# Category:Theorems in computational complexity theory

## Pages in category "Theorems in computational complexity theory"

The following 21 pages are in this category, out of 21 total. This list may not reflect recent changes (learn more).

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## Pages in category "Theorems in computational complexity theory"

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The following 21 pages are in this category, out of 21 total. This list may not reflect recent changes (learn more).

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1. No free lunch in search and optimization – No solution therefore offers a short cut. In computing, there are circumstances in which the outputs of all procedures solving a type of problem are statistically identical. Wolpert had previously derived no free lunch theorems for machine learning, in the no free lunch metaphor, each restaurant has a menu associating each lunch plate with a price. The menus of restaurants are identical except in one regard – the prices are shuffled from one restaurant to the next, for an omnivore who is as likely to order each plate as any other, the average cost of lunch does not depend on the choice of restaurant. But a vegan who goes to lunch regularly with a carnivore who seeks economy might pay an average cost for lunch. To methodically reduce the average cost, one must use advance knowledge of a) what one will order and that is, improvement of performance in problem-solving hinges on using prior information to match procedures to problems. In formal terms, there is no free lunch when the probability distribution on problem instances is such that all problem solvers have identically distributed results. In the case of search, an instance is an objective function. For typical interpretations of results, search is an optimization process, there is no free lunch in search if and only if the distribution on objective functions is invariant under permutation of the space of candidate solutions. This condition does not hold precisely in practice, but an no free lunch theorem suggests that it holds approximately, some computational problems are solved by searching for good solutions in a space of candidate solutions. A description of how to select candidate solutions for evaluation is called a search algorithm. On a particular problem, different search algorithms may obtain different results and it follows that if an algorithm achieves superior results on some problems, it must pay with inferiority on other problems. In this sense there is no free lunch in search, alternatively, following Schaffer, search performance is conserved. Usually search is interpreted as optimization, and this leads to the observation that there is no free lunch in optimization, the no free lunch results indicate that matching algorithms to problems gives higher average performance than does applying a fixed algorithm to all. Igel and Toussaint and English have established a general condition under which there is no free lunch, while it is physically possible, it does not hold precisely. Droste, Jansen, and Wegener have proved a theorem they interpret as indicating that there is no free lunch in practice, to make matters more concrete, consider an optimization practitioner confronted with a problem. Given some knowledge of how the problem arose, the practitioner may be able to exploit the knowledge in selection of an algorithm that will perform well in solving the problem. The authors of the no free lunch theorem say that the answer is essentially no, a problem is, more formally, an objective function that associates candidate solutions with goodness values