Metaheuristic

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A metaheuristic is a heuristic method for solving a very general class of computational problems by combining user given black-box procedures — usually heuristics themselves — in a hopefully efficient way. The name combines the Greek prefix "meta" ("beyond", here in the sense of "higher level") and "heuristic" (from ευρισκειν, heuriskein, "to find").

Metaheuristics are generally applied to problems for which there is no satisfactory problem-specific algorithm or heuristic; or when it is not practical to implement such a method. Most commonly used metaheuristics are targeted to combinatorial optimization problems, but of course can handle any problem that can be recast in that form, such as solving boolean equations.

In spite of overly-optimistic claims by some of their advocates, metaheuristics are not a panacea, and their indiscriminate use often is much less efficient than even the crudest problem-specific heuristic, by several orders of magnitude.

Contents 1 General concepts and nomenclature 2 Common meta-heuristics 3 General criticisms 4 Pragmatics 5 Bibliography 6 Links

[edit] General concepts and nomenclature

The goal of combinatorial optimization is to find a discrete mathematical object (such as a bit string or permutation) that maximizes (or minimizes) an arbitrary function specified by the user of the metaheuristic. These objects are generically called states, and the set of all candidate states is the search space. The nature of the states and the search space are usually problem-specific.

The function to be optimized is called the goal function, or objective function, and is usually provided by the user as a black-box procedure that evaluates the function on a given state. Depending on the meta-heuristic, the user may have to provide other black-box procedures that, say, produce a new random state, produce variants of a given state, pick one state among several, provide upper or lower bounds for the goal function over a set of states, and the like.

Some metaheuristics maintain at any instant a single current state, and replace that state by a new one. This basic step is sometimes called a state transition or move. The move is uphill or downhill depending on whether the goal function value increases or decreases. The new state may be constructed from scratch by a user-given generator procedure. Aternatively, the new state be derived from the current state by an user-given mutator procedure; in this case the new state is called a neighbor of the current one. Producers and mutators are often probabilistic procedures. The set of new states that can be produced by the mutator is the neighborhood of the current state.

More sophisticated meta-heuristics maintain, instead of a single current state, a current pool with several candidate states. The basic step then may add or delete states from this pool. User-given procedures may be called to select the states to be discarded, and to generate the new ones to be added. The latter may be generated by combination or crossover of two or more states from the pool.

A metaheuristic may also keep track of the current optimum, the optimum state among those already evaluated so far.

Since the set of candidates is usually very large, metaheuristics are typically implemented so that they can be interrupted after a client-specified time budget. If not interrupted, some exact metaheuristics will eventually check all candidates, and use heuristic methods only to choose the order of enumeration; therefore, they will always find the true optimum, if their time budget is large enough. Other metaheuristics give only a weaker probabilistic guarantee, namely that, as the time budget goes to infinity, the probability of checking every candidate tends to 1.

[edit] Common meta-heuristics

Some well-known meta heuristics are

• Random optimization
• Local search
• Greedy algorithm and hill-climbing
• Random-restart hill climbing
• Best-first search
• Simulated annealing
• Ant colony optimization
• Tabu search
• Genetic algorithms
• GRASP
• Stochastic Diffusion Search
• Generalized extremal optimization

Innumerable variants and hybrids of these techniques have been proposed, and many more applications of metaheuristics to specific problems have been reported. This is an active field of research, with a considerable literature, a large community of researchers and users, and a wide range of applications.

[edit] General criticisms

While there are many computer scientists who are enthusiastic advocates of metaheuristics, there are also many who are highly critical of the concept and have little regard for much of the research that is done on it.

Those critics point out, for one thing, that the general goal of the typical metaheuristic — the efficient optimization of an arbitrary black-box function — cannot be solved efficiently, since for any metaheuristic M one can easily build a function f that will force M to enumerate the whole search space (or worse). Further, the No-free-lunch theorem has proven that over the set of all mathematically possible problems, each optimization algorithm will do on average as well as any other. Thus, at best, a specific metaheuristic can be efficient only for restricted classes of goal functions (usually those that are partially "smooth" in some sense). However, when these restrictions are stated at all, they either exclude most applications of interest, or make the problem amenable to specific solution methods that are much more efficient than the meta-heuristic.

Moreover, the more advanced metaheuristics rely on auxiliary user-given black-box producers, mutators, etc.. It turns out that the effectiveness of a metaheuristic on a particular problem depends almost exclusively on these auxiliary functions, and very little on the heuristic itself. Given any two distinct metaheuristics M and N, and almost any goal function f, it is usually possible to write a set of auxiliary procedures that will make M find the optimum much more efficient than N, by many orders of magnitude; or vice-versa. In fact, since the auxiliary procedures are usually unrestricted, one can submit the basic step of metaheuristic M as the generator or mutator for N. Because of this extreme generality, one cannot say that any metaheuristic is better than any other, not even for a particular class of problems — in particular, better than brute force search, or the following "banal metaheuristic":

 1. Call the user-provided state generator.
 2. Print the resulting state.
 3. Stop.

Finally, all metaheuristic optimization techniques are extremely crude when evaluated by the standards of (continuous) nonlinear optimization. Within this area, it is well-known that to find the optimum of a smooth function on n variables one must essentially obtain its Hessian matrix, the n by n matrix of its second derivatives. If the function is given as a black-box procedure, then one must call it about n²/2 times, and solve an n by n system of linear equations, before one can make the first useful step towards the minimum. However, none of the common metaheuristics incorporate or accommodate this procedure. At best, they can be seen as computing some crude approximation to the local gradient of the goal function, and moving more or less "downhill". But gradient-descent can be inefficient for non-linear optimization and it is generally agreed upon that second order derivatives are necessary to yield efficient optimization methods.

[edit] Pragmatics

Independently of whether those criticisms are valid or not, metaheuristics can be terribly wasteful if used indiscriminately. Since their performance is critically dependent on the user-provided generators and mutators, one should concentrate on improving these procedures, rather than twiddling the parameters of sophisticated metaheuristics. A trivial metaheuristic with a good mutator will usually run circles around a sophisticated one with a poor mutator (and a good problem-specific heuristic will often do much better than both). In this area, more than in any other, a few hours of reading, thinking and programming can easily save months of computer time. On the other hand, this generalization does not necessarily extend equally to all problem domains. The use of genetic algorithms, for example, has produced evolved design solutions that exceed the best human-produced solutions despite years of theory and research. Problem domains falling into this category are often problems of combinatorial optimization and include the design of sorting networks, and evolved antennas, among others.

[edit] Bibliography

C. Blum and A. Roli (2003). Metaheuristics in combinatorial optimization: Overview and conceptual comparison. ACM Computing Surveys 35(3) 268–308.

[edit] Links

DGPF A distributed framework for randomized, heuristic searches like GA and Hill Climbing which comes with a specialization for Genetic Programming and allows to combine different search algorithms.

MHTB A toolbox of metaheuristic algorithms for MATLAB. It offers single-solution, population-based and hybrids metaheuristics. With this toolbox you can solve optimization problems defined in the MATLAB language using metaheuristic algorithms implemented in C++ and Java.