Constructing uniform designs: A heuristic integer programming method

In this paper, the wrap-around L2-discrepancy (WD) of asymmetrical design is represented as a quadratic form, thus the problem of constructing a uniform design becomes a quadratic integer programming problem. By the theory of optimization, some theoretic properties are obtained. Algorithms for constructing uniform designs are then studied. When the number of runs n is smaller than the number of all level-combinations m, the construction problem can be transferred to a zero-one quadratic integer programming problem, and an efficient algorithm based on the simulated annealing is proposed. When n≥m, another algorithm is proposed. Empirical study shows that when n is large, the proposed algorithms can generate designs with lower WD compared to many existing methods. Moreover, these algorithms are suitable for constructing both symmetrical and asymmetrical designs.

► The wrap-around L2-discrepancy (WD) can be represented as a quadratic form. ► Constructing a uniform design under WD is a quadratic integer programming problem. ► Some algorithms based on the simulated annealing for constructing UD are provided. ► The algorithms can construct asymmetrical designs with large number of runs. ► Empirical study shows that the algorithms are better than many existing methods.