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Global+Optimization+Algorithms+Theory+and+Application_Part7

# Global+Optimization+Algorithms+Theory+and+Application_Part7...

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2.4 Selection 121 Algorithm 2.5 : v ←− assignFitnessTournament q,r ( Pop , cmp F ) Input : q : the number of tournaments per individuals Input : r : the number of other contestants per tournament, normally 1 Input : Pop : the population to assign fitness values to Input : cmp F : the comparator function providing the prevalence relation Data : i,j,k,z : counter variables Data : b : a Boolean variable being true as long as a tournament isn’t lost Data : p : the individual currently examined Output : v : the fitness function begin 1 for i ←− len( Pop ) 1 down to 0 do 2 z ←− q 3 p ←− Pop [ i ] 4 for j ←− q down to 1 do 5 b ←− true 6 k ←− r 7 while ( k> 0) b do 8 b ←− Pop [ random u (0 , len( Pop )) ] . x p . x 9 k ←− k 1 10 if b then z ←− z 1 11 v ( p . x ) ←− z 12 return v 13 end 14 2.4 Selection 2.4.1 Introduction Definition 2.8 (Selection). In evolutionary algorithms, the selection 20 operation Mate = select( Pop ,v,ms ) chooses ms individuals according to their fitness values v from the popu- lation Pop and places them into the mating pool Mate [99, 1242, 232, 1431]. Mate = select( Pop ,v,ms ) ⇒∀ p Mate p Pop p Pop p G × X v ( p . x ) R + p Pop (len( Mate ) min { len( Pop ) ,ms } ) (len( Mate ) ms ) (2.12) On the mating pool, the reproduction operations discussed in Section 2.5 on page 137 will subsequently be applied. Selection may behave in a deterministic or in a randomized manner, depending on the algorithm chosen and its application-dependant implementation. Furthermore, elitist evolutionary algorithms may incorporate an archive Arc in the selection process, as sketched in Algorithm 2.2 . Generally, there are two classes of selection algorithms: such with replacement (anno- tated with a subscript r ) and such without replacement (annotated with a subscript w , see Equation 2.13 ) [1809]. In a selection algorithm without replacement, each individual from the population Pop is taken into consideration for reproduction at most once and therefore 20 http://en.wikipedia.org/wiki/Selection_%28genetic_algorithm%29 [accessed 2007-07-03]

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122 2 Evolutionary Algorithms also will occur in the mating pool Mate one time at most. The mating pool returned by algorithms with replacement can contain the same individual multiple times. Like in nature, one individual may thus have multiple offspring. Normally, selection algorithms are used in a variant with replacement. One of the reasons therefore is the number of elements to be placed into the mating pool (corresponding to the parameter ms ). If len( Pop ) < ms , the mating pool returned by a method without replacement contains less than ms individuals since it can at most consist of the whole population. Mate = select w ( Pop ,v,ms ) countOccurences( p, Mate ) = 1 p Mate (2.13) The selection algorithms have major impact on the performance of evolutionary algo- rithms. Their behavior has thus been subject to several detailed studies, conducted by, for instance, Goldberg and Deb [823], Blickle and Thiele [232], and Zhong et al. [2318], just to name a few.
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