Bounds for Maximin Latin Hypercube Designs

Abstract

Latin hypercube designs (LHDs) play an important role when approximating computer simula- tion models. To obtain good space-filling properties, the maximin criterion is frequently used. Unfortunately, constructing maximin LHDs can be quite time-consuming when the number of dimensions and design points increase. In these cases, we can use approximate maximin LHDs. In this paper, we construct bounds for the separation distance of certain classes of maximin LHDs. These bounds are useful for assessing the quality of approximate maximin LHDs. Until now only upper bounds are known for the separation distance of certain classes of unrestricted maximin designs, i.e. for maximin designs without a Latin hypercube struc- ture. The separation distance of maximin LHDs also satisfies these “unrestricted” bounds. By using some of the special properties of LHDs, we are able to find new and tighter bounds for maximin LHDs. Within the different methods used to determine the upper bounds, a vari- ety of combinatorial optimization techniques are employed. Mixed Integer Programming, the Travelling Salesman Problem, and the Graph Covering Problem are among the formulations used to obtain the bounds. Besides these bounds, also a construction method is described for generating LHDs that meet Baer’s bound for the ℓ1 distance measure for certain values of n.Latin hypercube design;maximin;space-filling;mixed integer programming;trav- elling salesman problem;graph covering.