A new measure of irregularity of distribution and quasi-Monte Carlo methods for global optimization

Measures of irregularity of distribution, such as discrepancy and dispersion, play a major role in quasi-Monte Carlo methods for integration and optimization. In this paper, a new measure of irregularity of distribution, called volume-dispersion, is introduced. Its relation to the discrepancy and traditional dispersion, and its applications in global optimization problems are investigated. Optimization errors are bounded in terms of the volume-dispersion. Also, the volume-dispersion is generalized to the so-called F-volume-dispersion and quasi-F-volume-dispersion. They are reasonable measures of representation of point sets for given probability distributions on general domains and have potential applications in optimization problems when prior knowledge about the possible location of the optimizer is known and in the problems of experimental designs. Methods of generating point sets with low quasi-F-volume-dispersion are described.