Despite the title of this blog, the term *metaheuristic* is not really well defined.

One of the first occurence of the term can (of course) be found in a paper by Fred Glover^{[1]}: *Future Paths for Integer Programming and Links to Artificial Intelligence*^{[2]}. In the section concerning *tabu search*, he talks about *meta-heuristic*:

Tabu search may be viewed as a "meta-heuristic" superimposed on another heuristic. The
approach undertakes to transcend local optimality by a strategy of forbidding (or, more broadly,
penalizing) certain moves.

In the AI field, a *heuristic* is a specific method that help solving a problem (from the greek for *to find*), but how must we understand the *meta* word ? Well, in greek, it means "after", "beyond" (like in *metaphysic*) or "about" (like in *metadata*). Reading Glover, *metaheuristics* seems to be *heuristics beyond heuristics*, which seems to be a good old definition, but what is the definition nowadays ? The litterature is really prolific on this subject, and the definitions are numerous.

There is at least three tendencies :

- one that consider that the most important part of
*metaheuristcs* is the gathering of several heuristics,
- one other that promotes the fact that
*meta*heuristics are designed as generalistic methods, that can tackle several problems without major changes in their design,
- the last one that use the term only for evolutionnary algorithms when they are hybridicized with local searches (methods that are called
*memetic algorithms* in the other points of vue).

The last one is quite minor in the generalistic litterature, it can mainly be found in the field of evolutionnary computation, separate out the two other tendencies is more difficult.

Here are some definitions gathered in more or less generalistic papers:

"iterative generation process which guides a subordinate heuristic by combining intelligently different concepts
for exploring and exploiting the search space"
(Osman and Laporte, 1996^{[3]})

"(metaheuristics) combine basic heuristic methods in higher level frameworks aimed at efficiently and effectively
exploring a search space"
(Blum and Roli, 2003^{[4]})

"a metaheuristic can be seen as a general-purpose heuristic method designed to guide an underlying
problem-specific heuristic (...) A metaheuristic is therefore a general algorithmic framework which can be applied to different optimization problems with relative few modifications to make them adapted to a specific problem."
(Dorigo and Stützle, 2004^{[5]})

"(metaheuristics) apply to all kinds of problems (...) are, at least to some extent, *stochastic* (...)
direct, i.e. they do not resort to the calculation of the gradients of the objective function (...)
inspired by *analogies*: with physics, biology or ethology"
(Dréo, Siarry, Petrowski and Taillard, 2006^{[6]})

One can summarize by enumerating the expected characteristics:

- optimization algorithms,
- with an iterative design,
- combining low level heuristics,
- aiming to tackle a large scale of "hard" problems.

As it is pointed out by the last reference, a large majority of metaheuristics (well, not to say *all*) use at least one stochastic (probabilistic) process and does not use more information than the solution and the associated value(s) of the objective function.

Talking about *combining* heuristics seems to be appropriate for *Ant Colony Optimization*, that specifically needs one (following Dorigo's point of vue), it can be less obvious for *Evolutionnary Algorithms*. One can consider that *mutation*, or even the method's strategy itself, is a heuristic, but isn't it too generalistic to be called a *heuristic* ?

If we forget the difficulty to demarcate what can be called a *heuristic* and what is the scope of the term *meta*, one can simply look at the use of the term among specialists. Despite the fact that the definition can be used in several fields (data mining, machine learning, etc.), the term is used for optimization algorithms. This is perhaps the best reason among others: the term permits to separate a research field from others, thus adding a little bit of marketing...

I would thus use this definition:

Metaheuristics are algorithms designed to tackle "hard" optimization problems, with the help of iterative
stochastic processes. These methods are manipulating direct samples of the objective function, and can be applied
to several problems without major changes in their design.