Edge Assembly Crossover 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 Search Space No Calculation  Search for the Solution 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 search space Local Search Techniques Adaptive Search Techniques Random search Ant Colony Simulated Annealing Neural Network search space

Standard Genetic Algorithm Step 0: Initialization Step 1: Selection Step 2 : Crossover Step 3 : Mutation Step 5 : Termination Test Step6: End Step 4: Evaluation Procedure GA begin t := 0 ; initialize P(t) ; evaluate P(t) ; while (not termination-condition) do t := t + 1 ; select P(t) from P(t-1) ; alter P(t) ; end

GAs: Terminology Representation : gene, chromosome, Population Step 0: Initialization Step 1: Selection Step 2 : Crossover Step 3 : Mutation Step 5 : Termination Test Step6: End Step 4: Evaluation Representation : gene, chromosome, Population Evaluation : objective function, fitness function Selection Operator : crossover, mutation Replacement : new Generation Termination : Generation count, Convergence

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 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 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 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 ; Expecting to have strings matched by schema S the average fitness of the population becomes ;

Crossover survives destroyed the string is matched by these two schemata survives destroyed

the probability of destruction of a schema S : the probability of survival of a schema S : (S) = 7, m = 8 ?

Selective probability of crossover The combined effect of selection and crossover a new schema growth equation :

Mutation All of the fixed positions of a schema must remain unchanged to survive mutation mutate at least one of these bits would destroy the schema the probability of destruction of a schema S : the probability of survival of a schema S :

TSP with GA

Path representation Crossover operators Mutation operators (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

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