Constraint Optimization with one parameter included in the constraint of the other - r

Question

I want to calculate the following
So I want to find Theta and Sigma that maximizes the function.
The constraints are:
> Theta>-Sigma
> -1<Sigma<1
So one of my problem is that I dont know how to deal with the fact that one parameter is included in the constraint of the other Parameter, that I want to optimize over.
I tried with optim(), constrOptim and dfoptim!
Using optim():
k=8
i=1:(k-1)
x=c(5,0.2)
n=24
nj=c(3,4,8,1,1,4,2,1)
EPPF <- function(x,n,nj) {
y=(x[1]+1):(x[1]+1+(n-1)-1)
z=-(prod(x[1]+i*x[2])/(prod(y))*prod(sapply(nj, hfun)))
return(z)}
hfun <- function(p){
h=(1-x[2]):((1-x[2])+p-1)
hfun=prod(h)
return(hfun)
}
> optim(c(6,0.3), fn=EPPF,method = "L-BFGS-B", n=n,nj=nj, lower = c(-x[1],-1), upper = c(Inf,1))
$par
[1] 6.0 0.3
$value
[1] -1.258458e-15
$counts
function gradient
2 2
$convergence
[1] 0
$message
[1] "CONVERGENCE: REL_REDUCTION_OF_F <= FACTR*EPSMCH"
I also tried using a constructor function:
make.EPPF <- function(n,nj,fixed=c(FALSE,FALSE)){
params <-fixed
function(p) {
hfun <- function(y){
h=(1-sigma):((1-sigma)+y-1)
hfun=prod(h)
return(hfun)
}
params[!fixed] <- p
theta <- params[1]
sigma <- params[2]
y=(theta+1):(theta+1+(n-1)-1)
z=(prod(theta+i*sigma)/(prod(y))*prod(sapply(nj, hfun)))
z
}
}
EPPF <-make.EPPF (n,nj)
> optim(c(theta=6, sigma=0.5), fn=EPPF,method = "L-BFGS-B",lower = c(-sigma,-1), upper = c(Inf,1))
Error in optim(c(theta = 6, sigma = 0.5), fn = EPPF, method = "L-BFGS-B", :
object 'sigma' not found
Using constrOptim():
> A <- matrix(c(1,1,0,1,0,-1),3,2,byrow=T)
> b <- c(0,-1,-1)
>
> constrOptim(c(3,0.3),EPPF,NULL,A,b, control=list(fnscale=-1))
$par
[1] 3.0 0.3
$value
[1] 9.712117e-16
$counts
[1] 0
$convergence
[1] 0
$message
NULL
$outer.iterations
[1] 1
$barrier.value
[1] 7.313452e-05
Using Package dfoptim:
> library(dfoptim)
> nmkb(x=c(6,0.3), EPPF, lower=c(-x[2],-1), upper=c(Inf, 1 ))
Error in par < lower :
comparison (3) is possible only for atomic and list types
Either there is for some reasons no convergence or some other Errors.
I am relativ new to programming and R and would really appreciate if someone could help me. Thanks!

Answer 1

These are 3 linear inequality constraints:
sigma + theta > 0
sigma + 1 > 0
-sigma + 1 > 0
You can do this in maxLik. But note that maxLik maximizes the function, hence remove the '-' in front of z. Here is the code that works for me (using Rscript):
k=8
i=1:(k-1)
x=c(5,0.2)
n=24
nj=c(3,4,8,1,1,4,2,1)
EPPF <- function(x,n,nj) {
theta <- x[1]
sigma <- x[2]
y=(x[1]+1):(x[1]+1+(n-1)-1)
z <- prod(x[1]+i*x[2])/(prod(y))*prod(sapply(nj, hfun))
z <- log(z)
return(z)
}
hfun <- function(p){
h=(1-x[2]):((1-x[2])+p-1)
hfun=prod(h)
return(hfun)
}
library(maxLik)
constraints <- list(ineqA=matrix(c(1,0,0,1,1,-1),3,2), ineqB=c(0,1,1))
m <- maxBFGS(EPPF, start=c(6,0.3), constraints=constraints, n=n, nj=nj)
print(summary(m))
I also took logarithm of the result as this leads to more "human" numbers. Otherwise you have to re-tune the stopping conditions. The answer seems to be -1, 1.