how to generate pseudo-random positive definite matrix with constraints on the off-diagonal elements?

Question

The user wants to impose a unique, non-trivial, upper/lower bound on the correlation between every pair of variable in a var/covar matrix.

For example: I want a vari

Answer 1

OK, fantastic Gregg: we're getting somewhere. Combining your idea with that of woodchips, yields this alternative approach. It's mathematically very dirty but it seems to work:

library(MCMCpack)
library(MASS)
p<-10
lb<-.6
ub<-.8
zupa<-function(theta){
    ac<-matrix(theta,p,p)
    fe<-rwish(100*p,ac%*%t(ac))
    det(fe)
}
ba<-optim(runif(p^2,-10,-5),zupa,control=list(maxit=10))
ac<-matrix(ba$par,p,p)
fe<-rwish(100*p,ac%*%t(ac))
me<-mvrnorm(p+1,rep(0,p),fe)
A<-cor(me)
bofi<-sqrt(diag(var(me)))%*%t(sqrt((diag(var(me)))))
va<-A[lower.tri(A)]
l1=100
while(l1>0){
    r1<-which(va>ub)
    l1<-length(r1)
    va[r1]<-va[r1]*.9
}
A[lower.tri(A)]<-va
A[upper.tri(A)]<-va
vari<-bofi*A
mk<-mvrnorm(10*p,rep(0,p),vari)
pc<-sign(runif(p,-1,1))
mf<-sweep(mk,2,pc,"*")
B<-cor(mf)
summary(abs(B[lower.tri(B)]))

Basically, this is the idea (say upper bound =.8 and lower it bound=.6), it has a good enough acceptance rate, which is not 100%, but it'll do at this stage of the project.