Hill Climbing and simulated annealing is an optimization technique that can be used to find the shortest route in the travelling salesman problem. I.e. it can minimize the length of a route through all cities by performing thermodynamical processes like heating or cooling to avoid getting stuck inside a local optima.
What if we look the other way around, and want to find the longest route, can one use the same optimization technique or are there any other techniques available? The reason I am asking this, is if we can use it to search for maximum trajectory points in the Collatz Conjecture. For example if it is possible, we could find $n=27$ to be the longest trajectory between $0$ and $54$ by not bruceforce, but by something like simulated annealing or the former.
The Collatz Conjecture asks wether the function $f(n)=3n+1 \quad\text{if $n$ is odd} \quad\text{or} \quad n/2 \quad \text{if $n$ is even}$ always reaches $1$ for any $n\in\mathbb{Z^+}$ when the function is iterated a finite number of times.
Is there a way to find the longest route or longest trajectory by using optimization techniques especially in the work of the Collatz Conjecture, or do we not have enough information about the factorization of the numbers and randomness in the party vectors? Or can we make a computer model that does this, or does a computer model allready exist?