Application of Quasi Monte Carlo Methods in Global Optimization

It has been recognized through theory and practice that uniformly distributed deterministic sequences provide more accurate results than purely random sequences. A quasi Monte Carlo (QMC) variant of a multi level single linkage (MLSL) algorithm for global optimization is compared with an original stochastic MLSL algorithm for a number of test problems of various complexities. An emphasis is made on high dimensional problems. Two different low-discrepancy sequences (LDS) are used and their efficiency is analysed. It is shown that application of LDS can significantly increase the efficiency of MLSL. The dependence of the sample size required for locating global minima on the number of variables is examined. It is found that higher confidence in the obtained solution and possibly a reduction in the computational time can be achieved by the increase of the total sample size N. N should also be increased as the dimensionality of problems grows. For high dimensional problems clustering methods become inefficient. For such problems a multistart method can be more computationally expedient.

[1]  I. Sobol,et al.  On quasi-Monte Carlo integrations , 1998 .

[2]  Aimo A. Törn,et al.  Global Optimization , 1999, Science.

[3]  G. T. Timmer,et al.  Stochastic global optimization methods part I: Clustering methods , 1987, Math. Program..

[4]  Zbigniew Ulanowski,et al.  Application of global optimization to particle identification using light scattering , 1998 .

[5]  M. Locatelli,et al.  Simple linkage: Analysis of a threshold-accepting global optimization method , 1996, J. Glob. Optim..

[6]  A. A. Zhigli︠a︡vskiĭ,et al.  Stochastic Global Optimization , 2007 .

[7]  I. Sobol.  On the Systematic Search in a Hypercube , 1979 .

[8]  J. Halton.  On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals , 1960 .

[9]  Panos M. Pardalos,et al.  Encyclopedia of Optimization , 2006 .

[10]  I. M. Sobol.  Determination of the extremal values of a function of several variables satisfying a generalized Lipschitz condition , 1989 .

[11]  Fabio Schoen,et al.  Random and Quasi-Random Linkage Methods in Global Optimization , 1998, J. Glob. Optim..

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

[13]  H. Niederreiter.  Point sets and sequences with small discrepancy , 1987 .

[14]  Leo Liberti.  Linearity Embedded in Nonconvex Programs , 2005, J. Glob. Optim..

[15]  Joseph F. Traub,et al.  Faster Valuation of Financial Derivatives , 1995 .

[16]  L. Dixon,et al.  Parallel algorithms for global optimization , 1993 .

[17]  Aimo A. Törn,et al.  Stochastic Global Optimization: Problem Classes and Solution Techniques , 1999, J. Glob. Optim..

[18]  A. G. Sukharev.  Optimal strategies of the search for an extremum , 1971 .

[19]  I. Sobol.  On the distribution of points in a cube and the approximate evaluation of integrals , 1967 .

[20]  Henryk Wozniakowski,et al.  A general theory of optimal algorithms , 1980, ACM monograph series.

[21]  P. Pardalos,et al.  State of the art in global optimization: computational methods and applications , 1996 .

[22]  G. T. Timmer,et al.  Stochastic global optimization methods part II: Multi level methods , 1987, Math. Program..

[23]  Harald Niederreiter,et al.  Implementation and tests of low-discrepancy sequences , 1992, TOMC.

[24]  Panos M. Pardalos,et al.  State of the Art in Global Optimization , 1996 .

[25]  P. Pardalos,et al.  Handbook of global optimization , 1995 .

[26]  N. Sahinidis,et al.  A Lagrangian Approach to the Pooling Problem , 1999 .