guarantees and computational cost for non-standard design sizes?
• RQ3 What is the trade-off between computational efficiency and design quality among GA, PSO, SA, and MaxPro in constructing MaxPro and maximin LHDs?
• RQ4 For which design sizes do algebraic constructions provide provably optimal solutions, and when must search-based methods be used?
• RQ5 How do tuning parameters and computational budgets affect the relative performance of GA, LaPSO, and other metaheuristics in LHD construction?

Key findings

• Genetic Algorithm (GA) consistently outperforms other search algorithms in terms of both design quality (low average and maximum |q|) and computational efficiency, especially for moderate to large designs.
• For MaxPro LHDs, the R package MaxPro is computationally fast but yields inferior results compared to GA implemented in the LHD package.
• LaPSO performs well in design quality but requires significantly more computational time than GA, making it less practical for large-scale applications.
• Algebraic constructions such as those based on Costas arrays and Williams transformation provide provably optimal designs for specific run sizes (e.g., n = p−1, p, or p−2 for prime p), but are limited in applicability to non-standard sizes.
• For orthogonal and nearly orthogonal LHDs, the Sun10 and Ye98 algebraic methods are recommended for n = 4r, 4r+1, 8r, or 8r+1, while GA is effective for other sizes.
• Table 12 to 14 provide practical recommendations: GA is recommended for maximin and maximum projection LHDs across most design sizes, while MaxPro is suitable only for small designs (n ≤ 5) due to performance limitations.