On optimal search techniques
The technique described in this paper is believed to offer a new dhnension to the nuclear physicist and the lack of a proof of optimality is understandable since the procedure cannot be guaranteed to converge with predetermined accuracy and with a bound on the number of function evaluations.
Abstract
IBM 7090 program actually computed the coeiIicie~ts of Table 1 to 8 significant digits. Using the regressio~ equation, some of the actual activities as well as those computed from the equation, are listed in Table 2. From this list it was determined that one-third of the activities fell within a probable error and two-thirds were outside of it. Other uses could of course be made of these data. An overall activity for the source could easily be determined by intercomparison with an exposure of a known uniform source developed under identical conditions. Also, the degree of homogeneity could easily be determined by some measure of deviation from the means. In an attempt to quantify the nonhomogeneity of the source, a frequency count of the levels in the second band of the digitized image was made. The results are tabulated in Table 3. It would also be possible to cot~struct a contour map of areas of like activity and to classify these areas as to size and intensity. Conclusions It is believed that the technique described in this paper offers a new dhnension to the nuclear physicist. In many areas, a photograph catx compete on a quantitative (and economic) basis with other measurement techniques of physics. The lack of a proof of optimality is understandable since the procedure cannot be guaranteed to converge with predetermined accuracy and with a bound on the number of function evaluations. A specific example of nonconvergence is given by the function f(x,y) =-(29x ~-96xy + 101y ~) where the domain of search is 0 =< x, y ~ Fn, the nth Fibbon-naccian number. If Fn is chosen greater than 5 and an accuracy of ~1 is desired, one should expect to start the process with Fn intervals. However, the process will then terminate with x >= 2 although the maximum is at (0, 0). If one were to plot this function, he would discover that it does have a narrow ridge and it is this ridge that prevents convergence despite the authors' claim that ridges do not disturb the technique. Actually, this is but a specific instance of a more general result due to Kiefer ["Optimal Sequential Search and Approximation Methods Under Minimum Regularity Assumptions," J. SIAM 5(1957), 105-136] where he shows, loosely speaking, that optimal search techniques do not exist for optinlizing functions of two or more variables. Dear Editor: Mr. McBeth's criticism of the …