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Global+Optimization+Algorithms+Theory+and+Application_Part18

# Global+Optimization+Algorithms+Theory+and+Application_Part18...

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21.2 Binary Problem Spaces 341 Algorithm 21.1 : r ←− f lp ( x ) Input : x : the solution candidate with an odd length Data : s : the current position in x Data : sign : the sign of the next position Data : isOnPath : true if and only if x is on the path Output : r : the objective value begin 1 sign ←− 1 2 s ←− len( x ) 1 3 r ←− 0 4 isOnPath ←− true 5 while ( s 0) isOnPath do 6 if s = 0 then 7 if x [0] = 1 then r ←− r + sign 8 sub ←− subList( x,s 2 , 2) 9 if sub = 11 then 10 r ←− r + sign parenleftBig 3 2 s 2 2 parenrightBig 11 sign ←−− sign 12 else 13 if sub negationslash = 00 then 14 if ( x [ s ] = 0 ) ( x [ s 1] = 1 ) ( x [ s 2] = 1 ) then 15 if ( s = 2) [( x [ s 3] = 1 ) (countOccurences( 1 , subList( x, 0 ,s 3)) = 0)] then 16 r ←− r + sign parenleftBig 3 2 s 2 1 1 parenrightBig else else isOnPath ←− false 17 else isOnPath ←− false 18 s ←− s 2 19 if isOnPath then r ←− r + len( x ) 20 else r ←− len( x ) countOccurences( 1 ,x ) 1 21 end 22 et al. [958] also introduce Fibonacci paths which are longer than the Root2paths. The prob- lem of finding maximum length paths in a l -dimensional hypercube is known as the snake- in-the-box 8 problem [1104, 483] which was first described by Kautz [1104] in the late 1950s. It is a very hard problem suffering from combinatorial explosion and currently, maximum snake lengths are only known for small values of l . 21.2.7 Tunable Model for Problematic Phenomena What is a good model problem? Which model fits best to our purposes? These questions should be asked whenever we apply a benchmark, whenever we want to use something for testing the ability of a global optimization approach. The mathematical functions intro- duced in Section 21.1 , for instance, are good for testing special mathematical reproduction operations like used in Evolution Strategies and for testing the capability of an evolutionary algorithm for estimating the Pareto frontier in multi-objective optimization. Kauffman’s NK fitness landscape (discussed in Section 21.2.1 ) was intended to be a tool for exploring the relation of ruggedness and epistasis in fitness landscapes but can prove very useful for finding out how capable an global optimization algorithm is to deal with problems exhibiting these phenomena. In Section 21.2.4 , we outlined the Royal Road functions, which were used to 8 http://en.wikipedia.org/wiki/Snake-in-the-box [accessed 2008-08-13]

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342 21 Benchmarks and Toy Problems investigate the ability of genetic algorithms to combine different useful formae and to test the Building Block Hypothesis. The Artificial Ant ( Section 21.3.1 ) and the GCD problem from Section 21.3.2 are tests for the ability of Genetic Programming of learning algorithms. All these benchmarks and toy problems focus on specific aspects of global optimization and will exhibit different degrees of the problematic properties of optimization problems to which we had devoted Section 1.4 : 1. premature convergence and multimodality ( Section 1.4.2 ), 2. ruggedness ( Section 1.4.3 ), 3. deceptiveness ( Section 1.4.4 ), 4. neutrality and redundancy ( Section 1.4.5 ), 5. overfitting and oversimplification (
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