If the minimum is not unique and the MLE is equal to zero over a range of functions, how can we find the minimum? I have read about golden section search algorithm or bisection algorithm, but I don't know if they would apply in this case or not? Is there any other better algorithm which is faster than these algorithms in R?
I think you should look at the quasi-newton optimization through the function
Ideally, you should try a few different methods and see which produces the best results for your problem. If you have several local critical points, it's important to use different starting values. Selecting random starting values over the range of the function usually works well.