Minimax designs using clustering

Minimax designs provide a uniform coverage of a design space $\mathcal{X}$ by minimizing the maximum distance from any point in $\mathcal{X}$ to its nearest design point. Although minimax designs have many useful applications, e.g., in optimal sensor allocation or as space-filling designs for computer experiments, there has been little work in developing algorithms to generate such designs. Motivated by the popular Linde-Buzo-Gray algorithm in vector quantization, we propose a new clustering-based approach for generating minimax designs on the unit hypercube and on general convex and bounded regions. A modification of Lloyd's algorithm forms a key step in the proposed algorithm, since it is computationally efficient for large datasets and is highly parallelizable. Integrating this clustering algorithm with the particle swarm optimization method enables us to obtain globally optimal minimax designs. Simulation studies show that our algorithm considerably outperforms existing methods in literature, particularly in high-dimensions.