Petr Pošík let an interesting comment on a previous post : "Metaheuristic" or "Stochastic Local Search"?. As bloggers always love worthwhile comments from insightful readers, I copy it here, as an article, along with my answer.

I would like to present here an idea that will take the "stochastic" and "local" idea even more further than Thomas and Dirk. In that view, even GAs are local search techniques! Why?

It has a lot to do with the definition of neighborhood of the current state of the algorithm. The state may be the position of an algorithm in the search space. In classical local search, the state is given by mere 1 point. In population-based techniques, the state is the whole population. New solutions are usually generated only by local modifications of the state, i.e. the search takes place only in the local neighborhood of the algorithm state.

Yes, of course, the local neighborhood of the population as a whole is very broad and such a local search is much less prone to premature convergence and exhibits a kind of global behaviour. But, IMHO, it is still a local search...

I completely understand your view: if there is a non-zero probability of generating any solution in the search space, then the search must eventually find the global optimum, and thus performs a global search.

This is the case for many stochastic local search algorithm in continuous domain which use e.g. normal distribution. But if you set the variance too high, you will get very slow convergence to that global optimum which is in practice not desirable. If you set variance too low, you will quickly find local optimum, but you will have to wait for the global one virtually infinitely long - again undesirable. Or, put it in another way, would you call a "global optimization" technique still "global" if it takes zilions of years to find that global solution?

In practice, I think, for continuous spaces we have to resort to algorithms that exhibit local behavior since we always have some hard constraints on the length of the optimization run. In my eyes, this perfectly justifies the name "stochastic local search".

In fact, as far as I comprehend them, both Dick Thierens and Thomas Stützle shares your point of view. For them, GA are a kind of stochastic local search.

I find the classification in "descent" algorithm and "population" ones quite artificial. For example, simulated annealing is implemented as a descent algorithm, but does actually perform a sampling of the objective function, which is commonly seen as a population algorithm characteristic.

More rigourously, I think that every *stochastic* metaheuristic does
try to avoid local optima. Stochastic processes are always here to do
so^{[1]}.

Maybe the problem is the definition of *local* search. In my humble
opinion, a local search is a method that search for a *local* optimum.
This can be opposed to *global* search, where one want to find a
*global* optimum.

You are right when you point out that this is related to ergodicity, as only
an ergodic^{[2]} algorithm may converge^{[3]}, and thus be able to reliably find a global
optimum. Thus, yes, I will say that a true and rigorous optimization method is
global if, and only if, it is at least quasi-ergodic^{[4]}. A pure random search is the basic global
optimization method, that a good metaheuristic should at least outperforms.

Here, it is the stochastic operators that manipulates the solutions that
permits the global behaviour. This is not related to the encoding of the
solutions/the neighbourhood structure, even if it is a really *crucial*
part of the algorithm.

Thus, a "stochastic local search" may be defined as a local search seeking a global optimum, which is a paradoxical definition. I'd rather prefer that we keep the implementation and the mathematical bases separated, and thus talk of "stochastic search", or "stochastic metaheuristic".

Or else, one may want to use a paradoxical definition, as a matter of fun, which is also a good reason to do so :-)

#### Notes

[1] There may be probabilistic choices that are not related
to such tasks, but they are not linked to the iterative aspects of the search
algorithms, thus not being *stochastics*

[2] i.e. that can evaluate any solution

[3] i.e. that have a non-null probability of finding the global optimum in a finite time, or to say it differently, that can find the global optimum after a time that may tends towards infinity (for continuous problems, for discrete ones it is bounded by the size of the instance)

[4] i.e. that can reach any solution in a *finite*
number of iterations