JOURNAL OF OPTIMIZATION THEORYAND APPLICATIONS: Vol. 16, Nos. 5]6, i975 Properties of the Random Search in Global Optimization ~,2 R. S. ANDERSSEN 8 AND P. BLOOMFIELD 4 Communicated by R. A. Howard Abstract. From theorems which we prove about the behavior of gaps in a set of N uniformly random points on the interval [0, 1], we determine properties of the random search procedure in one- dimensional global optimization. In particular, we show that the uniform grid search is better than the random search "when the optimum is chosen using the deterministic strategy, that a significant proportion of large gaps are contained in the uniformly random search, and that the error in the determination of the point at which the optimum occurs, assuming that it is unique, will on the average be twice as large using the uniformly random search compared with the uniform grid. In addition, some of the properties of the largest gap are verified numerically, and some extensions to higher dimen- sions are discussed. The latter show that not all of tile conclusions derived concerning the inadequacies of the one-dimensional random search extend to higher dimensions, and that on average the random search is better than the uniform grid for dimensions greater than 6. Key Words. Global optimization, uniformly random search, uniform grid search, gaps in random points. 1 This paper is based on work started in the Statistics Department of Princeton Uni- versity" when the first author was visiting as a Research Associate. Part of this research was supported by the Office of Naval Research, Contract No. 0014-67-A-0151-0017, and by the US Army Research Office--Durham, Contract No. DA-31-124-ARO-D-215. 2 The authors wish to thank B. Omodei for his careful work in preparing the programs for the results of Figs. I-2 and Table 1. The computations were performed on the IBM 360/50 of the Australian National University's Computer Centre. Thanks are also due to R. Miles for suggestions regarding the extension of the results to multidimensional regions, and to P. A. P. Moran and R. Brent for suggestions regarding the evaluation of the integral flo ..- fl o (xl s + ... + x~)~/~ dxl "'" dx~ . Staff Member, Computer Centre, Australian National University, Canberra, Australia. Professor, Statistics Department, Princeton University, Princeton, New Jersey. 383 © 1975Plenum PublishingCorporation,227 West 17th Street, New York, N.Y. 10011, No part of this publication may be reproduced, stored in a retrievalsystem, or transmitted,in any form or by any means, electronic,mechanical, photocopying, microfilming, recording,or otherwise withoutwrittenpermission of the publisher.