Many thanks for this very interesting email. I think there is a misunderstanding here though. Monte Carlo methods are a VERY broad family of numerical methods which deal with A LOT of different problems (the ones you mention are a very small part of this huge family of methods). Let me report here an extract of a paper of mine where I explain what this family is about:
"The purpose of Monte Carlo methods is to approximate the solution of problems in computational mathematics by using
random processes for each such problem. These methods give statistical estimates for any linear functional of the solution by
performing random sampling of a certain random variable whose mathematical expectation is the desired functional [46].
Essentially, they reduce a given problem to approximate calculations of some mathematical expectation. They represent
a very powerful tool when it comes to solve problems in mathematics, physics and engineering where the deterministic
methods hopelessly break down. Indeed Monte Carlo methods do not require any additional regularity of the solution and it
is always possible to control the accuracy of this solution in terms of the probability error. Another important advantage in
using Monte Carlo methods consists in the fact that they are very efficient in dealing with large and very large computational
problems such as multi-dimensional integration, very large linear systems, partial integro-differential equations in highly
dimensional spaces, etc. Finally, these methods are efficient on parallel processors and parallel machines. Thus, it is not
surprising that these methods have rapidly found a wide range of applications in applied Science."
I hope this somehow clarifies what I meant by Monte Carlo methods for optimization problems. Essentially, what I am looking for is a method which exploits the generation of (independent) random numbers to solve an optimization problem. This would represent an important feature of Gneural Network since it would extremely easy to parallelize and therefore useful for the training of "deep" neural networks.