Implement Levenberg-Marquardt algorithm in R - r

Question

I have been told to implement the Levenberg-Marquardt algorithm in R studio, considering lambda's initial value equals 10. The algorithm must stop when the gradient's norm is lower than the tolerance. I also need to print the values that x1, x2, λ, ∇f(x), d1 and d2 take for each iteration. Any ideas on how to do it? Many thanks in advance
This is what I have:
library(pracma)
library(matlib)
MetodeLM<-function(f,xi,t)
{
l=10
stop=FALSE
x<-xi
k=0
while (stop==FALSE){
dk<- inv(hessian(f,x)+l*diag(diag(hessian(f,x))))
x1<-x+dk
if (Norm(grad(f,x1))<t){
stop<-TRUE
}
else{
if (f(x1) < f(x)){
l<-l/10
k<-k+1
stop<-FALSE
}
else{
l<-l*10
stop<-FALSE
}
}
}
}

Answer 1

Correcting a few mistakes in your code, the following implementation of Levenberg Marquadt's algorithm should work (note that the update rule for the algorithm is shown in the following figure):
library(pracma)
# tolerance = t, λ = l
LM <- function(f, x0, t, l=10, r=10) {
x <- x0
k <- 0
while (TRUE) {
H <- hessian(f, x)
G <- grad(f, x)
dk <- inv(H + l * diag(nrow(H))) %*% G # dk <- solve(H + l * diag(nrow(H)), G)
x1 <- x - dk # update rule
print(k) # iteration
# print(l) # λ
print(x1) # x1, x2
print(G) # ∇f(x)
print(dk) # d1, d2
if (Norm(G) < t) break
l <- ifelse(f(x1) < f(x), l / r, l * r)
k <- k + 1
x <- x1 # update the old point
}
}
For example, with the following function, the non-linear optimization algorithm will quickly find a local minimum point (in the 10th iteration) as shown below
f <- function(x) {
return ((x[1]^2+x[2]-25)^2 + (x[1]+x[2]^2-25)^2)
}
x0 <- rep(0,2)
LM(f, x0, t=1e-3, l=400, r=2)
# [1] 0
# [,1]
# [1,] 0.165563
# [2,] 0.165563
# [1] -50 -50
# [,1]
# [1,] -0.165563
# [2,] -0.165563
# [1] 1
# [,1]
# [1,] 0.7986661
# [2,] 0.7986661
# [1] -66.04255 -66.04255
# [,1]
# [1,] -0.6331031
# [2,] -0.6331031
# ...
# [1] 10
# [,1]
# [1,] 4.524938
# [2,] 4.524938
# [1] 0.0001194898 0.0001194898
# [,1]
# [1,] 5.869924e-07
# [2,] 5.869924e-07
The following animation shows the convergence to the local minimum point for the function:
The following one is with LoG function