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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

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


Authors: Kai-Tai Fang, Dietmar Maringer, Yu Tang and Peter Winker
Journal: Math. Comp. 75 (2006), 859-878
MSC (2000): Primary 68Q17, 68Q15, 62K99
Published electronically: December 27, 2005
MathSciNet review: 2196996
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Abstract: New lower bounds for three- and four-level designs under the centered $ L_2$-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.


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Additional Information

Kai-Tai Fang
Affiliation: Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong, People’s Republic of China
Email: ktfang@math.hkbu.edu.hk

Dietmar Maringer
Affiliation: Faculty of Economics, Law and Social Sciences, University of Erfurt, Germany
Email: dietmar.maringer@uni-erfurt.de

Yu Tang
Affiliation: Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong, People’s Republic of China
Address at time of publication: Department of Mathematics, Suzhou University, Suzhou, 215006, People’s Republic of China
Email: ytang@math.hkbu.edu.hk

Peter Winker
Affiliation: Faculty of Economics, Law and Social Sciences, University of Erfurt, Germany
Email: peter.winker@uni-erfurt.de

DOI: http://dx.doi.org/10.1090/S0025-5718-05-01806-5
PII: S 0025-5718(05)01806-5
Keywords: Discrepancy, lower bound, uniform designs, stochastic optimization, threshold accepting
Received by editor(s): November 3, 2004
Published electronically: December 27, 2005
Additional Notes: The work was partially supported by the Grants GER/JRS/03-04/01, RGC/HKBU 200804, FRG/03-04/II-711, and DAAD D/03/314145.
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.