Abstract
A computer method to find vectors s that minimize (ci>0 constants) subject to a probability constraint P{μi≤si, i=1,…,r}≥1-α (≤α≤1) where v ,…,vr have a joint multinomial distribution, is obtained by solving the corresponding optimization problem through the usual normal approximation. Thus vectors are sought that minimize (bi>0 constants) subject to a multivariate normal probability constraint where v1,…,vr have a joint singular multivariate normal distribution. The singular normal probability integral is expressed in various computer-ready formulas as: (a) one integral over a simplex, (b) a sum of integral over multidimensional rectangular regions, and (c) a sum of integrals over multidimensional right triangles or plane orthoschemes. The optimization of F, and thereby of G, is accomplished using a known nonlinear program in conjunction with also known numerical multivariate normal distribution computer codes which work well for r=3. Binomial tables and a bisection method may be used for r=2. However fo...
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