Constructing uniform designs: A heuristic integer programming method

In this paper, the wrap-around L"2-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.

[1]  M. Gilli,et al.  Heuristic Optimization Methods in Econometrics , 2009 .

[2]  Bahram Alidaee,et al.  A scatter search approach to unconstrained quadratic binary programs , 1999 .

[3]  Kai-Tai Fang,et al.  Lower bounds for wrap-around L2-discrepancy and constructions of symmetrical uniform designs , 2005, J. Complex..

[4]  F. J. Hickernell.  Lattice rules: how well do they measure up? in random and quasi-random point sets , 1998 .

[5]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[6]  Runze Li,et al.  Design and Modeling for Computer Experiments (Computer Science & Data Analysis) , 2005 .

[7]  Erricos John Kontoghiorghes,et al.  Handbook of Computational Econometrics , 2009 .

[8]  Panos M. Pardalos,et al.  Introduction to Global Optimization , 2000, Introduction to Global Optimization.

[9]  B. Freisleben,et al.  Genetic algorithms for binary quadratic programming , 1999 .

[10]  John E. Beasley,et al.  Heuristic algorithms for the unconstrained binary quadratic programming problem , 1998 .

[11]  Liansheng Zhang,et al.  Global optimality conditions for quadratic 0-1 optimization problems , 2010, J. Glob. Optim..

[12]  Fred J. Hickernell,et al.  A generalized discrepancy and quadrature error bound , 1998, Math. Comput..

[13]  Riccardo Poli,et al.  New ideas in optimization , 1999 .

[14]  Harald Niederreiter,et al.  Random number generation and Quasi-Monte Carlo methods , 1992, CBMS-NSF regional conference series in applied mathematics.

[15]  Runze Li,et al.  Design and Modeling for Computer Experiments , 2005 .

[16]  Leonidas D. Iasemidis,et al.  Quadratic Binary Programming and Dynamical System Approach to Determine the Predictability of Epileptic Seizures , 2001, J. Comb. Optim..

[17]  Gintaras Palubeckis,et al.  Multistart Tabu Search Strategies for the Unconstrained Binary Quadratic Optimization Problem , 2004, Ann. Oper. Res..

[18]  Chang-Xing Ma,et al.  Wrap-Around L2-Discrepancy of Random Sampling, Latin Hypercube and Uniform Designs , 2001, J. Complex..

[19]  P. Merz,et al.  Memetic algorithms for the unconstrained binary quadratic programming problem. , 2004, Bio Systems.

[20]  Reiner Horst,et al.  Introduction to Global Optimization (Nonconvex Optimization and Its Applications) , 2002 .

[21]  Arnold J. Stromberg,et al.  Number-theoretic Methods in Statistics , 1996 .

[22]  Yong-Dao Zhou,et al.  Lower bounds of the wrap-around L2-discrepancy and relationships between MLHD and uniform design with a large size , 2008 .

[23]  Kengo Katayama,et al.  Performance of simulated annealing-based heuristic for the unconstrained binary quadratic programming problem , 2001, Eur. J. Oper. Res..

[24]  Bernd Freisleben,et al.  Greedy and Local Search Heuristics for Unconstrained Binary Quadratic Programming , 2002, J. Heuristics.

[25]  K. Fang,et al.  Application of Threshold-Accepting to the Evaluation of the Discrepancy of a Set of Points , 1997 .