Sliced Latin hypercube designs (SLHDs) are widely used in computer experiments with both quantitative and qualitative factors and in batches. Optimal SLHDs achieve better space-filling property on the whole experimental region. However, most existing methods for constructing optimal SLHDs have restriction on the run sizes. In this paper, we propose a new method for constructing SLHDs with arbitrary run sizes, and a new combined space-filling measurement describing the space-filling property for both the whole design and its slices. Furthermore, we develop general algorithms to search the optimal SLHD with arbitrary run sizes under the proposed measurement. Examples are presented to illustrate that effectiveness of the proposed methods.

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This article presents a new DIRECT-type SCABALL (scattering balls) algorithm with a new partition method for derivation-free optimization problems. It does not focus on dividing the region of interest into specific geometric shapes, but rather scatters several balls to cover it. In SCABALL, several potential optimal regions are selected at each iteration, and they are covered by smaller balls sequentially. In this way, the SCABALL ensures the everywhere dense convergence. The center points and radii of the scattered balls significantly influence the efficiency of SCABALL; therefore, the minimax designs are used in the initial and sequential stages to obtain better coverage. The SCABALL parameters, including the number of balls and their radii, were analyzed by numerical investigation. We provided the empirical choices for those parameters and found that the balls’ radii can be contracted to balance efficiency and global convergence. Numerical experiments show that the SCABALL algorithm is locally biased and robust.

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