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An Analysis of Edge Assembly Crossover for the Traveling Salesman Problem Yuichi Nagata and Shigenobu Kobayashi IEEE, Conference on Evolutionary Computation, 1999 Genetic Algorithm Holland, 1975 Imitation of Evolution of life form in the Nature individuals - members of species - in the nature model of evolution processes where the basic operations are natural selection, crossover and mutation Schema Theorem - analysis of reproduction model Nothing to do with the real genetic organism Problem Solving Polynomial time function y = ax2 + b • Given constant a, b, know x, calculate y • Find x that gives Maximum y in a given range of x No Calculation Search for the Solution Search Space • • • • For small space, use classical exhaustive techniques For larger space, need special techniques Analysis of space Global Optima vs Local Optima Stochastic Search Local Search Techniques Adaptive Search Techniques Random search Ant Colony Simulated Annealing Neural Network search space Standard Genetic Algorithm Step 0: Initialization Procedure GA begin t := 0 ; initialize P(t) ; evaluate P(t) ; while (not termination-condition) do begin t := t + 1 ; select P(t) from P(t-1) ; alter P(t) ; evaluate P(t) ; end end Step 1: Selection Step 2 : Crossover Step 3 : Mutation Step 4: Evaluation Step 5 : Termination Test Step6: End GAs: Terminology Step 0: Initialization – – – – – – Representation : gene, chromosome, Population Evaluation : objective function, fitness function Selection Operator : crossover, mutation Replacement : new Generation Termination : Generation count, Convergence Step 1: Selection Step 2 : Crossover Step 3 : Mutation Step 4: Evaluation Step 5 : Termination Test Step6: End Representation • Very crucial step • representation should satisfy the presumption that the whole chromosome is decomposable to building blocks • String of genes of given alphabet: – Binary – Float – Integer • More complex representation – matrices – rules – trees Initialization of the Starting Population • Aspects affecting a performance of GA – schemata sampled in the initial population • Initialization mechanisms – random – informed - uses prior knowledge of the desired solution shape • Pre-processing – runs several short pre-processing runs – samples the promising areas of the search space identified during the foregone pre-processing runs Selection • Models nature’s survival-of-the-fittest principle • Selection strategies: – Roulette wheel (proportionate) – Ranking – Tournament • Selection process: – determination of Expected values: EVi = fitnessi / fitnessavg – sampling algorithm - conversion of EVi to the actual number of individuals Roulette Wheel Selection Crossover • Provides random information exchange works on couples of individuals • Simple 1-point crossover Mutation • Mutation - preserves population diversity – works on single individual Replacement Strategy • Replacement strategy defines: – how big portion of the old population will be replaced in each generation of the new population, and – the rule that determines which individuals from the old population will be replaced and which individuals will be placed in the new population • Generational - the old population is entirely rebuilt in each generation (short-lived species) • Steady-state - just a few individuals are replaced in each generation (longer-lived species) Premature Convergence • The ratio of the best-fit individual’s reproduction rate to the average reproduction rate is too high • selection kills ‘worse’ individuals too early Theory of GAs Schema Theorem (S ) ( S , t 1) ( S , t ) eval( S , t ) / F (t ) 1 pc o ( s ) pm m 1 • Schema = Pattern • Schema Theorem – Short, low-order, above-average schemata – receive exponentially increasing trials in subsequent generations of a genetic algorithm • Building Block Hypothesis – GA seeks near-optimal performance through short, low-order, high-performance schemata • Schema In binary representation - 2L strings, 3L schemata L = 7, S = (**0*1*1) - covers 24 strings – {0,1, *} – S = {*1*01***, 1*0*11*0, 10111011, *******1, ****0*** } • Fitness of a schema - average fitness computed over all covered strings • Schema property – order o(S ) • the length of string minus the number of * • defining the specialty of a schema • 8 bits : 11010011, schema and building block 1*010*1* – defining length (S ) • the distance between the first and the last fixed string positions • defining the compactness of information contained in a schema • (*11**1*0) = 6, (1******1) = 7 Selection • eval(S,t) is the average fitness of all strings in the population matched by the schema S at time t ; eval( S , t ) j 1 eval(v j ) / p p • Expecting to have ( S , t 1) strings matched by schema S ( S , t 1) ( S , t ) | p(t ) | eval( S , t ) / F (t ) – the average fitness of the population F (t ) F (t ) / | p(t ) | – becomes ; ( S , t 1) ( S , t ) eval( S , t ) / F (t ) Crossover – the string is matched by these two schemata S 0 (* * * *111* * * * * * * * * * * * * * * * * * * * * * * * * *) S1 (111* * * * * * * * * * * * * * * * * * * * * * * * * * * *10) survives va (111011111010001000110000001000110) , vb (000101000010010101001010111111011) va (111011111010001000111010111111011) , vb (000101000010010101000000001000110) destroyed – the probability of destruction of a schema S : pd ( S ) (S ) m 1 (S) = 7, m = 8 ? – the probability of survival of a schema S : ps ( S ) 1 (S ) m 1 (S ) ( S , t 1) ( S , t ) eval( S , t ) / F (t ) 1 pc o ( s ) pm m 1 • Selective probability of crossover pc ps ( S ) 1 pc (S ) m 1 • The combined effect of selection and crossover • a new schema growth equation : (S ) ( S , t 1) ( S , t ) eval( S , t ) / F (t ) 1 pc m 1 Mutation • All of the fixed positions of a schema must remain unchanged to survive mutation va (111011101101110000100011111011110) • mutate at least one of these bits would destroy the schema • the probability of destruction of a schema S : o( s ) p m • the probability of survival of a schema S : 1 o(s) pm (S ) ( S , t 1) ( S , t ) eval( S , t ) / F (t ) 1 pc o ( s ) pm m 1 TSP with GA Path representation (5 1 7 8 9 4 6 2 3) 5-1-7-8-9-4-6-2-3 • Crossover operators Node Orientation vs Edge Orientation • Mutation operators –insertion 5-2-1-7-8-9-4-6-2-3 –Reciprocal Exchange 5-9-7-8-1-4-6-2-3 –Inversion 5-9-8-7-1-4-6-2-3 Information of the parents transferred to offsprings Node crossover = simple but information discarded Edge crossover = tough but information enclosed TSP: Edge-Recombination Operator a-b-c-d-e b-d-e-c-a b-c-e-a-d Edge Assembly Crossover Edge Assembly Crossover Previous work Exx crossover Ex crossover Test Library EXX Crossover EX Crossover EXX Crossover att532