Lower bounds and stochastic optimization algorithms for uniform designs with three or four levels

Kai Tai Fang*, Dietmar Maringer, Yu Tang, Peter Winker

*Corresponding author for this work

Research output: Contribution to journalJournal articlepeer-review

44 Citations (Scopus)


New lower bounds for three- and four-level designs under the centered L2-discrepancy are provided. We describe necessary conditions for the existence of a uniform design meeting these lower bounds. We consider several modifications of two stochastic optimization algorithms for the problem of finding uniform or close to uniform designs under the centered L 2-discrepancy. Besides the threshold accepting algorithm, we introduce an algorithm named balance-pursuit heuristic. This algorithm uses some combinatorial properties of inner structures required for a uniform design. Using the best specifications of these algorithms we obtain many designs whose discrepancy is lower than those obtained in previous works, as well as many new low-discrepancy designs with fairly large scale. Moreover, some of these designs meet the lower bound, i.e., are uniform designs.

Original languageEnglish
Pages (from-to)859-878
Number of pages20
JournalMathematics of Computation
Issue number254
Early online date27 Dec 2005
Publication statusPublished - Apr 2006
Externally publishedYes

Scopus Subject Areas

  • Algebra and Number Theory
  • Computational Mathematics
  • Applied Mathematics

User-Defined Keywords

  • Discrepancy
  • Lower bound
  • Stochastic optimization
  • Threshold accepting
  • Uniform designs


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