FINDING THE EXTREMUM OF A SPECIAL FUNCTION (Ⅰ)

The problem of the optimal allocation of experiments, i. e., the selection of the optimal number of experiments to place in each block in locating the maximum of a one dimensional unimodal function by variable block search, proposed in [1] and [2], can be stated as the following problem of non-linear integer programming: for any given positive integers n, N（N>n>1）and real number δ（0 ≤δ≤1/2 ）. find dan optimal solution （k,k,…, k） of the equation k+k+…k= N, k≥1 （i=1, 2,…, n） all integers, （1） in the sense that the function L,k,...,k（δ） defined in the text attains its maximum value at（k, k"", k）. We shall prove in this paper that for any n, N（N>n>1） and δ（0≤δ≤1/2）, one can always find an optimal solution of （1） from those solutions: （ⅰ） kis even, （ⅱ）|k-k|≤2 （1≤i,j≤n）, and （ⅲ） all the even k appearing in k,k,…,khave the same value. Furthermore, an optimal solution of （1） is obtained for the cases N=n（e-1）,δ=0 and 1/2 where e≥4 is