Higher order functions and Walsh coefficients

Iglesias, Marcos; Peñaranda, V. S.; Verschoren, A.

doi:10.36045/bbms/1168957340

Cited by 3 publications

(5 citation statements)

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“…As an application, using this and so-called "partition coefficients" (see [3] for details), one easily calculates the order zero Walsh coefficient (7) w 0 = the first order Walsh coefficients…”

Section: Em 2l (Z)mentioning

confidence: 99%

“…It has been shown in [4,7,10], that there exists a strong correlation between epistasis, order and hardness for certain large classes of functions, in particular for those that may be described by a limited number of control parameters. Royal Road and template functions (studied in sections 3 and 4) are of this type.…”

Section: Introductionmentioning

confidence: 99%

“…= W ( Tj -(iO( Tj . )o,t/;( T> )i,... ,tO(r,)2'-i) the Walsh vector of T,, it follows from(7), that So, we have to calculate the first order Walsh coefficients of r, (0 ^ j' ^ £ -n). In order to do this, denote by Aj the subset consisting of n consecutive indices, starting in j , that is, A, = {j,j + l,...,j + n-1}.Fix an index i (0 ^ i < £), and consider the schema % = # .…”

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confidence: 99%

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The 2-epistasis of fitness functions

Iglesias

Peñaranda

Vidal

et al. 2007

Bull. Austral. Math. Soc.

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In this note about Genetic Algorithms(GA-), we study the2-epistasisof a fitness function over a search space. This concept is a natural generalisation of that ofepistasis, previously considered by Davidor in 1991 Suys and Verschoren in 1996 and Van Hove and Verschoren in 1994 for example. We completely characterise fitness functions whose 2-epistasis is minimal: these are exactly the second order functions. The validity of 2-epistasis as a measure of hardness with respect to genetic algorithms is checked over some classical laboratory functions. Finally, we obtain an upper bound of the maximal value of the 2-epistasis when we restrict attention to non-negative functions.

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Section: Em 2l (Z)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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confidence: 99%

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The 2-epistasis of fitness functions

Iglesias

Peñaranda

Vidal

et al. 2007

Bull. Austral. Math. Soc.

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“…, i r . In the next result, proved in [5], the weight u(t) of t is the number of ones in the string t. THEOREM 3.10. For any function f : → R with Walsh coefficients w t , the following statements are equivalent:…”

Section: A Linear Algebra Approach To K-epistasismentioning

confidence: 97%

“…In fact, there usually exists a reasonable correlation between these factors and GA-hardness for functions described by a limited number of control parameters, such as Forrest and Mitchell's Royal Road functions [2]. We refer the reader to [5,7,9] for examples and details.…”

Section: Introductionmentioning

confidence: 99%

Higher Epistasis in Genetic Algorithms

Iglesias

Peñaranda²,

Vidal³

et al. 2008

Bulletin of the Australian Mathematical Society

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We study the k-epistasis of a fitness function over a search space. This concept is a natural generalization of that of epistasis, previously considered by Davidor, Suys We completely characterize fitness functions whose k-epistasis is minimal: these are exactly the functions of order k. We also obtain an upper bound for the k-epistasis of nonnegative fitness functions.2000 Mathematics subject classification: 68R99.

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